Renormalization and Effective Field Theory · 2018-12-04 · Renormalization and Effective . Field Theory . Kevin Costello . This is a preliminary version of the book Renormalization

Renormalization and Effective Field Theory Kevin Costello This is a preliminary version of the book Renormalization and Effective Field Theory published by the American Mathematical Society (AMS). This preliminary version is made available with the permission of the AMS and may not be changed, edited, or reposted at any other website without explicit written permission from the author and the AMS.

Author's preliminary version made available with permission of the publisher, the American Mathematical Society

http://www.ams.org/bookstore-getitem/item=surv-170

Contents

Chapter 1. Introduction 11. Overview 12. Functional integrals in quantum field theory 43. Wilsonian low energy theories 64. A Wilsonian definition of a quantum field theory 135. Locality 136. The main theorem 167. Renormalizability 198. Renormalizable scalar field theories 219. Gauge theories 2310. Observables and correlation functions 2711. Other approaches to perturbative quantum field theory 27Acknowledgements 29

Chapter 2. Theories, Lagrangians and counterterms 311. Introduction 312. The e!ective interaction and background field functional integrals 323. Generalities on Feynman graphs 344. Sharp and smooth cut-o!s 425. Singularities in Feynman graphs 446. The geometric interpretation of Feynman graphs 477. A definition of a quantum field theory 538. An alternative definition 559. Extracting the singular part of the weights of Feynman graphs 5710. Constructing local counterterms 6211. Proof of the main theorem. 6712. Proof of the parametrix formulation of the main theorem 6913. Vector-bundle valued field theories 7114. Field theories on non-compact manifolds 81

Chapter 3. Field theories on Rn 911. Some functional analysis 922. The main theorem on Rn 993. Vector-bundle valued field theories on Rn 1044. Holomorphic aspects of theories on Rn 107

Chapter 4. Renormalizability 113

3


4 CONTENTS

1. The local renormalization group flow 1132. The Kadano!-Wilson picture and asymptotic freedom 1223. Universality 1254. Calculations in 4 theory 1265. Proofs of the main theorems 1316. Generalizations of the main theorems 135

Chapter 5. Gauge symmetry and the Batalin-Vilkovisky formalism 1391. Introduction 1392. A crash course in the Batalin-Vilkovisky formalism 1413. The classical BV formalism in infinite dimensions 1554. Example: Chern-Simons theory 1605. Example : Yang-Mills theory 1626. D-modules and the classical BV formalism 1647. BV theories on a compact manifold 1708. E!ective actions 1739. The quantum master equation 17510. Homotopies between theories 17811. Obstruction theory 18612. BV theories on Rn 18913. The sheaf of BV theories on a manifold 19614. Quantizing Chern-Simons theory 203

Chapter 6. Renormalizability of Yang-Mills theory 2071. Introduction 2072. First-order Yang-Mills theory 2073. Equivalence of first-order and second-order formulations 2104. Gauge fixing 2135. Renormalizability 2146. Universality 2177. Cohomology calculations 218

Appendix. Appendix 1: Asymptotics of graph integrals 2271. Generalized Laplacians 2272. Polydi!erential operators 2293. Periods 2294. Integrals attached to graphs 2305. Proof of Theorem 4.0.12 233

Appendix. Appendix 2 : Nuclear spaces 2431. Basic definitions 2432. Examples 2443. Subcategories 2454. Tensor products of nuclear spaces from geometry 2475. Algebras of formal power series on nuclear Frechet spaces 247


CONTENTS 5

Bibliography 249



CHAPTER 1

Introduction

1. Overview

Quantum field theory has been wildly successful as a framework for thestudy of high-energy particle physics. In addition, the ideas and techniquesof quantum field theory have had a profound influence on the developmentof mathematics.

There is no broad consensus in the mathematics community, however,as to what quantum field theory actually is.

This book develops another point of view on perturbative quantum fieldtheory, based on a novel axiomatic formulation.

Most axiomatic formulations of quantum field theory in the literaturestart from the Hamiltonian formulation of field theory. Thus, the Segal(Seg99) axioms for field theory propose that one assigns a Hilbert space ofstates to a closed Riemannian manifold of dimension d ! 1, and a unitaryoperator between Hilbert spaces to a d-dimensional manifold with boundary.In the case when the d- dimensional manifold is of the form M " [0, t], weshould view the corresponding operator as time evolution.

The Haag-Kastler (Haa92) axioms also start from the Hamiltonian for-mulation, but in a slightly di!erent way. They take as the primary objectnot the Hilbert space, but rather a C! algebra, which will act on a vacuumHilbert space.

I believe that the Lagrangian formulation of quantum field theory, usingFeynman’s sum over histories, is more fundamental. The axiomatic frame-work developed in this book is based on the Lagrangian formalism, and onthe ideas of low-energy e!ective field theory developed by Kadano! (Kad66),Wilson (Wil71), Polchinski (Pol84) and others.

1.1. The idea of the definition of quantum field theory I use is verysimple. Let us assume that we are limited, by the power of our detectors,to studying physical phenomena that occur below a certain energy, say ".The part of physics that is visible to a detector of resolution " we will callthe low-energy e!ective field theory. This low-energy e!ective field theoryis succinctly encoded by the energy " version of the Lagrangian, which iscalled the low-energy e!ective action Seff ["].

The notorious infinities of quantum field theory only occur if we con-sider phenomena of arbitrarily high energy. Thus, if we restrict attention to

1


2 1. INTRODUCTION

phenomena occurring at energies less than ", we can compute any quantitywe would like in terms of the e!ective action Seff ["].

If "! < ", then the energy "! e!ective field theory can be deducedfrom knowledge of the energy " e!ective field theory. This leads to anequation expressing the scale "! e!ective action Seff ["!] in terms of thescale " e!ective action Seff ["]. This equation is called the renormalizationgroup equation.

If we do have a continuum quantum field theory (whatever that is!)we should, in particular, have a low-energy e!ective field theory for everyenergy. This leads to our definition : a continuum quantum field theory isa sequence of low-energy e!ective actions Seff ["], for all " < #, which arerelated by the renormalization group flow. In addition, we require that theSeff ["] satisfy a locality axiom, which says that the e!ective actions Seff ["]become more and more local as "$#.

This definition aims to be as parsimonious as possible. The only as-sumptions I am making about the nature of quantum field theory are thefollowing:

(1) The action principle: physics at every energy scale is described by aLagrangian, according to Feynman’s sum-over-histories philosophy.

(2) Locality: in the limit as energy scales go to infinity, interactionsbetween fields occur at points.

1.2. In this book, I develop complete foundations for perturbative quan-tum field theory in Riemannian signature, on any manifold, using this defi-nition.

The first significant theorem I prove is an existence result: there are asmany quantum field theories, using this definition, as there are Lagrangians.

Let me state this theorem more precisely. Throughout the book, I willtreat ! as a formal parameter; all quantities will be formal power series in!. Setting ! to zero amounts to passing to the classical limit.

Let us fix a classical action functional Scl on some space of fields E ,which is assumed to be the space of global sections of a vector bundle on amanifold M1. Let T (n)(E , Scl) be the space of quantizations of the classicaltheory that are defined modulo !n+1. Then,

Theorem 1.2.1.

T (n+1)(E , Scl) $ T (n)(E , Scl)

is a torsor for the abelian group of Lagrangians under addition (modulo thoseLagrangians which are a total derivative).

Thus, any quantization defined to order n in ! can be lifted to a quan-tization defined to order n + 1 in !, but there is no canonical lift; any twolifts di!er by the addition of a Lagrangian.

1The classical action needs to satisfy some non-degeneracy conditions


1. OVERVIEW 3

If we choose a section of each torsor T (n+)(E , Scl) $ T (n)(E , Scl) wefind an isomorphism

T (")(E , Scl) %= series Scl + !S(1) + !2S(2) + · · ·

where each S(i) is a local functional, that is, a functional which can bewritten as the integral of a Lagrangian. Thus, this theorem allows one toquantize the theory associated to any classical action functional Scl. How-ever, there is an ambiguity to quantization: at each term in !, we are freeto add an arbitrary local functional to our action.

1.3. The main results of this book are all stated in the context of thistheorem.

In Chapter 4, I give a definition of an action of the group R>0 on thespace of theories on Rn. This action is called the local renormalization groupflow, and is a fundamental part of the concept of renormalizability developedby Wilson and others. The action of group R>0 on the space of theories onRn simply arises from the action of this group on Rn by rescaling.

The coe#cients of the action of this local renormalization group flow onany particular theory are the functions of that theory. I include explicitcalculations of the function of some simple theories, including the 4

theory on R4.This local renormalization group flow leads to a concept of renormaliz-

ability. Following Wilson and others, I say that a theory is perturbativelyrenormalizable if it has “critical” scaling behaviour under the renormaliza-tion group flow. This means that the theory is fixed under the renormal-ization group flow except for logarithmic corrections. I then classify allpossible renormalizable scalar field theories, and find the expected answer.For example, the only renormalizable scalar field theory in four dimensions,invariant under isometries and under the transformation $ ! , is the 4

theory.In Chapter 5, I show how to include gauge theories in my definition of

quantum field theory, using a natural synthesis of the Wilsonian e!ectiveaction picture and the Batalin-Vilkovisky formalism. Gauge symmetry, inour set up, is expressed by the requirement that the e!ective action Seff ["]at each energy " satisfies a certain scale " Batalin-Vilkovisky quantummaster equation. The renormalization group flow is compatible with theBatalin-Vilkovisky quantum master equation: the flow from scale " to scale"! takes a solution of the scale " master equation to a solution to the scale"! equation.

I develop a cohomological approach to constructing theories which arerenormalizable and which satisfy the quantum master equation. Given anyclassical gauge theory, satisfying the classical analog of renormalizability, Iprove a general theorem allowing one to construct a renormalizable quan-tization, providing a certain cohomology group vanishes. The dimension


4 1. INTRODUCTION

of the space of possible renormalizable quantizations is given by a di!erentcohomology group.

In Chapter 6, I apply this general theorem to prove renormalizabilityof pure Yang-Mills theory. To apply the general theorem to this example,one needs to calculate the cohomology groups controlling obstructions anddeformations. This turns out to be a lengthy (if straightforward) exercise inGel’fand-Fuchs Lie algebra cohomology.

Thus, in the approach to quantum field theory presented here, to proverenormalizability of a particular theory, one simply has to calculate theappropriate cohomology groups. No manipulation of Feynman graphs isrequired.

2. Functional integrals in quantum field theory

Let us now turn to giving a detailed overview of the results of this book.First I will review, at a basic level, some ideas from the functional inte-

gral point of view on quantum field theory.

2.1. Let M be a manifold with a metric of Lorentzian signature. Wewill think of M as space-time. Let us consider a quantum field theory of asingle scalar field : M $ R.

The space of fields of the theory is C"(M). We will assume that wehave an action functional of the form

S( ) =!

x#ML ( )(x)

where L ( ) is a Lagrangian. A typical Lagrangian of interest would be

L ( ) = !12 (D+m2) + 1

4!4

where D is the Lorentzian analog of the Laplacian operator.A field & C"(M, R) can describes one possible history of the universe

in this simple model.Feynman’s sum-over-histories approach to quantum field theory says

that the universe is in a quantum superposition of all states & C"(M, R),each weighted by eiS( )/!.

An observable – a measurement one can make – is a function

O : C"(M, R) $ C.

If x &M , we have an observable Ox defined by evaluating a field at x:

Ox( ) = (x).

More generally, we can consider observables that are polynomial functionsof the values of and its derivatives at some point x & M . Observables ofthis form can be thought of as the possible observations that an observer atthe point x in the space-time manifold M can make.


2. FUNCTIONAL INTEGRALS IN QUANTUM FIELD THEORY 5

The fundamental quantities one wants to compute are the correlationfunctions of a set of observables, defined by the heuristic formula

'O1, . . . , On( =!

#C!(M)eiS( )/!O1( ) · · ·On( )D .

Here D is the (non-existent!) Lebesgue measure on the space C"(M).The non-existence of a Lebesgue measure (i.e. a non-zero translation

invariant measure) on an infinite dimensional vector space is one of thefundamental di#culties of quantum field theory.

We will refer to the picture described here, where one imagines theexistence of a Lebesgue measure on the space of fields, as the naive functionalintegral picture. Since this measure does not exist, the naive functionalintegral picture is purely heuristic.

2.2. Throughout this book, I will work in Riemannian signature, in-stead of the more physical Lorentzian signature. Quantum field theory inRiemannian signature can be interpreted as statistical field theory, as I willnow explain.

Let M be a compact manifold of Riemannian signature. We will takeour space of fields, as before, to be the space C"(M, R) of smooth functionson M . Let S : C"(M, R) $ R be an action functional, which, as before, weassume is the integral of a Lagrangian. Again, a typical example would bethe 4 action

S( ) = !12

!

x#M(D+m2) + 1

4!4.

Here D denotes the non-negative Laplacian. 2

We should think of this field theory as a statistical system of a randomfield & C"(M, R). The energy of a configuration is S( ). The behaviourof the statistical system depends on a temperature parameter T : the systemcan be in any state with probability

e$S( )/T .

The temperature T plays the same role in statistical mechanics as the pa-rameter ! plays in quantum field theory.

I should emphasize that time evolution does not play a role in this pic-ture: quantum field theory on d-dimensional space-time is related to statis-tical field theory on d-dimensional space. We must assume, however, thatthe statistical system is in equilibrium.

As before, the quantities one is interested in are the correlation functionsbetween observables, which one can write (heuristically) as

'O1, . . . , On( =!

#C!(M)e$S( )/T O1( ) · · ·On( )D .

2Our conventions are such that the quadratic part of the action is negative-definite.


6 1. INTRODUCTION

The only di!erence between this picture and the quantum field theory for-mulation is that we have replaced i! by T .

If we consider the limiting case, when the temperature T in our statis-tical system is zero, then the system is “frozen” in some extremum of theaction functional S( ). In the dictionary between quantum field theory andstatistical mechanics, the zero temperature limit corresponds to classicalfield theory. In classical field theory, the system is frozen at a solution tothe classical equations of motion.

Throughout this book, I will work perturbatively. In the vocabularyof statistical field theory, this means that we will take the temperatureparameter T to be infinitesimally small, and treat everything as a formalpower series in T . Since T is very small, the system will be given by a smallexcitation of an extremum of the action functional.

In the language of quantum field theory, working perturbatively meanswe treat ! as a formal parameter. This means we are considering smallquantum fluctuations of a given solution to the classical equations of mo-tions.

Throughout the book, I will work in Riemannian signature, but willotherwise use the vocabulary of quantum field theory. Our sign conventionsare such that ! can be identified with the negative of the temperature.

3. Wilsonian low energy theories

Wilson (Wil71; Wil72), Kadano! (Kad66), Polchinski (Pol84) and othershave studied the part of a quantum field theory which is seen by detectorswhich can only measure phenomena of energy below some fixed ". Thispart of the theory is called the low-energy e!ective theory.

There are many ways to define “low energy”. I will start by giving adefinition which is conceptually very simple, but di#cult to work with. Inthis definition, the low energy fields are those functions on our manifold Mwhich are sums of low-energy eigenvectors of the Laplacian.

In the body of the book, I will use a definition of e!ective field theorybased on length rather than energy. The great advantage of this defini-tion is that it relates better to the concept of locality. I will explain therenormalization group flow from the length-scale point of view shortly.

In this introduction, I will only discuss scalar field theories on compactRiemannian manifolds. This is purely for expository purposes. In the bodyof the book I will work with a general class of theories on a possibly non-compact manifold, although always in Riemannian signature.

3.1. Let M be a compact Riemannian manifold. For any subset I )[0,#), let C"(M)I ) C"(M) denote the space of functions which are sumsof eigenfunctions of the Laplacian with eigenvalue in I. Thus, C"(M)%!denotes the space of functions that are sums of eigenfunctions with eigen-value * ". We can think of C"(M)%! as the space of fields with energy atmost ".


3. WILSONIAN LOW ENERGY THEORIES 7

Detectors that can only see phenomena of energy at most " can berepresented by functions

O : C"(M)%! $ R[[!]],

which are extended to C"(M) via the projection C"(M)$ C"(M)%!.Let us denote by Obs%! the space of all functions on C"(M) that arise

in this way. Elements of Obs%! will be referred to as observables of energy* ".

The fundamental quantities of the low-energy e!ective theory are thecorrelation functions 'O1, . . . , On( between low-energy observables Oi &Obs%!. It is natural to expect that these correlation functions arise fromsome kind of statistical system on C"(M)%!. Thus, we will assume thatthere is a measure on C"(M)%!, of the form

eSeff [!]/!D

where D is the Lebesgue measure, and Seff ["] is a function on Obs%!,such that

'O1, . . . , On( =!

#C!(M)"!

eSeff [!]( )/!O1( ) · · ·On( )D

for all low-energy observables Oi & Obs%!.The function Seff ["] is called the low-energy e!ective action. This ob-

ject completely describes all aspects of a quantum field theory that can beseen using observables of energy * ".

Note that our sign conventions are unusual, in that Seff ["] appears inthe functional integral via eSeff [!]/!, instead of e$Seff [!]/! as is more usual.We will assume the quadratic part of Seff ["] is negative-definite.

3.2. If "! * ", any observable of energy at most "! is in particular anobservable of energy at most ". Thus, there are inclusion maps

Obs%!# !$ Obs%!if "! * ".

Suppose we have a collection O1, . . . , On & Obs%!# of observables ofenergy at most "!. The correlation functions between these observablesshould be the same whether they are considered to lie in Obs%!# or Obs%!.That is,!

#C!(M)"!#

eSeff [!#]( )/!O1( ) · · ·On( )d

=!

#C!(M)"!

eSeff [!]( )/!O1( ) · · ·On( )d .

It follows from this that

Seff ["!]( L) = ! log

"!

H#C!(M)(!#,!]

exp#

1!Seff ["]( L + H)

$%


8 1. INTRODUCTION

where the low-energy field L is in C"(M)%!# . This is a finite dimensionalintegral, and so (under mild conditions) is well defined as formal power seriesin !.

This equation is called the renormalization group equation. It says that if"! < ", then Seff ["!] is obtained from Seff ["] by averaging over fluctuationsof the low-energy field L & C"(M)%!# with energy between "! and ".

3.3. Recall that in the naive functional-integral point of view, there issupposed to be a measure on the space C"(M) of the form

eS( )/!d ,

where d refers to the (non-existent) Lebesgue measure on the vector spaceC"(M), and S( ) is a function of the field .

It is natural to ask what role the “original” action S plays in the Wilso-nian low-energy picture. The answer is that S is supposed to be the “energyinfinity e!ective action”. The low energy e!ective action Seff ["] is supposedto be obtained from S by integrating out all fields of energy greater than ",that is

Seff ["]( L) = ! log

"!

H#C!(M)(!,!)

exp#

1!S( L + H)

$%.

This is a functional integral over the infinite dimensional space of fields withenergy greater than ". This integral doesn’t make sense; the terms in itsFeynman graph expansion are divergent.

However, one would not expect this expression to be well-defined. Theinfinite energy e!ective action should not be defined; one would not expect tohave a description of how particles behave at infinite energy. The infinitiesin the naive functional integral picture arise because the classical actionfunctional S is treated as the infinite energy e!ective action.

3.4. So far, I have explained how to define a renormalization groupequation using the eigenvalues of the Laplacian. This picture is very easy toexplain, but it has many disadvantages. The principal disadvantage is thatthis definition is not local on space-time. Thus, it is di#cult to integratethe locality requirements of quantum field theory into this version of therenormalization group flow.

In the body of this book, I will use a version of the renormalization groupflow that is based on length rather than on energy. A complete account ofthis will have to wait until Chapter 2, but I will give a brief description here.

The version of the renormalization group flow based on length is notderived directly from Feynman’s functional integral formulation of quantumfield theory. Instead, it is derived from a di!erent (though ultimately equiv-alent) formulation of quantum field theory, again due to Feynman (Fey50).

Let us consider the propagator for a free scalar field , with actionSfree( ) = Sk( ) = !1

2

&(D+m2) . This propagator P is defined to be

the integral kernel for the inverse of the operator D+m2 appearing in the



action. Thus, P is a distribution on M2. Away from the diagonal in M2,P is a distribution. The value P (x, y) of P at distinct points x, y in thespace-time manifold M can be interpreted as the correlation between thevalue of the field at x and the value at y.

Feynman realized that the propagator can be written as an integral

P (x, y) =! "

=0e$ m2

K (x, y)d

where K (x, y) is the heat kernel. The fact that the heat kernel can beinterpreted as the transition probability for a random path allows us towrite the propagator P (x, y) as an integral over the space of paths in Mstarting at x and ending at y:

P (x, y) =! "

=0e$ m2

!

f :[0, ]&Mf(0)=x,f( )=y

exp#!!

0+df+2

$.

(This expression can be given a rigorous meaning using the Wiener measure).From this point of view, the propagator P (x, y) represents the proba-

bility that a particle starts at x and transitions to y along a random path(the worldline). The parameter is interpreted as something like the propertime: it is the time measured by a clock travelling along the worldline. (Thisexpression of the propagator is sometimes known as the Schwinger repre-sentation).

Any reasonable action functional for a scalar field theory can be decom-posed into kinetic and interacting terms,

S( ) = Sfree( ) + I( )

where Sfree( ) is the action for the free theory discussed above. Fromthe space-time point of view on quantum field theory, the quantity I( )prescribes how particles interact. The local nature of I( ) simply says thatparticles only interact when they are at the same point in space-time. Fromthis point of view, Feynman graphs have a very simple interpretation: theyare the “world-graphs” traced by a family of particles in space-time movingin a random fashion, and interacting in a way prescribed by I( ).

This point of view on quantum field theory is the one most closely relatedto string theory (see e.g. the introduction to (GSW88)). In string theory,one replaces points by 1-manifolds, and the world-graph of a collection ofinteracting particles is replaced by the world-sheet describing interactingstrings.

3.5. Let us now briefly describe how to treat e!ective field theory fromthe world-line point of view.

In the energy-scale picture, physics at scales less than " is described bysaying that we are only allowed fields of energy less than ", and that theaction on such fields is described by the e!ective action Seff ["].

In the world-line approach, instead of having an e!ective action Seff ["]at each energy-scale ", we have an e!ective interaction Ieff [L] at each


10 1. INTRODUCTION

length-scale L. This object encodes all physical phenomena occurring atlengths greater than L. (The e!ective interaction can also be considered inthe energy-scale picture also: the relationship between the e!ective actionSeff ["] and the e!ective interaction Ieff ["] is simply

Seff ["]( ) = !12

!

MD + Ieff ["]( )

for fields & C"(M)[0,!). The reason for introducing the e!ective interac-tion is that the world-line version of the renormalization group flow is betterexpressed in these terms.

In the world-line picture of physics at lengths greater than L, we canonly consider paths which evolve for a proper time greater than L, andthen interact via Ieff [L]. All processes which involve particles moving for aproper time of less than L between interactions are assumed to be subsumedinto Ieff [L].

The renormalization group equation for these e!ective interactions canbe described by saying that quantities we compute using this prescriptionare independent of L. That is,

Definition 3.5.1. A collection of e!ective interactions Ieff [L] satisfiesthe renormalization group equation if, when we compute correlation func-tions using Ieff [L] as our interaction, and allow particles to travel for aproper time of at least L between any two interactions, the result is indepen-dent of L.

If one works out what this means, one sees that the scale L e!ective in-teraction Ieff [L] can be constructed in terms of Ieff [ ] by allowing particlesto travel along paths with proper-time between and L, and then interactusing Ieff [ ].

More formally, Ieff [L] can be expressed as a sum over Feynman graphs,where the edges are labelled by the propagator

P ( , L) =! L

e$ m2K

and where the vertices are labelled by Ieff [ ].This e!ective interaction Ieff [L] is an !-dependent functional on the

space C"(M) of fields. We can expand Ieff [L] as a formal power series

Ieff [L] ='

i,k'0

!iIeffi,k [L]

whereIeffi,k [L] : C"(M)$ R

is homogeneous of order k. Thus, we can think of Ii,k[L] as being a symmetriclinear map

Ieffi,k [L] : C"(M)(k $ R.



Figure 1. The first few expressions in the renormalizationgroup flow from scale to scale L. The dotted lines indicateincoming particles. The blobs indicate interactions betweenthese particles. The symbol Ieff

i,k [L] indicates the !i term inthe contribution to the interaction of k particles at length-scale L. The solid lines indicate the propagation of a particlebetween two interactions; particles are allowed to propagatewith proper time between and L.

We should think of Ieffi,k [L] as being a contribution to the interaction of k

particles which come together in a region of size around L.Figure 1 shows how to express, graphically, the world-line version of the

renormalization group flow.

3.6. So far in this section, we have sketched the definition of two versionsof the renormalization group flow: one based on energy, and one based onlength. There is a more general definition of the renormalization group flowwhich includes these two as special cases. This more general version is basedon the concept of parametrix.


12 1. INTRODUCTION

Definition 3.6.1. A parametrix for the Laplacian D on a manifold isa symmetric distribution P on M2 such that (D,1)P ! M is a smoothfunction on M2 (where M refers to the -distribution on the diagonal ofM .

This condition implies that the operator $P : C"(M) $ C"(M) asso-ciated to the kernel P is an inverse for D, up to smoothing operators: bothP -D! Id and D -P ! Id are smoothing operators.

If Kt is the heat kernel for the Laplacian D, then Plength(0, L) =& L0 Ktdt

is a parametrix. This family of parametrices arises when one considers theworld-line picture of quantum field theory.

Similarly, we can define an energy-scale parametrix

Penergy[",#) ='

'!

1e , e

where the sum is over an orthonormal basis of eigenfunctions e for theLaplacian D, with eigenvalue .

Thus, we see that in either the world-line picture or the momentum scalepicture one has a family of parametrices ( Plength(0, L) and Penergy[",#),respectively) which converge (in the L$ 0 and "$# limits, respectively)to the zero distribution. The renormalization group equation in either caseis written in terms of the one-parameter family of parametrices.

This suggests a more general version of the renormalization group flow,where an arbitrary parametrix P is viewed as defining a “scale” of the the-ory. In this picture, one should have an e!ective action Ieff [P ] for eachparametrix. If P, P ! are two di!erent parametrices, then Ieff [P ] and Ieff [P !]must be related by a certain renormalization group equation, which expressesIeff [P ] in terms of a sum over graphs whose vertices are labelled by Ieff [P !]and whose edges are labelled by P ! P !.

If we restrict such a family of e!ective interactions to parametrices ofthe form Plength(0, L) one finds a solution to the world-line version of therenormalization group equation. If we only consider parametrices of theform Penergy[",#), and then define

Seff ["]( ) = !12

!

MD + Ieff [Penergy[",#)]( ),

for & C"(M)[0,!), one finds a solution to the energy scale version of therenormalization group flow.

A general definition of a quantum field theory along these lines is ex-plained in detail in Chapter 2, Section 8. This definition is equivalent toone based only on the world-line version of the renormalization group flow,which is the definition used for most of the book.


5. LOCALITY 13

4. A Wilsonian definition of a quantum field theory

Any detector one could imagine has some finite resolution, and so onlyprobes some low-energy e!ective theory, described by some Seff ["]. How-ever, one could imagine building detectors of arbitrarily high (but finite)resolution, and so one could imagine probing Seff ["] for arbitrarily high(but finite) ".

As is usual in physics, one should only consider those objects which canin principle be observed. Thus, one should say that all aspects of a quantumfield theory are encoded in its various low-energy e!ective theories.

Let us make this into a (rough) definition. A more precise version of thisdefinition is given later in this introduction; a completely precise version isgiven in the body of the book.

Definition 4.0.2. A (continuum) quantum field theory is:(1) An e!ective action

Seff ["] : C"(M)[0,!] $ R[[!]]

for all " & (0,#). More precisely, Seff ["] should be a formalpower series both in the field & C"(M)[0,!] and in the variable!.

(2) Modulo !, each Seff ["] must be of the form

Seff ["]( ) = !12

!

MD + cubic and higher terms.

where D is the positive-definite Laplacian. (If we want to considera massive scalar field theory, we can replace D by D+m2).

(3) If "! < ", Seff ["!] is determined from Seff ["] by the renormaliza-tion group equation (which makes sense in the formal power seriessetting).

(4) The e!ective actions Seff ["] satisfy a locality axiom, which we willsketch below.

Earlier I described several di!erent versions of the renormalization groupequation; one based on the world-line formulation of quantum field theory,and one defined by considering arbitrary parametrices for the Laplacian.One gets an equivalent definition of quantum field theory using either ofthese versions of the renormalization group flow.

5. Locality

Locality is one of the fundamental principles of quantum field theory.Roughly, locality says that any interaction between fundamental particlesoccurs at a point. Two particles at di!erent points of space-time cannotspontaneously a!ect each other. They can only interact through the mediumof other particles. The locality requirement thus excludes any “spooky ac-tion at a distance”.


14 1. INTRODUCTION

Locality is easily understood in the naive functional integral picture.Here, the theory is supposed to be described by a functional measure of theform

eS( )/!d ,

where d represents the non-existent Lebesgue measure on C"(M). In thispicture, locality becomes the requirement that the action function S is alocal action functional.

Definition 5.0.3. A function

S : C"(M)$ R[[!]]

is a local action functional, if it can be written as a sum

S( ) ='

Sk( )

where Sk( ) is of the form

Sk( ) =!

M(D1 )(D2 ) · · · (Dk )dV olM

where Di are di!erential operators on M .Thus, a local action functional S is of the form

S( ) =!

x#ML ( )(x)

where the Lagrangian L ( )(x) only depends on Taylor expansion of at x.

5.1. Of course, the naive functional integral picture doesn’t make sense.If we want to give a definition of quantum field theory based on Wilson’sideas, we need a way to express the idea of locality in terms of the finiteenergy e!ective actions Seff ["].

As "$#, the e!ective action Seff ["] is supposed to encode more andmore “fundamental” interactions. Thus, the first tentative definition is thefollowing.

Definition 5.1.1 (Tentative definition of asymptotic locality). A col-lection of low-energy e!ective actions Seff ["] satisfying the renormalizationgroup equation is asymptotically local if there exists a large " asymptoticexpansion of the form

Seff ["]( ) .'

fi(")%i( )

where the %i are local action functionals. (The "$# limit of Seff ["] doesnot exist, in general).

This asymptotic locality axiom turns out to be a good idea, but with afundamental problem. If we suppose that Seff ["] is close to being local forsome large ", then for all "! < ", the renormalization group equation impliesthat Seff ["!] is entirely non-local. In other words, the renormalization groupflow is not compatible with the idea of locality.


5. LOCALITY 15

This problem, however, is an artifact of the particular form of the renor-malization group equation we are using. The notion of “energy” is verynon-local: high-energy eigenvalues of the Laplacian are spread out all overthe manifold. Things work much better if we use the version of the renor-malization group flow based on length rather than energy.

The length-based version of the renormalization group flow was sketchedearlier. It will be described in detail in Chapter 2, and used throughout therest of the book.

This length scale version of the renormalization group equation is essen-tially equivalent to the version based on energy, in the following sense:

Any solution to the length-scale RGE can be translatedinto a solution to the energy-scale RGE and conversely3.

Under this transformation, large length will correspond (roughly) to lowenergy, and vice-versa.

The great advantage of working with length scales, however, is that onecan make sense of locality. Unlike the energy-scale renormalization groupflow, the length-scale renormalization group flow di!uses from local to non-local. We have seen earlier that it is more convenient to describe the length-scale version of an e!ective field theory by an e!ective interaction Ieff [L]rather than by an e!ective action. If the length-scale L e!ective interactionIeff [L] is close to being local, then Ieff [L + ] is slightly less local, and soon.

As L $ 0, we approach more “fundamental” interactions. Thus, thelocality axiom should say that Ieff [L] becomes more and more local asL $ 0. Thus, one can correct the tentative definition asymptotic locality tothe following:

Definition 5.1.2 (Asymptotic locality). A collection of low-energy ef-fective actions Ieff [L] satisfying the length-scale version of the renormaliza-tion group equation is asymptotically local if there exists a small L asymp-totic expansion of the form

Seff [L]( ) .'

fi(L)%i( )

where the %i are local action functionals. (The actual L $ 0 limit will notexist, in general).

Because solutions to the length scale and energy scale RGEs are in bi-jection, this definition applies to solutions to the energy scale RGE as well.

We can now update our definition of quantum field theory:

Definition 5.1.3. A (continuum) quantum field theory is:(1) An e!ective action

Seff ["] : C"(M)[0,!] $ R[[!]]

3The converse requires some growth conditions on the energy-scale e!ective actionsSeff ["].


16 1. INTRODUCTION

for all " & (0,#). More precisely, Seff ["] should be a formalpower series both in the field & C"(M)[0,!] and in the variable!.

(2) Modulo !, each Seff ["] must be of the form

Seff ["]( ) = !12

!


where D is the positive-definite Laplacian. (If we want to considera massive scalar field theory, we can replace D by D+m2).

(3) If "! < ", Seff ["!] is determined from Seff ["] by the renormaliza-tion group equation (which makes sense in the formal power seriessetting).

(4) The e!ective actions Seff ["], when translated into a solution tothe length-scale version of the RGE, satisfy the asymptotic localityaxiom.

Since solutions to the energy and length-scale versions of the RGE areequivalent, one can base this definition entirely on the length-scale versionof the RGE. We will do this in the body of the book.

Earlier we sketched a very general form of the RGE, which uses anarbitrary parametrix to define a “scale” of the theory. In Chapter 2, Section8, we will give a definition of a quantum field theory based on arbitraryparametrices, and we will show that this definition is equivalent to the onedescribed above.

6. The main theorem

Now we are ready to state the first main result of this book.

Theorem A. Let T (n) denote the set of theories defined modulo !n+1.Then, T (n+1) is a principal bundle over T (n) for the abelian group of localaction functionals S : C"(M) $ R.

Recall that a functional S is a local action functional if it is of the form

S( ) =!

ML ( )

where L is a Lagrangian. The abelian group of local action functionals isthe same as that of Lagrangians up to the addition of a Lagrangian whichis a total derivative.

Choosing a section of each principal bundle T (n+1) $ T (n) yields anisomorphism between the space of theories and the space of series in ! whosecoe#cients are local action functionals.

A variant theorem allows one to get a bijection between theories andlocal action functionals, once one has made an additional universal (butunnatural) choice, that of a renormalization scheme. A renormalization


6. THE MAIN THEOREM 17

scheme is a way to extract the singular part of certain functions of onevariable. We construct a certain subalgebra

P((0, 1)) ) C"((0, 1))

consisting of functions f( ) of a “motivic” nature. Functions in P((0, 1))arise as the periods of families of algebraic varieties over Zariski open subsetsU ) A1

Q, such that U(R) contains (0, 1). (For more details, see Chapter 2,Section 9).

Definition 6.0.4. A renormalization scheme is a subspace

P((0, 1))<0 )P((0, 1))

of “purely singular” functions, complementary to the subspace

P((0, 1))'0 )P((0, 1))

of functions whose r $# limit exists.

The choice of a renormalization scheme gives us a way to extract thesingular part of functions in P((0, 1)).

The variant theorem is the following.

Theorem B. The choice of a renormalization scheme leads to a bijec-tion between the space of theories and the space of local action functionals

S : C"(M)$ R[[!]].

Equivalently, there is a bijection between the space of theories and thespace of Lagrangians up to the addition of a Lagrangian which is a totalderivative.

Theorem B implies theorem A, but theorem A is the more natural for-mulation.

There are certain caveats:(1) Like the e!ective actions Seff ["], the local action functional S is a

formal power series both in & C"(M) and in !.(2) Modulo !, we require that S is of the form

S( ) = !12

!


There is a more general formulation of this theorem, where the space offields is allowed to be the space of sections of a graded vector bundle. Inthe more general formulation, the action functional S must have a quadraticterm which is elliptic in a certain sense.

6.1. Let me sketch how to prove theorem A. Given the action S, weconstruct the low-energy e!ective action Seff ["] by renormalizing a certainfunctional integral. The formula for the functional integral is

Seff ["]( L) = ! log

(!

H#C!(M)(!,!)

eS( L+ H)/!)

.


18 1. INTRODUCTION

This expression is the renormalization group flow from infinite energy toenergy ". This is an infinite dimensional integral, as the field H has un-bounded energy.

This functional integral is renormalized using the technique of coun-terterms. This involves first introducing a regulating parameter r into thefunctional integral, which tames the singularities arising in the Feynmangraph expansion. One choice would be to take the regularized functionalintegral to be an integral only over the finite dimensional space of fields& C"(M)(!,r].Sending r $ # recovers the original integral. This limit won’t exist,

but one renormalizes this limit by introducing counterterms. Countertermsare functionals SCT (r, ) of both r and the field , such that the limit

limr&"

!

H#C!(M)(!,r]

exp#

1!S( L + H)! 1

!SCT (r, L + H)$

exists. These counterterms are local, and are uniquely defined once onechooses a renormalization scheme.

The e!ective action Seff ["] is then defined by this limit:

Seff ["]( L) =

limr&"

!

H#C!(M)(!,r]

exp#

1!S( L + H)! 1

!SCT (r, L + H)$

6.2. In practise, we don’t use the energy-scale regulator r but rathera length-scale regulator . The reason is the same as before: it is easierto construct local theories using the length-scale regulator than the energy-scale regulator. In what follows, I will ignore this rather technical point; tomake the following discussion completely accurate, the reader should replacethe energy-scale regulator r by the length-scale regulator we will use later.

The counterterms SCT are constructed by a simple inductive procedure,and are local action functionals of the field & C"(M).

Once we have chosen such a renormalization scheme, we find a set ofcounterterms SCT (r, ) for any local action functional S. These countert-erms are uniquely determined by the requirements that firstly, the r $ #limit above exists, and secondly, they are purely singular as a function ofthe regulating parameter r.

6.3. What we see from this is that the bijection between theories andlocal action functionals is not canonical, but depends on the choice of arenormalization scheme. Thus, theorem A is the most natural formulation:there is no natural bijection between theories and local action functionals.Theorem A implies that the space of theories is an infinite dimensionalmanifold, modelled on the topological vector space of R[[!]]-valued localaction functionals on C"(M).


7. RENORMALIZABILITY 19

7. Renormalizability

We have seen that the space of theories is an infinite dimensional man-ifold, modelled on the space of R[[!]]-valued local action functionals onC"(M).

A physicist would find this unsatisfactory. Because the space of theoriesis infinite dimensional, to specify a particular theory, it would take an infinitenumber of experiments. Thus, we can’t make any predictions.

We need to find a natural finite-dimensional submanifold of the spaceof all theories, consisting of “well-behaved” theories. These well-behavedtheories will be called renormalizable.

7.1. An old-fashioned viewpoint is the following:

A local action functional (or Lagrangian) is renormalizable ifit has only finitely many counterterms:

SCT (r) ='

finite

fi(r)SCTi

In general, this definition picks out a finite dimensional subspace of the in-finite dimensional space of theories. However, it is not natural: the spe-cific counterterms will depend on the choice of renormalization scheme,and therefore this definition may depend on the choice of renormalizationscheme.

More fundamentally, any definitions one makes should be directly interms of the only physical quantities one can measure, namely the low-energye!ective actions Seff ["]. Thus, we would like a definition of renormalizabil-ity using only the Seff ["].

The following is the basic idea of the definition we suggest, followingWilson and others.

Definition 7.1.1 (Rough definition). A theory, defined by e!ective ac-tions Seff ["], is renormalizable if the Seff ["] don’t grow too fast as "$#.However, we must measure Seff ["] in units appropriate to energy scale ".

For instance, if Seff [1] is measured in joules, then Seff [103] should bemeasured in kilo-joules, and so on.

However, this change of units only makes sense on Rn. Since we canidentify energy with length$2, changing the units of energy amounts torescaling Rn. In addition, the field & C"(Rn) can have its own energy(which should be thought of as giving the target of the map : Rn $ Rsome weight). Once we incorporate both of these factors, the procedure ofchanging units (in a scalar field theory) is implemented by the map

Rl : C"(Rn)$ C"(Rn)

(x) /$ l1$n/2 (l$1x).


20 1. INTRODUCTION

7.2. As our definition of renormalizability only makes sense on Rn, wewill now restrict to considering scalar field theories on Rn. We want to mea-sure Seff ["] as " $ #, after we have changed units. Define RGl(Seff ["])by

RGl(Seff ["])( ) = Seff [l$2"](Rl( ))Thus, RGl(Seff ["]) is the e!ective action Seff [l2"], but measured in unitsthat have been rescaled by l.

We can use the map RGl to implement precisely the definition of renor-malizability suggested above.

Definition 7.2.1. A theory {Seff ["]} is renormalizable if RGl(Seff ["])grows at most logarithmically as l $ 0.

7.3. It turns out that the map RGl defines a flow on the space of theories.

Lemma 7.3.1. If {Seff ["]} satisfies the renormalization group equation,then so does {RGl(Seff ["]}.

Thus, sending{Seff ["]}$ {RGl(Seff ["])}

defines a flow on the space of theories: this is the local renormalization groupflow.

Recall that the choice of a renormalization scheme leads to a bijectionbetween the space of theories and Lagrangians. Under this bijection, thelocal renormalization group flow acts on the space of Lagrangians. The con-stants appearing in a Lagrangian (the coupling constants) become functionsof l; the dependence of the coupling constants on the parameter l is calledthe function. Renormalizability means these coupling constants have atmost logarithmic growth in l.

The local renormalization group flow RGl, as l $ 0, can be interpretedgeometrically as focusing on smaller and smaller regions of space-time, whilealways using units appropriate to the size of the region one is considering.In energy terms, applying RGl as l $ 0 amounts to focusing on phenomenaof higher and higher energy.

The logarithmic growth condition thus says the theory doesn’t breakdown completely when we probe high-energy phenomena. If the e!ectiveactions displayed polynomial growth, for instance, then one would find thatthe perturbative description of the theory wouldn’t make sense at high en-ergy, because the terms in the perturbative expansion would increase withthe energy.

7.4. The definition of renormalizability given above can be viewed as aperturbative approximation to an ideal non-perturbative definition.

Definition 7.4.1 (Ideal definition). A non-perturbative theory is renor-malizable if, as we flow the theory under RGl and let l $ 0, we converge toa fixed point.


8. RENORMALIZABLE SCALAR FIELD THEORIES 21

This fixed point, if it exists, would be a scaling limit of the theory; itwould necessarily be a scale-invariant theory. For instance, it is expectedthat Yang-Mills theory is renormalizable in this sense, and that the scalinglimit is a free theory.

This ideal definition is di#cult to make sense of perturbatively (when wetreat ! as a formal parameter). For instance, suppose a coupling constant cchanges to

c /$ l!c = c + !c log l + · · ·

Non-perturbatively, we might think that ! > 0, so that this flow convergesto a fixed point. Perturbatively, however, ! is a formal parameter, so itappears to have logarithmic growth.

Our perturbative definition can be interpreted as saying that a pertur-bative theory is renormalizable if, at first sight, it looks like it might benon-perturbatively renormalizable in this sense. For instance, if it containscoupling constants which are of polynomial growth in l, these will proba-bly persist at the non-perturbative level, implying that the theory does notconverge to a fixed point.

One can make a more refined perturbative definition by requiring thatthe logarithmic growth which does appear is of the correct sign (thus distin-guishing between c /$ l!c and c /$ l$!c). This more refined definition leadsto asymptotic freedom, which is the statement that a theory converges to afree theory as l $ 0.

8. Renormalizable scalar field theories

Now that we have a definition of renormalizability, the next question toask is: which theories are renormalizable?

It turns out to be straightforward to classify all renormalizable scalarfield theories.

8.1. Suppose we have a local action functional S of a scalar field on Rn,and suppose that S is translation invariant. We say that S is of dimensionk if

S(Rl( )) = lkS( ).

Recall that Rl( )(x) = l1$n/2 ($1lx).


22 1. INTRODUCTION

Every translation invariant local action functional S is a finite sum ofterms of dimension. For instance:!

R4D is of dimension 0

!

R4

4 is of dimension 0!

R4

3 "

"xiis of dimension ! 1

!

R4

2 is of dimension 2

Now let us state how one classifies scalar field theories, in general.

Theorem 8.1.1. Let R(k)(Rn) denote the space of renormalizable scalarfield theories on Rn, invariant under translation, defined modulo !n+1.

Then,R(k+1)(Rn) $ R(k)(Rn)

is a torsor for the vector space of local action functionals S( ) which are asum of terms of non-negative dimension.

Further, R(0)(Rn) is canonically isomorphic to the space of local actionfunctionals of the form

S( ) = !12

!

RnD + cubic and higher terms, of non-negative dimension.

As before, the choice of a renormalization scheme leads to a section ofeach of the torsors R(k+1)(Rn) $ R(k)(Rn), and so to a bijection betweenthe space of renormalizable scalar field theories and the space of series

!12

!D +

'!iSi

where each Si is a translation invariant local action functional of non-negative dimension, and S0 is at least cubic.

Applying this to R4, we find the following.

Corollary 8.1.2. Renormalizable scalar field theories on R4, invariantunder SO(4) " R4 and under $ ! , are in bijection with Lagrangians ofthe form

L ( ) = a D + b 4 + c 2

for a, b, c & R[[!]], where a = !12 modulo ! and b = 0modulo !.

More generally, there is a finite dimensional space of non-free renormal-izable theories in dimensions n = 3, 4, 5, 6, an infinite dimensional space indimensions n = 1, 2, and none in dimensions n > 6. ( “Finite dimensional”means as a formal scheme over Spec R[[!]]: there are only finitely manyR[[!]]-valued parameters).


9. GAUGE THEORIES 23

Thus we find that the scalar field theories traditionally considered to be“renormalizable” are precisely the ones selected by the Wilsonian definitionadvocated here. However, in this approach, one has a conceptual reason forwhy these particular scalar field theories, and no others, are renormalizable.

9. Gauge theories

We would like to apply the Wilsonian philosophy to understand gaugetheories. In Chapter 5, we will explain how to do this using a synthesis ofWilsonian ideas and the Batalin-Vilkovisky formalism.

9.1. In mathematical parlance, a gauge theory is a field theory wherethe space of fields is a stack. A typical example is Yang-Mills theory, wherethe space of fields is the space of connections on some principal G-bundleon space-time, modulo gauge equivalence.

It is important to emphasize the di!erence between gauge theories andfield theories equipped with some symmetry group. In a gauge theory, thegauge group is not a group of symmetries of the theory. The theory doesnot make any sense before taking the quotient by the gauge group.

One can see this even at the classical level. In classical U(1) Yang-Millstheory on a 4-manifold M , the space of fields (before quotienting by thegauge group) is &1(M). The action is S( ) =

&M d 0d . The highly degen-

erate nature of this action means that the classical theory is not predictive:a solution to the equations of motion is not determined by its behaviour ona space-like hypersurface. Thus, classical Yang-Mills theory is not a sensibletheory before taking the quotient by the gauge group.

9.2. Let us now discuss gauge theories in e!ective field theory. Naively,one could imagine that to give a gauge theory would be to give an e!ectivegauge theory at every energy level, in a way related by the renormalizationgroup flow.

One immediate problem with this idea is that the space of low-energygauge symmetries is not a group. The product of low-energy gauge sym-metries is no longer low-energy; and if we project this product onto itslow-energy part, the resulting multiplication on the set of low-energy gaugesymmetries is not associative.

For example, if g is a Lie algebra, then the Lie algebra of infinitesi-mal gauge symmetries on a manifold M is C"(M) , g. The space of low-energy infinitesimal gauge symmetries is then C"(M)%! , g. In general,the product of two functions in C"(M)%! can have arbitrary energy; sothat C"(M)<! , g is not closed under the Lie bracket.

This problem is solved by a very natural union of the Batalin-Vilkoviskyformalism and the e!ective action philosophy.

9.3. The Batalin-Vilkovisky formalism is widely regarded as being themost powerful and general way to quantize gauge theories. The first stepin the BV procedure is to introduce extra fields – ghosts, corresponding


24 1. INTRODUCTION

to infinitesimal gauge symmetries; anti-fields dual to fields; and anti-ghostsdual to ghosts – and then write down an extended classical action functionalon this extended space of fields.

This extended space of fields has a very natural interpretation in ho-mological algebra: it describes the derived moduli space of solutions to theEuler-Lagrange equations of the theory. The derived moduli space is ob-tained by first taking a derived quotient of the space of fields by the gaugegroup, and then imposing the Euler-Lagrange equations of the theory in aderived way. The extended classical action functional on the extended spaceof fields arises from the di!erential on this derived moduli space.

In more pedestrian terms, the extended classical action functional en-codes the following data:

(1) the original action functional on the original space of fields;(2) the Lie bracket on the space of infinitesimal gauge symmetries,(3) the way this Lie algebra acts on the original space of fields.

In order to construct a quantum theory, one asks that the extended actionsatisfies the quantum master equation. This is a succinct way of encodingthe following conditions:

(1) The Lie bracket on the space of infinitesimal gauge symmetriessatisfies the Jacobi identity.

(2) This Lie algebra acts in a way preserving the action functional onthe space of fields.

(3) The Lie algebra of infinitesimal gauge symmetries preserves the“Lebesgue measure” on the original space of fields. That is, thevector field on the original space of fields associated to every infin-itesimal gauge symmetry is divergence free.

(4) The adjoint action of the Lie algebra on itself also preserves the“Lebesgue measure”. Again, this says that a vector field associatedto every infinitesimal gauge symmetry is divergence free.

Unfortunately, the quantum master equation is an ill-defined expression.The 3rd and 4th conditions above are the source of the problem: the diver-gence of a vector field on the space of fields is a singular expression, involvingthe same kind of singularities as those appearing in one-loop Feynman dia-grams.

9.4. This form of the quantum master equation violates our philosophy:we should always express things in terms of the e!ective actions. The quan-tum master equation above is about the original “infinite energy” action, sowe should not be surprised that it doesn’t make sense.

The solution to this problem is to combine the BV formalism with thee!ective action philosophy. To give an e!ective action in the BV formalismis to give a functional Seff ["] on the energy * " part of the extended spaceof fields (i.e., the space of ghosts, fields, anti-fields and anti-ghosts). This


9. GAUGE THEORIES 25

energy " e!ective action must satisfy a certain energy " quantum masterequation.

The reason that the e!ective action philosophy and the BV formalismwork well together is the following.

Lemma. The renormalization group flow from scale " to scale "! carriessolutions of the energy " quantum master equation into solutions of theenergy "! quantum master equation.

Thus, to give a gauge theory in the e!ective BV formalism is to givea collection of e!ective actions Seff ["] for each ", such that Seff ["] satis-fies the scale " QME, and such that Seff ["!] is obtained from Seff ["] bythe renormalization group flow. In addition, one requires that the e!ectiveactions Seff ["] satisfy a locality axiom, as before.

This picture also solves the problem that the low energy gauge symme-tries are not a group. The energy " e!ective action Seff ["], satisfying theenergy " quantum master equation, gives the extended space of low-energyfields a certain homotopical algebraic structure, which has the following in-terpretation:

(1) The space of low-energy infinitesimal gauge symmetries has a Liebracket.

(2) This Lie algebra acts on the space of low-energy fields.(3) The space of low-energy fields has a functional, invariant under the

bracket.(4) The action of the Lie algebra on the space of fields, and on itself,

preserves the Lebesgue measure.However, these axioms don’t hold on the nose, but hold up to a sequence ofcoherent higher homotopies.

9.5. Let us now formalize our definition of a gauge theory. As we haveseen, whenever we have the data of a classical gauge theory, we get anextended space of fields, that we will denote by E . This is always the spaceof sections of a graded vector bundle on the manifold M . As before, let E%!denote the space of low-energy extended fields.

Definition 9.5.1. A theory in the BV formalism consists of a set oflow-energy e!ective actions

Seff ["] : E%! $ R[[!]],

which is a formal series both in E%! and !, and which is such that:(1) The renormalization group equation is satisfied.(2) Each Seff ["] satisfies the energy " quantum master equation.(3) The same locality axiom as before holds.(4) There is one more technical restriction : modulo !, each Seff ["] is

of the form

Seff ["](e) = 'e,Qe(+ cubic and higher terms


26 1. INTRODUCTION

where '!,!( is a certain canonical pairing on E , and Q : E $ Esatisfies certain ellipticity conditions.

As before, the locality axiom needs to be expressed in length-scale terms.The main theorem holds in this context also, but in a slightly modified

form. If we remove the requirement that the e!ective actions satisfy thequantum master equation, we find a bijection between theories and localaction functionals, depending on the choice of a renormalization scheme,as before. Requiring that the e!ective actions satisfy the QME leads to aconstraint on the corresponding local action functional, which is called therenormalized quantum master equation. This renormalized QME replacesthe ill-defined QME appearing in the naive BV formalism.

Physicists often say that a theory satisfying the quantum master equa-tion is free of “gauge anomalies”. In general, an anomaly is a symmetry ofthe classical theory which fails to be a symmetry of the quantum theory. Inmy opinion, this terminology is misleading: the gauge group action on thespace of fields is not a symmetry of the theory, but rather an inextricablepart of the theory. The presence of gauge anomalies means that the theorydoes not exist in a meaningful way.

9.6. Renormalizing gauge theories. It is straightforward to gener-alize the Wilsonian definition of renormalizability (Definition 7.2.1) to applyto gauge theories in the BV formalism. As before, this definition only workson Rn, because one needs to rescale space-time. This rescaling of space-timeleads to a flow on the space of theories, which we call the local renormal-ization group flow. (This flow respects the quantum master equation). Atheory is defined to be renormalizable if it exhibits at most logarithmicgrowth under the local renormalization group flow.

Now we are ready to state one of the main results of this book.

Theorem. Pure Yang-Mills theory on R4, with coe"cients in a simpleLie algebra g, is perturbatively renormalizable.

That is, there exists a theory {SeffY M ["]}, which is renormalizable, which

satisfies the quantum master equation, and which modulo ! is given by theclassical Yang-Mills action.

The moduli space of such theories is isomorphic to !R[[!]].

Let me state more precisely what I mean by this. At the classical level(modulo !) there are no di#culties with renormalization, and it is straight-forward to define pure Yang-Mills theory in the BV formalism4. Becausethe classical Yang-Mills action is conformally invariant in four dimensions,it is a fixed point of the local renormalization group flow.

One is then interested in quantizing this classical theory in a renormal-izable way.

4For technical reasons, we use a first-order formulation of Yang-Mills, which is equiv-alent to the usual formulation.


11. OTHER APPROACHES TO PERTURBATIVE QUANTUM FIELD THEORY 27

The theorem states that one can do this, and that the set of all suchrenormalizable quantizations is isomorphic (non-canonically) to !R[[!]].

This theorem is proved by obstruction theory. A lengthy (but straight-forward) calculation in Lie algebra cohomology shows that the group ofobstructions to finding a renormalizable quantization of Yang-Mills theoryvanishes; and that the corresponding deformation group is one-dimensional.Standard obstruction theory arguments then imply that the moduli spaceof quantizations is R[[!]], as desired.

This calculation uses the following strange “coincidence” in Lie algebracohomology: although H5(su(3)) is one-dimensional, the outer automor-phism group of su(3) acts on this space in a non-trivial way. A more directconstruction of Yang-Mills theory, not relying on obstruction theory, is de-sirable.

10. Observables and correlation functions

The key quantities one wants to compute in a quantum field theory arethe correlation functions between observables. These are the quantities thatcan be more-or-less directly related to experiment.

The theory of observables and correlation functions is addressed in thework in progress (CG10), written jointly with Owen Gwilliam. In this sequel,we will show how the observables of a quantum field theory (in the sense ofthis book) form a rich algebraic structure called a factorization algebra. Theconcept of factorization algebra was introduced by Beilinson and Drinfeld(BD04), as a geometric formulation of the axioms of a vertex algebra. Thefactorization algebra associated to a quantum field theory is a completeencoding of the theory: from this algebraic object one can reconstruct thecorrelation functions, the operator product expansion, and so on.

Thus, a proper treatment of correlation functions requires a great dealof preliminary work on the theory of factorization algebras and on the fac-torization algebra associated to a quantum field theory. This is beyond thescope of the present work.

11. Other approaches to perturbative quantum field theory

Let me finish by comparing briefly the approach to perturbative quantumfield theory developed here with others developed in the literature.

11.1. In the last ten years, the perturbative version of algebraic quan-tum field theory has been developed by Brunetti, Dutsch, Fredenhagen,Hollands, Wald and others: see (BF00; BF09; DF01; HW10). In this work,the authors investigate the problem of constructing a solution to the axiomsof algebraic quantum field theory in perturbation theory. These authorsprove results which have a very similar form to those proved in this book:term by term in !, there is an ambiguity in quantization, described by acertain class of Lagrangians.


28 1. INTRODUCTION

The proof of these results relies on a version of the Epstein-Glaser(EG73) construction of counterterms. This construction of counterterms re-lies, like the approach used in this book, on working directly on real space,as opposed to on momentum space. In the Epstein-Glaser approach torenormalization, as in the approach described here, the proof that the coun-terterms are local is easy. In contrast, in momentum-space approaches toconstructing counterterms – such as that developed by Bogoliubov-Parasiuk(BP57) and Hepp (Hep66) – the problem of constructing local countertermsinvolves complicated graph combinatorics.

11.2. Another, related, approach to perturbative quantum field the-ory on Riemannian space-times was developed by Hollands (Hol09) andHollands-Olbermann (HO09). In this approach the field theory is encodedin a vertex algebra on the space-time manifold.

This seems to be philosophically very closely related to my joint workwith Owen Gwilliam (CG10), which uses the renormalization techniquesdeveloped in this paper to produce a factorization algebra on the space-timemanifold. Thus, the quantization results proved by Hollands-Olbermannshould be close analogues of the results presented here.

11.3. In a lecture at the conference “Renormalization: algebraic, geo-metric and probabilistic aspects” in Lyon in 2010, Maxim Kontsevich pre-sented an approach to perturbative renormalization which he developedsome years before. The output of Kontsevich’s construction is (as in (HO09)and (CG10)) a vertex algebra on the space-time manifold. The form of Kont-sevich’s theorem is very similar to the main theorem of this book: order byorder in !, the space of possible quantizations is a torsor for an Abeliangroup constructed from certain Lagrangians. Kontsevich’s work relies on anew construction of counterterms which, like the Epstein-Glaser construc-tion and the construction developed here, relies on working in real spacerather than on momentum space.

Again, it is natural to speculate that there is a close relationship betweenKontsevich’s work and the construction of factorization algebras presentedin (CG10).

11.4. Let me finally mention an approach to perturbative renormaliza-tion developed initially by Connes and Kreimer (CK98; CK99), and furtherdeveloped by (among others) Connes-Marcolli (CM04; CM08b) and vanSuijlekom (vS07). The first result of this approach is that the Bogoliubov-Parasiuk-Hepp-Zimmermann (BP57; Hep66) algorithm has a beautiful in-terpretation in terms of the Birkho! decomposition for loops in a certainpro-algebraic group constructed combinatorially from graphs.

In this book, however, counterterms have no intrinsic importance: theyare simply a technical tool used to prove the main results. Thus, it is notclear to me if there is any relationship between Connes-Kreimer Hopf algebraand the results of this book.


ACKNOWLEDGEMENTS 29

Acknowledgements

Many people have contributed to the material in this book. Withoutthe constant encouragement and insightful editorial suggestions of LaurenWeinberg this project would probably never have been finished. Particularthanks are also due to Owen Gwilliam, for many discussions which havegreatly clarified the ideas in this book; and to Dennis Sullivan, for his en-couragement and many helpful comments. I would also like to thank AlbertoCattaneo, Jacques Distler, Mike Douglas, Giovanni Felder, Arthur Green-spoon, Si Li, Pavel Mnev, Josh Shadlen, Yuan Shen, Stephan Stolz, PeterTeichner, and A.J. Tolland. Of course, any errors are solely the responsibil-ity of the author.



CHAPTER 2

Theories, Lagrangians and counterterms

1. Introduction

In this chapter, we will make precise the definition of quantum fieldtheory we sketched in Chapter 1. Then, we will show the main theorem:

Theorem A. Let T (n)(M) denote the space of scalar field theories ona manifold M , defined modulo !n+1.

Then T (n+1)(M) $ T (n)(M) is (in a canonical way) a principal bundlefor the space of local action functionals.

Further, T (0)(M) is canonically isomorphic to the space of local actionfunctionals which are at least cubic.

This theorem has a less natural formulation, depending on an additionalchoice, that of a renormalization scheme. A renormalization scheme is anobject of a “motivic” nature, defined in Section 9.

Theorem B. The choice of a renormalization scheme leads to a sectionof each principal bundle T (n+1)(M) $ T (n)(M), and thus to an isomor-phism between the space of theories and the space of local action functionalsof the form

*!iSi, where S0 is at least cubic.

1.1. Let me summarize the contents of this chapter.The first few sections explain, in a leisurely fashion, the version of the

renormalization group flow we use throughout this book. Sections 2 and 4introduce the heat kernel version of high-energy cut-o! we will use through-out the book. Section 3 contains a general discussion of Feynman graphs,and explains how certain finite dimensional integrals can be written as sumsover graphs. Section 5 explains why infinities appear in the naive func-tional integral formulation of quantum field theory. Section 6 shows howthe weights attached to Feynman graphs in functional integrals can be in-terpreted geometrically, as integrals over spaces of maps from graphs to amanifold.

In Section 7, we finally get to the precise definition of a quantum fieldtheory and the statement of the main theorem. Section 8 gives a variant ofthis definition which doesn’t rely on the heat kernel, but instead works withan arbitrary parametrix for the Laplacian. This variant definition is equiv-alent to the one based on the heat kernel. Section 9 introduces the conceptof renormalization scheme, and shows how the choice of renormalizationscheme allows one to extract the singular part of the weights attached to

31


32 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

Feynman graphs. Section 10 uses this to construct the local counterterms as-sociated to a Lagrangian, which are needed to render the functional integralfinite. Section 11 gives the proof of theorems A and B above.

Finally, we turn to generalizations of the main results. Section 13 showshow everything generalizes, mutatis mutandis, to the case when our fieldsare no longer just functions, but sections of some vector bundle. Section 14shows how we can further generalize to deal with theories on non-compactmanifolds, as long as an appropriate infrared cut-o! is introduced.

2. The e!ective interaction and background field functionalintegrals

As in the introduction, a quantum field theory in our Wilsonian defini-tion will be given by a collection of e!ective actions, related by the renor-malization group flow. In this section we will write down a version of therenormalization group flow, based on the e!ective interaction, which we willuse throughout the book.

2.1. Let us assume that our energy " e!ective action can be written as

S["]( ) = !12

+, (D+m2)

,+ I["]( )

where:(1) The function I["] is a formal series in !, I["] = I0["]+!I1["]+ · · · ,

where the leading term I0 is at least cubic. Each Ii is a formal powerseries on the vector space C"(M) of fields (later, I will explain whatthis means more precisely).

The function I["] will be called the e!ective interaction.(2) ' , ( denotes the L2 inner product on C"(M, R) defined by ' , ( =&

M .(3) D denotes1 the Laplacian on M , with signs chosen so that the

eigenvalues of D are non-negative; and m & R>0.Recall that the renormalization group equation relating S["] and S["!]

can be written

S["!]( L) = ! log

"!

H#C!(M)[!#,!)

eS[!]( L+ H)/!%

.

We can rewrite this in terms of the e!ective interactions, as follows. Thespaces C"(M)<!# and C"(M)[!#,!) are orthogonal. It follows that

S["]( L + H)

= !12

+L, (D+m2) L

,! 1

2

+H , (D+m2) H

,+ I["]( L + H).

1The symbol # will be reserved for the BV Laplacian


2. THE EFFECTIVE ACTION 33

Therefore the e!ective interaction form of the renormalization group equa-tion (RGE) is

I["!](a)

= ! log

"!

#C!(M)[!#,!)

exp#! 1

2!+

, (D+m2),

+1!I["]( + a)

$%.

Note that in this expression the field a no longer has to be low-energy.We obtain a variant of the renormalization group equation by consideringe!ective interactions I["] which are functionals of all fields, not just low-energy fields, and using the equation above. This equation is invertible; itis valid even if "! > ".

We will always deal with this invertible e!ective interaction form of theRGE. Henceforth, it will simply be referred to as the RGE.

2.2. We will often deal with integrals of the form!

x#Uexp ($(x)/! + I(x + a)/!)

over a vector space U , where $ is a quadratic form (normally negativedefinite) on U . We will use the convention that the “measure” on U will bethe Lebesgue measure normalised so that

!

x#Uexp ($(x)/!) = 1.

Thus, the measure depends on !.

2.3. Normally, in quantum field theory textbooks, one starts with anaction functional

S( ) = !12

+, (D+m2)

,+ I( ),

where the interacting term I( ) is a local action functional. This meansthat it can be written as a sum

I( ) ='

!iIi,k( )

where

Ii,k(a) =s'

j=1

!

MD1,j(a) · · ·Dk,j(a)

and the Di,j are di!erential operators on M . We also require that I( ) isat least cubic modulo !.

As I mentioned before, the local interaction I is supposed to be thoughtof as the scale# e!ective interaction. Then the e!ective interaction at scale" is obtained by applying the renormalization group flow from energy #down to energy ". This is expressed in the functional integral

I["](a) = ! log

"!

#C!(M)[!,!)

exp#! 1

2!+

, (D+m2),

+1!I( + a)

$%.



This functional integral is ill-defined.

3. Generalities on Feynman graphs

In this section, we will describe the Feynman graph expansion for func-tional integrals of the form appearing in the renormalization group equation.This section will only deal with finite dimensional vector spaces, as a toymodel for the infinite dimensional functional integrals we will be concernedwith for most of this book. For another mathematical description of theFeynman diagram expansion in finite dimensions, one can consult, for ex-ample, (Man99).

3.1.

Definition 3.1.1. A stable2 graph is a graph , possibly with externaledges (or tails); and for each vertex v of an element g(v) & Z'0, calledthe genus of the vertex v; with the property that every vertex of genus 0 isat least trivalent, and every vertex of genus 1 is at least 1-valent (0-valentvertices are allowed, provided they are of genus > 1).

If is a stable graph, the genus g( ) of is defined by

g( ) = b1( ) +'

v#V ( )

g(v)

where b1( ) is the first Betti number of .

More formally, a stable graph is determined by the following data.(1) A finite set H( ) of half-edges of .(2) A finite set V ( ) of vertices of .(3) An involution : H( ) $ H( ). The set of fixed points of this

involution is denoted T ( ), and is called the set of tails of . Theset of two-element orbits is denoted E( ), and is called the set ofedges.

(4) A map : H( )$ V ( ), which sends a half-edge to the vertex towhich it is attached.

(5) A map g : V ( $ Z'0.From this data we construct a topological space | | which is the quotient of

V ( )1-H( )" [0, 1

2 ].

by the relation which identifies (h, 0) & H( )" [0, 12 ] with (h) & V ( ); and

identifies (h, 12) with ( (h), 1

2). We say is connected if | | is. A graph isstable, as above, if every vertex v of genus 0 is at least trivalent, and everyvertex of genus 1 is at least univalent.

We are also interested in automorphisms of stable graphs. It is helpfulto give a formal definition. An element of g & Aut( ) of the group Aut( ) is

2The term “stable” comes from algebraic geometry, where such graphs are used tolabel the strata of the Deligne-Mumford moduli space of stable curves.


3. GENERALITIES ON FEYNMAN GRAPHS 35

a pair of maps H(g) : H( ) $ H( ), V (g) : V ( ) $ V ( ), such that H(g)commutes with , and such that the diagram

H( )H(g)

!!

""

H( )

""V ( )

V (g)!! V ( )

commutes.

3.2. Let U be a finite-dimensional super vector space, over a groundfield K. Let O(U) denote the completed symmetric algebra on the dualvector space U). Thus, O(U) is the ring of formal power series in a variableu & U .

LetO+(U)[[!]] ) O(U)[[!]]

be the subspace of those functionals which are at least cubic modulo !.For an element I & O(U)[[!]], let us write

I ='

i,k'0

!iIi,k

where Ii,k & O(U) is homogeneous of degree k as a function of u & U .If f & O(U) is homogeneous of degree k, then it defines an Sk-invariant

linear map

Dk f : U(k $ K

u1 , · · ·, uk $#"

"u1· · · ""uk

f

$(0).

Thus, if we expand I & O(U)[[!]] as a sum I =*

!iIi,k as above, then wehave collection of Sk invariant elements Dk Ii,k & (U))(k.

Let be a stable graph, with n tails. Let : {1, . . . , n} %= T ( ) be anordering of the set of tails of . Let P & Sym2 U ) U(2, let I & O+(U)[[!]],and let a1, . . . , an & U By contracting the tensors P and ai with the dualtensor I according to a rule given by , we will define

w , (P, I)(a1, . . . , aT ( )) & K.

The rule is as follows. Let H( ), T ( ), E( ), and V ( ) refer to the setsof half-edges, tails, internal edges, and vertices of , respectively. Recallthat we have chosen an isomorphism : T ( ) %= {1, . . . , n}. Putting apropagator P at each internal edge of , and putting ai at the ith tail of ,gives an element of

U(E( ) , U(E( ) , U(T ( ) %= U(H( ).

Putting Dk Ii,k at each vertex of valency k and genus i gives us an elementof

Hom(U(H( ), K).



Figure 1. The first few graphs in the expansion of W (P, I).The variable a & U is placed at each external edge.

Contracting these two elements yields the weight w , (P, I)(a1, . . . , an).Define a function

w (P, I) & O(U)

by

w (P, I)(a) = w , (P, I)(a, . . . , a)

where is any ordering of the set of tails of . Note that w (P, I) ishomogeneous of degree n, and has the property that, for all a1, . . . , an & U ,

"

"a1· · · ""an

w (P, I) ='

:{1,...,n}*=T ( )

w , (P, I)(a1, . . . , an)

where the sum is over ways of ordering the set of tails of .Let vi,k denote the graph with one vertex of genus i and valency k, and

with no internal edges. Then our definition implies that

wvi,k(P, I) = k!Ii,k.



3.3. Now that we have defined the functions w (P, I) & O(U), we willarrange them into a formal power series.

Let us define

W (P, I) =' 1

|Aut( )|!g( )w (P, I) & O+(U)[[!]]

where the sum is over connected stable graphs , and g( ) is the genus ofthe graph . The condition that all genus 0 vertices are at least trivalentimplies that this sum converges. Figure 1 illustrates the first few terms ofthe graphical expansion of W (P, I).

Our combinatorial conventions are such that, for all a1, . . . , ak & U ,#"

"a1· · · ""ak

W (P, I)$

(0) ='

,

!g( )

|Aut| ( , )w , (P, I)(a1, . . . , ak)

where the sum is over graphs with k tails and an isomorphism :{1, . . . , k} %= T ( ), and the automorphism group Aut( , ) preserves theordering of the set of tails.

Lemma 3.3.1.W (0, I) = I.

Proof. Indeed, when P = 0 only graphs with no edges can contribute,so that

W (0, I) ='

i,k

!i

|Aut(vi,k)|wvi,k(0, I).

Here, as before, vi,k is the graph with a single vertex of genus i and valency k,and with no internal edges. The automorphism group of vi,k is the symmetricgroup Sk, and we have seen that wvi,k(0, I) = k!Ii,k. Thus, W (0, I) =*

i,k !iIi,k, which (by definition) is equal to I.!

3.4. As before, let P & Sym2 U , and let us write P =*

P !,P !!. Definean order two di!erential operator "P : O(U)$ O(U) by

"P =12

' "

"P !!"

"P ! .

A convenient way to summarize the Feynman graph expansion W (P, I) isthe following.

Lemma 3.4.1.

W (P, I) (a) = ! log {exp(!"P ) exp(I/!)} (a) & O+(U)[[!]].

The expression exp(!"P ) exp(I/!) is the exponential of a di!erentialoperator on U applied to a function on U ; thus, it is a function on U .



Proof. We will prove this by first verifying the result for P = 0, andthen checking that both sides satisfy the same di!erential equation as afunction of P . When P = 0, we have seen that

W (0, I) = I

which of course is the same as ! log exp(!"P ) exp(I/!).Now let us turn to proving the general case. It is easier to consider the

exponentiated version: so we will actually verify that

exp-!$1W (P, I)

.= exp(!"P #) exp(I/!).

We will do this by verifying that, if is a parameter of square zero, andP ! & Sym2 U ,

exp-!$1W

-P + P !, I

..= (1 + ! "P ) exp

-!$1W (P, I)

..

Let a1, . . . , ak & U , and let us consider#"

"a1· · · ""ak

exp-!$1W (P, I)

.$(0).

It will su#ce to prove a similar di!erential equation for this expression.It follows immediately from the definition of the weight function w (P, I)

of a graph that#"

"a1· · · ""ak

eW (P,I)/!$

(0) ='

,

!g( )

|Aut( , )|w , (P, I)(a1, . . . , ak)

where the sum is over all possibly disconnected stable graphs with anisomorphism : {1, . . . , k} %= T ( ). The automorphism group Aut( , )consists of those automorphisms preserving the ordering of the set of tailsof .

Let be a parameter of square zero, and let P ! & Sym2 U . Let usconsider varying P to P + P !. We find that

dd

#"

"a1· · · ""ak

exp-!$1W

-P + P !, I

..$(0)

='

,e,

!g( )

|Aut( , e, )|w ,e, (P, I)(a1, . . . , ak).

Here, the sum is over possibly disconnected stable graphs with a distin-guished edge e & E( ). The weight w ,e, is defined in the same way as w ,

except that the distinguished edge e is labelled by P !, whereas all other edgesare labelled by P . The automorphism group considered must preserve theedge e as well as .

Given any graph with a distinguished edge e, we can cut along thisedge to get another graph ! with two more tails. These tails can be ordered



in two di!erent ways. If we write P ! =*

u! , u!! where u!, u!! & U , theseextra tails are labelled by u! and u!!. Thus we find that

dd

#"

"a1· · · ""ak

exp-!$1W

-P + P !, I

..$(0)

=12

'

,

!g( )

|Aut( , )|w , (P, I)(a1, . . . , ak, u!, u!!).

Here the sum is over graphs with k + 2 tails, and an ordering of thesek + 2 tails. The factor of 1

2 arises because of the two di!erent ways to orderthe new tails. Comparing this to the previous expression, we find that

dd

#"

"a1· · · ""ak

exp-!$1W

-P + P !, I

..$(0)

=12"

"a1· · · ""ak

"

"u!"

"u!! exp-!$1W (P, I)

.(0)

Since

"P # =12

' "

"u!"

"u!!

this completes the proof. !

This expression makes it clear that, for all P1, P2 & Sym2 U ,

W (P1,W (P2, I)) = W (P1 + P2, I) .

3.5. Now suppose that U is a finite dimensional vector space over R,equipped with a non-degenerate negative definite quadratic form $. LetP & Sym2 U be the inverse to !$. Thus, if ei is an orthonormal basis for!$, P =

*ei , ei. (When we return to considering scalar field theories, U

will be replaced by the space C"(M), the quadratic form $ will be replacedby the quadratic form !

+, (D + m2)

,, and P will be the propagator for

the theory).The Feynman diagram expansion W (P, I) described above can also be

interpreted as an asymptotic expansion for an integral on U .

Lemma 3.5.1.

W (P, I) (a) = ! log!

x#Uexp

#12!$(x, x) +

1!I(x + a)

$.

The integral is understood as an asymptotic series in !, and so makessense whatever the signature of $. As I mentioned before, we use the con-vention that the measure on U is normalized so that

!

x#Uexp

#12!$(x, x)

$= 1.

This normalization accounts for the lack of a graph with one loop and zeroexternal edges in the expansion.



If U is a complex vector space and $ is a non-degenerate complex linearinner product on U , then the same formula holds, where we integrate overany real slice UR of U .

Proof. It su#ces to show that, for all functions f & O(U),!

x#Ue(2!)$1"(x,x)f(x + a) = e!"P f

(where both sides are regarded as functions of a & U).The result is clear when f = 1. Let l & U). Note that

e!"P (lf)! le!"P (lf) = !["P , l]e!"P (lf).

In this expression, !["P , l] is viewed as an order 1 di!erential operator onO(U).

The quadratic form $ on U provides an isomorphism U $ U). If u & U ,let u) & U) be the corresponding element; and, dually, if l & U), let l) & Ube the corresponding element.

Note that

["P , l] = ! "

"l).

It su#ces to verify a similar formula for the integral. Thus, we need to checkthat

!

x#Ue(2!)$1"(x,x)l(x)f(x + a) = !! "

"l)a

!

x#Ue(2!)$1"(x,x)f(x + a).

(The subscript in l)a indicates we are applying this di!erential operator tothe a variable).

Note that"

"l)xe(2!)$1"(x,x) = !$1l(x)e"(x,x)/!.

Thus,!

x#Ue(2!)$1"(x,x)l(x)f(x + a) = !

!

x#U

#"

"l)xe(2!)$1"(x,x)

$f(x + a)

= !!!

x#Ue(2!)$1"(x,x) "

"l)xf(x + a)

= !!!

x#Ue(2!)$1"(x,x)l(x)

"

"l)af(x + a)

= !! ""l)a

!

x#Ue(2!)$1"(x,x)f(x + a)

as desired.!



3.6. One can ask, how much of this picture holds if U is replaced byan infinite dimensional vector space? We can’t define the Lebesgue measurein this situation, thus we can’t define the integral directly. However, onecan still contract tensors using Feynman graphs, and one can still definethe expression W (P, I), as long as one is careful with tensor products anddual spaces. (As we will see later, the singularities in Feynman graphs arisebecause the inverse to the quadratic forms we will consider on infinite di-mensional vector spaces do not lie in the correct completed tensor product.)

Let us work over a ground field K = R or C. Let M be a manifold andE be a super vector bundle on M over K. Let E = '(M,E) be the supernuclear Frechet space of global sections of E. Let , denote the completedprojective tensor product, so that E ,E = '(M "M,E "E). (Some detailsof the symmetric monoidal category of nuclear spaces, equipped with thecompleted projective tensor product, are presented in Appendix 2).

Let O(E ) denote the algebra of formal power series on E ,

O(E ) =/

n'0

Hom(E (n, K)Sn

where Hom denotes continuous linear maps and the subscript Sn denotescoinvariants. Note that O(E ) is an algebra: direct product of distributionsdefines a map

Hom(E (n, K)"Hom(E (m, K) $ Hom(E (n+m, K).

These maps induce an algebra structure on O(E ).We can also regard O(E ) as simply the completed symmetric algebra of

the dual space E ), that is,

O(E ) = 0Sym+(E )).

Here, E ) is the strong dual of E , and is again a nuclear space. The completedsymmetric algebra is taken in the symmetric monoidal category of nuclearspaces, as detailed in Appendix 2.

As before, letO+(E )[[!]] ) O(E )[[!]]

be the subspace of those functionals I which are at least cubic modulo !.Let Symn E denote the Sn-invariants in E (n. If P & Sym2 E and I &

O+(E )[[!]] then, for any stable graph , one can define

w (P, I) & O(E ).

The definition is exactly the same as in the finite dimensional situation. LetT ( ) be the set of tails of , H( ) the set of half-edges of , V ( ) the set ofvertices of , and E( ) the set of internal edges of . The tensor productsof interactions at the vertices of define an element of

Hom(E (H( ), R).



We can contract this tensor with the element of E (2E( ) given by the tensorproduct of the propagators; the result of this contraction is

w (P, I) & Hom(E (T ( ), R).

Thus, one can define

W (P, I) =' 1

|Aut( )|w (P, I) & O+(E )[[!]]

exactly as before.The interpretation in terms of di!erential operators works in this situa-

tion too. As in the finite dimensional situation, we can define an order twodi!erential operator

"P : O(E )$ O(E ).

On the direct factor

Hom(E (n, K)Sn = Symn E )

of O(E ), the operator "P comes from the map

Hom(E (n, K) $ Hom(E (n$2, K)

given by contracting with the tensor P & E (2.Then,

W (P, I) = ! log {exp(!"P ) exp(I/!)}as before.

4. Sharp and smooth cut-o!s

4.1. Let us return to our scalar field theory, whose action is of the form

S( ) = !12

+, (D+m2)

,+ I( ).

The propagator P is the kernel for the operator (D+m2)$1. There areseveral natural ways to write this propagator. Let us pick a basis {ei}of C"(M) consisting of orthonormal eigenvectors of D, with eigenvalues

i & R'0. Then,

P ='

i

12i + m2

ei , ei.

There are natural cut-o! propagators, where we only sum over some of theeigenvalues. For a subset U ) R'0, let

PU ='

i such that i#U

12i + m2

ei , ei.

Note that, unlike the full propagator P , the cut-o! propagator PU is asmooth function on M "M as long as U is a bounded subset of R'0.


4. SHARP AND SMOOTH CUT-OFFS 43

By analogy with the case of finite-dimensional integrals, we have theformal identity

W (P, I) (a) = ! log!

#C!(M)exp

#! 1

2!+

, (D+m2),

+1!I( + a)

$

Both sides of this equation are ill-defined. The propagator P is not a smoothfunction on C"(M"M), but has singularities along the diagonal; this meansthat W (P, I) is not well defined. And, of course, the integral on the righthand side is infinite dimensional.

In a similar way, we have the following (actual) identity, for any func-tional I & O+(C"(M))[[!]]:

(†) W-P[!#,!), I

.(a)

= ! log!

#C!(M)[!#,!)

exp#! 1

2!+

, (D+m2),

+1!I( + a)

$.

Both sides of this identity are well-defined; the propagator P[!#,!) is a smoothfunction on M "M , so that W

-P[!#,!), I

.is well-defined. The right hand

side is a finite dimensional integral.The equation (†) says that the map

O+(C"(M))[[!]] $ O+(C"(M))[[!]]

I /$W-P[!#,!), I

.

is the renormalization group flow from energy " to energy "!.

4.2. In this book we will use a cut-o! based on the heat kernel, ratherthan the cut-o! based on eigenvalues of the Laplacian described above.

For l & R>0, let K0l & C"(M "M) denote the heat kernel for D; thus,

!

y#MK0

l (x, y) (y) =1e$l D

2(x)

for all & C"(M).We can write K0

l in terms of a basis of eigenvalues for D as

K0l =

'e$l iei , ei.

LetKl = e$lm2

K0l

be the kernel for the operator e$l(D+m2). Then, the propagator P can bewritten as

P =! "

l=0Kldl.

For , L & [0,#], let

P ( , L) =! L

l=Kldl.



This is the propagator with an infrared cut-o! L and an ultraviolet cut-o!. Here and L are length scales rather than energy scales; length behaves

as the inverse to energy. Thus, is the high-energy cut-o! and L is thelow-energy cut-o!.

The propagator P ( , L) damps down the high energy modes in the prop-agator P . Indeed,

P ( , L) ='

i

e$ i ! e$L i

2i + m2

ei , ei

so that the coe#cient of ei , ei decays as $2i e$ i for i large.

Because P ( , L) is a smooth function on M " M , as long as > 0,the expression W (P ( , L), I) is well-defined for all I & O+(C"(M))[[!]].(Recall the superscript + means that I must be at least cubic modulo !.)

Definition 4.2.1. The map

O+(C"(M))[[!]] $ O+(C"(M))[[!]]I /$W (P ( , L), I)

is defined to be the renormalization group flow from length scale to lengthscale L.

From now on, we will be using this length-scale version of the renormal-ization group flow.

Figure 2 illustrates the first few terms of the renormalization group flowfrom scale to scale L.

As the e!ective interaction I[L] varies smoothly with L, there is an infin-itesimal form of the renormalization group equation, which is a di!erentialequation in I[L]. This is illustrated in figure 3.

The expression for the propagator in terms of the heat kernel has avery natural geometric/physical interpretation, which will be explained inSection 6.

5. Singularities in Feynman graphs

5.1. In this section, we will consider explicitly some of the simple Feyn-man graphs appearing in W (P ( , L), I) where I( ) = 1

3!

&M

3, and try totake the limit as $ 0. We will see that, for graphs which are not trees,the limit in general won’t exist. The 1

3! present in the interaction term sim-plifies the combinatorics of the Feynman diagram expansion. The Feynmandiagrams we will consider are all trivalent, and as explained in Section 3,each vertex is labelled by the linear map

C"(M)(3 $R

1 , 2 , 3 /$"

" 1

"

" 2

"

" 3I

=!

M1(x) 2(x) 3(x)


5. SINGULARITIES IN FEYNMAN GRAPHS 45

Figure 2. The first few expressions in the renormalizationgroup flow from scale to scale L.

Consider the graph 1, given by

We see that

w 1(P ( , L), I)(a) =!

l#[ ,L]

!

x#Ma(x)Kl(x, x)dVolM dl

where dVolM is the volume form associated to the metric on M , and a &C"(M) is the field we place at the external edge of the graph. Since

Kl(x, x) . l$ dim M/2 + higher order terms

for l small, the limit lim &0 w 1(P ( , L), I)(a) doesn’t exist.



Figure 3. The renormalization group di!erential equation.

Next consider 2, given by

Since there are no external edges there is no dependence on a & C"(M),and we find

w 2(P ( , L), I) =!

l1,l2,l3#[ ,L]

!

x,y#MKl1(x, y)Kl2(x, y)Kl3(x, y)dVolM,M .

Using the fact that

Kl(x, y) . l$dim M/2e$d(x$y)2/l + higher order terms

for l small, we can see that the limit of w 2(P ( , L), I) as $ 0 is singular.Let 3 be the graph


6. THE GEOMETRIC INTERPRETATION OF FEYNMAN GRAPHS 47

This graph is a tree; the integrals associated to graphs which are trees alwaysadmit $ 0 limits. Indeed,

w 3(P ( , L), I)(a) =!

l#[ ,L]

!

x,y#Ma(x)2Kl(x, y)a(y)2

=3

a2,

! L 1e$l Da2

2dl

4

and the second expression clearly admits an $ 0 limit.All of the calculations above are for the interaction I( ) = 1

3!

&3. For

more general interactions I, one would have to apply a di!erential operatorto both a and Kl(x, y) in the integrands.

6. The geometric interpretation of Feynman graphs

From the functional integral point of view, Feynman graphs are justgraphical tools which help in the perturbative calculation of certain func-tional integrals. In the introduction, we gave a brief account of the world-lineinterpretation of quantum field theory. In this picture, Feynman graphs de-scribe the trajectories taken by some interacting particles. The length scaleversion of the renormalization group flow becomes very natural from thispoint of view.

As before, we will work in Euclidean signature; Lorentzian signaturepresents significant additional analytical di#culties. We will occasionallycomment on the formal picture in Lorentzian signature.

6.1. Let us consider a massless scalar field theory on a compact Rie-mannian manifold M . Thus, the fields are C"(M) and the action is

S( ) = !12

!

MD .

The propagator P (x, y) is a distribution on M2, which can be expressed asa functional integral

P (x, y) =!

#C!(M)eS( ) (x) (y).

Thus, the propagator encodes the correlation between the values of the fieldsat the points x and y.We will derive an alternative expression of the propagator as a one-

dimensional functional integral.



Recall that we can write the propagator of a massless scalar field theoryon a Riemannian manifold M as an integral of the heat kernel. If x, y & Mare distinct points, then

P (x, y) =! "

0Kt(x, y)dt

where Kt & C"(M2) is the heat kernel. This expression of the propagatoris sometimes known as the Schwinger representation, and the parameter t asthe Schwinger parameter. One can also interpret the parameter t as propertime, as we will see shortly.

The heat kernel Kt(x, y) is the probability density that a particle inBrownian motion on M , which starts at x at time zero, lands at y at timet. Thus, we can rewrite the heat kernel as

Kt(x, y) =!

f :[0,t]&Mf(0)=x,f(t)=y

DWienerf

where DWienerf is the Wiener measure on the path space.We can think of the Wiener measure as the measure for a quantum field

theory of mapsf : [0, t]$ M

with action given by

E(f) =! t

0'df,df( .

Thus, we will somewhat loosely write

Kt(x, y) =!

f :[0,t]&Mf(0)=x,f(t)=y

e$E(f)

where we understand that the integral can be given rigorous meaning usingthe Wiener measure.

Combining these expressions, we find the desired expression for the prop-agator as a one-dimensional functional integral:

P (x, y) =!

#C!(M)eS( ) (x) (y) =

! "

t=0

!

f :[0,t]&Mf(0)=x,f(t)=y

e$E(f).

This expression is the core of the world-line formulation of quantum fieldtheory. This expression tells us that the correlation between the values ofthe fields at points x and y can be expressed in terms of an integral overpaths in M which start at x and end at y.

If we work in Lorentzian signature, we find the (formal) identity!

#C!(M)eS( )i (x) (y) =

! "

t=0

!

f :[0,t]&Mf(0)=x,f(t)=y

eiE(f).

This expression is di#cult to make rigorous sense of; I don’t know of arigorous treatment of the Wiener measure when the target manifold hasLorentzian signature.



6.2. We should interpret these identities as follows. We should think ofparticles moving through space-time as equipped with an “internal clock”;as the particle moves, this clock ticks at a rate independent of the time pa-rameter on space-time. The world-line of such a particle is a parameterizedpath in space-time, that is, a map f : R $ M . This path is completelyarbitrary: it can go backwards or forwards in time. Two world-lines whichdi!er by a translation on the source R should be regarded as the same. Inother words, the internal clock of a particle doesn’t have an absolute startingpoint.

If I = [0, ] is a closed interval, and if f : I $ M is a path describingpart of the world-line of a particle, then the energy of f is, as before,

E(f) =!

[0, ]'df,df( .

In quantum field theory, everything that can happen will happen, but witha probability amplitude of eiE where E is the energy. Thus, to calculatethe probability that a particle starts at the point x in space-time and endsat the point y, we must integrate over all paths f : [0, ] $ M , startingat x and ending at y. We must also integrate over the parameter , whichis interpreted as the time taken on the internal clock of the particle as itmoves from x to y. This leads to the expression (in Lorentzian signature)we discussed earlier,

P (x, y) =! "

t=0

!

f :[0,t]&Mf(0)=x,f(t)=y

eiE(f).

6.3. We would like to have a similar picture for interacting theories. Wewill consider, for simplicity, the 3 theory, where the fields are & C"(M),and the action is

S( ) =!

M!1

2D +

16

3.

We will discuss the heuristic picture first, ignoring the di#culties of renor-malization. At the end, we will explain how the renormalization group flowand the idea of e!ective interactions can be explained in the world-line pointof view.

The fundamental quantities one is interested in are the correlation func-tions, defined by the heuristic functional integral formula

E(x1, . . . , xn) =!

#C!(M)eS( )/! (x1) · · · (xn).

We would like to express these correlation functions in the world-line pointof view.

6.4. The 3 theory corresponds, in the world-line point of view, to atheory where three particles can fuse at a point in M . Thus, world-linesin the 3 theory become world-graphs; further, just as the world-lines forthe free theory are parameterized, the world-graphs arising in the 3 theory



have a metric, that is, a length along each edge. This length on the edge ofthe graph corresponds to time traversed by the particle on this edge in itsinternal clock.

Measuring the value of a field at a point x & M corresponds, in theworld-line point of view, to observing a particle at a point x. We would like tofind an expression for the correlation function E(x1, . . . , xn) in the world-linepoint of view. As always in quantum field theory, one should calculate thisexpectation value by summing over all events that could possibly happen.Such an event is described by a world-graph with end points at x1, . . . , xn.Since only three particles can interact at a given point in space-time, suchworld-graphs are trivalent. Thus, the relevant world-graphs are trivalent,have n external edges, and the end points of these external edges maps tothe points x1, . . . , xn.

Thus, we find that

E(x1, . . . , xn) =' 1

|Aut |!$ ( )

!

g#Met

!

f : &Me$E(f).

Here, the sum runs over all world-graphs , and the integral is over thosemaps f : $ M which take the endpoints of the n external edges of tothe points x1, . . . , xn.

The symbol Met( ) refers to the space of metrics on , in other words,to the space RE( )-T ( )

>0 where E( ) is the set of internal edges of , andT ( ) is the set of tails.

If is a metrized graph, and f : $ M is a map, then E(f) is the sumof the energies of f restricted to the edges of , that is,

E(f) ='

e#E( )

! l(e)

0'df,df( .

The space of maps f : $ M is given a Wiener measure, constructed fromthe usual Wiener measure on path space.

This graphical expansion for the correlation functions is only a formalexpression: if has a non-zero first Betti number, then the integral overMet( ) will diverge, as we will see shortly. However, this graphical expansionis precisely the expansion one finds when formally applying Wick’s lemmato the functional integral expression for E(x1, . . . , xn). The point is that werecover the propagator when we consider the integral over all possible mapsfrom a given edge.

In Lorentzian signature, of course, one should use eiE(f) instead ofe$E(f).

6.5. As an example, we will consider the path integral f : $ M whereis the metrized graph



Then has two vertices, labelled by interactions I0,3.The integral !

f : &Me$E(f)

is obtained by putting the heat kernel Kl on each edge of of length l, andintegrating over the position of the two vertices. Thus, we find

!

f : &Me$E(f) =

!

x,y,#MKl1(x, x)Kl2(x, y)Kl3(y, y).

However, the second integral, over the space of metrized graphs, does notmake sense. Indeed, the heat kernel Kl(x, y) has a small l asymptotic ex-pansion of the form

Kl(x, y) = l$n/2e$.x$y.2/4l'

lifi(x, y).

This implies that the integral!

l1,l2,l3

!

f : &Me$E(f) =

!

x,y,#MKl1(x, x)Kl2(x, y)Kl3(y, y)

does not converge.This second integral is the weight attached to the graph in the Feyn-

man diagram expansion of the functional integral for the 13!

3 interaction.

6.6. Next, we will explain (briefly and informally) how to construct thecorrelation functions from a general scale L e!ective interaction I[L]. Wewill not need this construction of the correlation functions elsewhere in thisbook. A full treatment of observables and correlation functions will appearin (CG10).

The correlation functions will allow us to give a world-line formulationfor the renormalization group equation on a collection {I[L]} of e!ectiveinteractions: the renormalization group equation is equivalent to the state-ment that the correlation functions computed using I[L] are independent ofL.

The correlations EnI[L](f1, . . . , fn) we will define will take, as their input,

functions f1, . . . , fn & C"(M). Thus, the correlation functions will give acollection of distributions on Mn:

EnI[L] : C"(Mn)$ R[[!]].



Figure 4. The weight C (I[ ], f1, f2, f3) attached to a graphwith 3 external vertices.

These correlation functions will be defined as a sum over graphs.Let 'n denote the set of graphs with n univalent vertices, which are

labelled as v1, . . . , vn. These vertices will be referred to as the externalvertices. The remaining vertices will be called the internal vertices. Theinternal vertices of a graph & 'n can be of any valency, and are labelledby a genus g(v) & Z'0. The internal vertices of genus 0 must be at leasttrivalent.

For a graph & 'n, and smooth functions f1, . . . , fn & C"(M), we willdefine

C (I[L])(f1, . . . , fn) & Rby contracting certain tensors attached to the edges and the vertices.

We will label each internal vertex v of genus i and valency k by

Ii,k[L] : C"(M)(H(v) $ R.

Let f1, . . . , fn & C"(M) be smooth functions on M . The external vertexvi of will be labelled by the distribution

C"(M)$ R

/$!

Mf .

Any edge e joining two internal vertices will be labelled by

P (L,#) & C"(M2).

The remaining edges, which join two external vertices or join an externaland an internal vertex, will be labelled by P (0,#), which is a distributionon M2.

As usual, we can contract all these tensors to define an element

C (I[L])(f1, . . . , fn) & R.

One may worry that because some of the edge are labelled by the distributionP (0,#), this expression is not well defined. However, because the externaledges are labelled by smooth functions fi, there are no problems. Figure 4describes C (I[ ], f1, f2, f3) for a particular graph .


7. A DEFINITION OF A QUANTUM FIELD THEORY 53

Then, the correlation function for the e!ective interaction I[L] is definedby

EI[L](f1, . . . , fn) ='

##n

1Aut( )

!n$g( )$1C (I[L])(f1, . . . , fn).

Unlike the heuristic graphical expansion we gave for the correlation functionsof the 3 theory, this expression is well-defined.

We should interpret this expansion as saying that we can compute thecorrelation functions from the e!ective interaction I[L] by allowing particlesto propagate in the usual way, and to interact by I[L]; except that in betweenany two interactions, particles must travel for a proper time of at least L.This accounts for the fact that edges which join to internal vertices arelabelled by P (L,#).

If we have a collection {I[L] | L & (0,#)} of e!ective interactions, thenthe renormalization group equation is equivalent to the statement that allthe correlation functions constructed from I[L] using the prescription givenabove are independent of L.

7. A definition of a quantum field theory

7.1. Now we have some preliminary definitions and an understandingof why the terms in the graphical expansion of a functional integral diverge.This book will describe a method for renormalizing these functional integralsto yield a finite answer.

This section will give a formal definition of a quantum field theory, basedon Wilson’s philosophy of the e!ective action; and a precise statement of themain theorem, which says roughly that there’s a bijection between theoriesand Lagrangians.

Definition 7.1.1. A local action functional I & O(C"(M)) is a func-tional which arises as an integral of some Lagrangian. More precisely, if weTaylor expand I as I =

*k Ik where

Ik( a) = kIk(a)

(so that Ik is homogeneous of degree k of the variable a & C"(M)), then Ik

must be of the form

Ik(a) =s'

j=1

!

MD1,j(a) · · ·Dk,j(a)

where Di,j are arbitrary di!erential operators on M .Let Oloc(C"(M)) ) O(C"(M)) be the subspace of local action function-

als.As before, let

O+loc(C

"(M))[[!]] ) Oloc(C"(M))[[!]]



be the subspace of those local action functionals which are at least cubicmodulo !.

Thus, local action functionals are the same as Lagrangians modulo thoseLagrangians which are a total derivative.

Definition 7.1.2. A perturbative quantum field theory, with space offields C"(M) and kinetic action !1

2

+, (D+m2)

,, is given by a set of

e!ective interactions I[L] & O+(C"(M))[[!]] for all L & (0,#], such that(1) The renormalization group equation

I[L] = W (P ( , L), I[ ])

is satisfied, for all , L & (0,#].(2) For each i, k, there is a small L asymptotic expansion

Ii,k[L] .'

r#Z%0

gr(L)$r

where gr(L) & C"((0,#)L) and $r & Oloc(C"(M)).Let T (") denote the set of perturbative quantum field theories, and let T (n)

denote the set of theories defined modulo !n+1. Thus, T (") = lim2!T (n).

Let me explain more precisely what I mean by saying there is a small Lasymptotic expansion

Ii,k[L] .'

j#Z%0

gr(L)$r.

Without loss of generality, we can require that the local action functionals$r appearing here are homogeneous of degree k in the field a.

Then, the statement that there is such an asymptotic expansion meansthat there is a non-decreasing sequence dR & Z, tending to infinity, suchthat for all R, and for all fields a & C"(M),

limL&0

L$dR

"Ii,k[L](a)!

R'

r=0

gr(L)$r(a)

%= 0.

In other words, we are asking that the asymptotic expansion exists in theweak topology on Hom(C"(M)(k, R)Sk .

The main theorem of this chapter (in the case of scalar field theories) isthe following.

Theorem A. Let T (n) denote the set of perturbative quantum field the-ories defined modulo !n+1. Then T (n+1) is, in a canonical way, a principalbundle over T (n) for the abelian group Oloc(C"(M)) of local action func-tionals on M .

Further, T (0) is canonically isomorphic to the space O+loc(C

"(M)) oflocal action functionals which are at least cubic.


8. AN ALTERNATIVE DEFINITION 55

There is a variant of this theorem, which states that there is a bijectionbetween theories and Lagrangians once we choose a renormalization scheme,which is a way to extract the singular part of certain functions of one vari-able. The concept of renormalization scheme will be discussed in Section 9;this is a choice that only has to be made once, and then it applies to alltheories on all manifolds.

Theorem B. Let us fix a renormalization scheme.Then, we find a section of each torsor T (n+1) $ T (n), and so a bijection

between the set of perturbative quantum field theories and the set of localaction functionals I & O+

loc(C"(M))[[!]]. (Recall the superscript + means

that I must be at least cubic modulo !.)

7.2. We will first prove theorem B, and deduce theorem A (which is themore canonical formulation) as a corollary.

In one direction, the bijection in theorem B is constructed as follows. IfI & O+

loc(C"(M))[[!]] is a local action functional, then we will construct a

canonical series of counterterms ICT ( ). These are local action functionals,depending on a parameter & (0,#) as well as on !. The countertermsare zero modulo !, as the tree-level Feynman graphs all converge. Thus,ICT ( ) & !Oloc(C"(M))[[!]], C"((0,#)) where , denotes the completedprojective tensor product.

These counterterms are constructed so that the limit

lim&0

W-P ( , L), I ! ICT ( )

.

exists. This limit defines the scale L e!ective interaction I[L].Conversely, if we have a perturbative QFT given by a collection of ef-

fective interactions I[l], the local action functional I is obtained as a cer-tain renormalized limit of I[l] as l $ 0. The actual limit doesn’t exist;to construct the renormalized limit we again need to subtract o! certaincounterterms.

A detailed proof of the theorem, and in particular of the construction ofthe local counterterms, is given in Section 10.

8. An alternative definition

In the previous section I presented a definition of quantum field theorybased on the heat-kernel cut-o!. In this section, I will describe an alter-native, but equivalent, definition, which allows a much more general classof cut-o!s. This alternative definition is a little more complicated, but isconceptually more satisfying. One advantage of this alternative definition isthat it does not rely on the heat kernel.

As before, we will consider a scalar field theory where the quadratic termof the action is 1

2

&(D +m2) .

Definition 8.0.1. A parametrix for the operator D+m2 is a distribu-tion P on M "M , which is symmetric, smooth away from the diagonal, and



is such that((D+m2), 1)P ! M & C"(M "M)

is smooth; where M refers to the delta distribution along the diagonal inM "M .

For any L > 0, the propagator P (0, L) is a parametrix. In the alternativedefinition of a quantum field theory presented in this section, we can useany parametrix as the propagator.

Note that if P, P ! are two parametrices, the di!erence P ! P ! betweenthem is a smooth function. We will give the set of parametrices a partialorder, by saying that

P * P !

if Supp(P ) ) Supp(P !). For any two parametrices P, P !, we can find someP !! with P !! < P and P !! < P !.

8.1. Before we introduce the alternative definition of quantum fieldtheory, we need to introduce a technical notation. Given any functionalJ & O(C"(M)), we get a continuous linear map

C"(M) $ O(C"(M))

/$ dJ

d.

Definition 8.1.1. A function J has smooth first derivative if this mapextends to a continuous linear map

D(M)$ O(C"(M)),

where D(M) is the space of distributions on M .

Lemma 8.1.2. Let $ & C"(M)(2 and suppose that J & O+(C"(M))[[!]]has smooth first derivative. Then so does W ($, J) & O+(C"(M))[[!]].

Proof. Recall that

W ($, J) = ! log1e!"P eJ/!

2.

Thus, it su#ces to verify two things. Firstly, that the subspace O(C"(M))consisting of functionals with smooth first derivative is a subalgebra; this isclear. Secondly, we need to check that "" preserves this subalgebra. This isalso clear, because "" commutes with d

d for any & C"(M)). !

8.2. The alternative definition of quantum field theory is as follows.

Definition 8.2.1. A quantum field theory is a collection of functionals

I[P ] & O+(C"(M))[[!]],

one for each parametrix P , such that the following properties hold.


9. EXTRACTING THE SINGULAR PART 57

(1) If P, P ! are parametrices, then

W-P ! P !, I[P !]

.= I[P ].

This expression makes sense, because P ! P ! is a smooth functionon M "M .

(2) The functionals I[P ] satisfy the following locality axiom. For any(i, k), the support of

Supp Ii,k[P ] )Mk

can be made as close as we like to the diagonal by making theparametrix P small. More precisely, we require that, for any openneighbourhood U of the small diagonal M ) Mk, we can find someP such that

Supp Ii,k[P !] ) U

for all P ! * P .(3) Finally, the functionals I[P ] all have smooth first derivative.

Theorem 8.2.2. This definition of a quantum field theory is equivalentto the previous one, presented in definition 7.1.2.

More precisely, if I[L] is a set of e!ective interactions satisfying theheat-kernel definition of a QFT, then if we set

I[P ] = W (P ! P (0, L), I[L])

for any parametrix P , the functionals I[P ] satisfy the parametrix definitionof quantum field theory presented in this section. Further, every collectionI[P ] of functionals satisfying the parametrix definition arises uniquely inthis way.

The proof of this theorem will be presented in Section 12, after the proofof Theorem A.

9. Extracting the singular part of the weights of Feynman graphs

9.1. In order to construct the local counterterms needed for theorem A,we need a method for extracting the singular part of the finite-dimensionalintegrals w (P ( , L), I) attached to Feynman graphs. This section will de-scribe such a method, which relies on an understanding of the behaviour ofof the functions w (P ( , L), I) as $ 0. We will see that w (P ( , L), I) hasa small asymptotic expansion

w (P ( , L), I)(a) .'

gi( )$i(L, a),

where the $i(L, a) are well-behaved functions of the field a and of L. Fur-ther, the $i(L, a) have a small L asymptotic expansion in terms of localaction functionals.

The functionals gi( ) appearing in this expansion are of a very specialform: they are periods of algebraic varieties. For the purposes of this book,the fact that these functions are periods is not essential. Thus, the reader



may skip the definition of periods without any loss. However, given theinterest in the relationship between periods and quantum field theory in themathematics literature (see (KZ01), for example) I felt that this point isworth mentioning.

Before we state the theorem precisely, we need to explain what makes afunction of a period.

9.2. According to Kontsevich and Zagier (KZ01), most or all constantsappearing in mathematics should be periods.

Definition 9.2.1. A number & C is a period if there exists an alge-braic variety X of dimension d, a normal crossings divisor D ) X, and aform & &d(X) vanishing on D, all defined over Q; and a homology class

& Hd(X(C), D(C)),Qsuch that

=!

.

We are interested in periods which depend on a variable & (0, 1). Suchfamilies of periods arise from families of algebraic varieties over the a#neline.

Suppose we have the following data.(1) an algebraic variety X over Q;(2) a normal crossings divisor D ) X;(3) a Zariski open subset U ) A1

Q, defined over Q, such that U(R)containts (0, 1).

(4) a smooth map X $ U , of relative dimension d, also defined overQ, whose restriction to D is flat.

(5) a relative d-form & &d(X/U), defined over Q, and vanishingalong D.

(6) a homology class & Hd((X1/2(C), D1/2(C)), Q), where X1/2 andD1/2 are the fibres of X and D over 1/2 & U(R). We assumethat is invariant under the complex conjugation map on the pair(X1/2(C), D1/2(C)).

Let us assume that the maps

X(C)$ U(C)D(C)$ U(C)

are locally trivial fibrations. For t & (0, 1) ) U(R), we will let Xt(C) andDt(C) denote the fibre over s(t) & U(R).

We can transfer the homology class & H+(X1/2(C), D1/2(C)) to anyfibre (Xt(C), Dt(C)) for t & (0, 1). This allows us to define a function f on(0, 1) by

f(t) =!

t

t.



The function f is real analytic. Further, f is real valued, because the cycle

t & Hd(X1/2(C), D1/2(C))

is invariant under complex conjugation.

Definition 9.2.2. Let PQ((0, 1)) ) C"((0, 1)) be the subalgebra offunctions of this form. Elements of this subalgebra will be called rationalperiods.

Note that PQ((0, 1)) is, indeed, a subalgebra; the sum of functions cor-responds to the disjoint union of the pairs (X,D) of algebraic varieties, andthe product of functions corresponds to fibre product of the algebraic va-rieties (X,D) over U . Further, PQ((0, 1)) is of countable dimension overQ.

Note that, if f is a rational period, then, for every rational numbert & Q 3 (0, 1), f(t) is a period in the sense of Kontsevich and Zagier.

Definition 9.2.3. Let

P((0, 1)) = PQ((0, 1)), R ) C"((0, 1))

be the real vector space spanned by the space of rational periods. Elementsof P((0, 1)) will be called periods.

9.3. Now we are ready to state the theorem on the small asymptoticexpansions of the functions w (P ( , L), I)(a).

We will regard the functional w (P ( , L), I)(a) as a function of the threevariables , L and a & C"(M). It is an element of the space of functionals

O(C"(M), C"((0, 1) ), C"((0,#)L))).

The subscripts and L indicate the coordinates on the intervales (0, 1) and(0,#). If we fix but allow L and a to vary, we get a functional

w (P ( , L), I) & O(C"(M), C"((0,#)L)).

This is a topological vector space; we are interested in the behaviour ofw (P ( , L), I) as $ 0.

The following theorem describes the small behaviour of w (P ( , L), I).

Theorem 9.3.1. Let I & Oloc(C"(M))[[!]] be a local functional, and letbe a connected stable graph.

(1) There exists a small asymptotic expansion

w (P ( , L), I) ."'

i=0

gi( )(i

where thegi &P((0, 1) )

are periods, and (i & O(C"(M), C"((0,#)L)).



The precise meaning of “asymptotic expansion” is as follows:there is a non-decreasing sequence dR & Z, indexed by R & Z>0,such that dR $# as R $#, and such that for all R,

lim&0

$dR

"w (P ( , L), I)!

R'

i=0

gi( )(i

%= 0.

where the limit is taken in the topological vector space

O(C"(M), C"((0,#)L)).

(2) The gi( ) appearing in this asymptotic expansion have a finite orderpole at zero: for each i there is a k such that lim &0

kgi( ) = 0.(3) Each (i appearing in the asymptotic expansion above has a small

L asymptotic expansion of the form

(i ."'

j=0

fi,j(L)$i,j

where the $i,j are local action functionals, that is, elements ofOl(C"(M)); and each fi,k(L) is a smooth function of L & (0,#).

This theorem is proved in Appendix 1. All of the hard work required toconstruct counterterms is encoded in this theorem. The theorem is provedby using the small t asymptotic expansion for the heat kernel to approximateeach w (P ( , L), I) for small .

9.4. These results allow us to extract the singular part of the finite-dimensional integral w (P ( , L), I). Of course, the notion of singular partis not canonical, but depends on a choice.

Definition 9.4.1. Let P((0, 1))'0 )P((0, 1)) be the subspace of thosefunctions f of which are periods and which admit a limit as $ 0.

A renormalization scheme is a complementary subspace

P((0, 1))<0 )P((0, 1))

to P((0, 1))'0.

Thus, once we have chosen a renormalization scheme we have a directsum decomposition

P((0, 1)) = P((0, 1))'0 4P((0, 1))<0.

A renormalization scheme is the data one needs to define the singularpart of a function in P((0, 1)).

Definition 9.4.2. If f & P((0, 1)), define the singular Sing(f) of f tobe the projection of f onto P((0, 1))<0.



9.5. We can now use this definition to extract the singular part of thefunctions w (P ( , L), I). As before, let us think of w (P ( , L), I) as a distri-bution on Mk. Then, Theorem 9.3.1 shows that w (P ( , L), I) has a small

asymptotic expansion of the form

w (P ( , L), I) ."'

i=0

gi( )$i

where the gi( ) are periods and the $i & Hom(C"(Mk), C"((0,#)L)).Theorem 9.3.1 also implies that there exists an N & Z'0 such that, for alln > N , gn( ) admits an $ 0 limit.

Denote the N th partial sum of the asymptotic expansion by

(N ( ) =N'

i=0

gi( )$i.

Then, we can define the singular part of w (P ( , L), I) simply by

Sing w (P ( , L), I) = Sing (N ( ) =N'

i=0

(Sing gi( ))$i.

This singular part is independent of N , because if N is increased the function(N ( ) is modified only by the addition of functions of which are periodsand which tend to zero as $ 0.

Theorem 9.3.1 implies that Sing w (P ( , L), I) has the following prop-erties.

Theorem 9.5.1. Let I & Oloc(C"(M))[[!]] be a local functional, and letbe a connected stable graph.

(1) Sing w (P ( , L), I) is a finite sum of the form

Sing w (P ( , L), I) ='

fi( )$i

where$i & Oloc(C"(M), C"((0,#)L)),

andfi &P((0, 1))<0

are purely singular periods.(2) The limit

lim&0

(w (P ( , L), I)! Sing w (P ( , L), I))

exists in the topological vector space Oloc(C"(M), C"((0,#)L)).(3) Each $i appearing in the finite sum above has a small L asymptotic

expansion

$i ."'

j=0

fi,j(L)(i,j



where (i,j & Oloc(C"(M)) is local, and fi,j(L) is a smooth functionof L & (0,#).

10. Constructing local counterterms

10.1. The heart of the proof of theorem A is the construction of localcounterterms for a local interaction I & Oloc(C"(M)). This construction issimple and inductive, without the complicated graph combinatorics of theBPHZ algorithm.

The theorem on the existence of local counterterms is the following.

Theorem 10.1.1. There exists a unique series of local counterterms

ICTi,k ( ) & Oloc(C"(M)),alg P((0,#))<0,

for all i > 0, k 5 0, with ICTi,k homogeneous of degree k as a function of

a & C"(M), such that, for all L & (0,#], the limit

lim&0

W

5

6P ( , L), I !'

i,k

!iICTi,k ( )

7

8

exists.

Here the symbol ,alg denotes the algebraic tensor product, so only finitesums are allowed.

10.2. We will construct our counterterms using induction on the genusand number of external edges of the Feynman graphs. Later, we will see avery short (though unilluminating) construction of the counterterms, whichdoes not use Feynman graphs. For reasons of exposition, we will introducethe Feynman graph picture first.

Let 'i,k denote the set of all stable graphs of genus i with k externaledges. Let

Wi,k (P, I) ='

##i,k

w (P ( , L), I).

Thus,W (P, I) =

'!iWi,k (P, I) .

If the graph is of genus zero, and so is a tree, then lim &0 w (P ( , L), I)converges. Thus, the first counterterms we need to construct are those fromgraphs with one loop and one external edge. Let us define

ICT1,1 ( , L) = Sing W1,1 (P ( , L), I) .

Section 9 explains the meaning of the singular part Sing of W1,1 (P ( , L), I).We need to check that this has the desired properties. It is immediate

from the definition that

W1,1-P ( , L), I ! !ICT

1,1 ( , L).

= W1,1 (P ( , L), I)! ICT1,1 ( , L)

and so the limit lim &0 W1,1-P ( , L), I ! !ICT

1,1 ( , L).

exists.


10. CONSTRUCTING LOCAL COUNTERTERMS 63

Figure 5. Explanation of why ICT1,1 ( , L) is independent of

L.

Next, we need to check that

Lemma. ICT1,1 ( , L) is local.

First we will show that

Lemma. ICT1,1 ( , L) is independent of L.

Figure 5 illustrates ddLW1,1 (P ( , L), I). This expression is non-singular,

as it is obtained by contracting the distribution I0,3 on C"(M3) with thesmooth function KL & C"(M2). Therefore ICT

1,1 ( , L), which we defined tobe the singular part of W1,1 (P ( , L), I), is independent of L.

Since ICT1,1 ( , L) is independent of L, to verify that it is local we only

need to examine the behaviour of W1,1 (P ( , L), I) at small L. Theorem9.5.1 implies that Sing W1,1 (P ( , L), I) has a small L asymptotic expansionin terms of local action functionals. Therefore, since we know ICT

1,1 ( , L) isindependent of L, it follows that it is local.

Now that we know ICT1,1 ( , L) is independent of L, we will normally drop

L from the notation.

10.3. The next step is to construct ICT1,2 ( , L). However, it is just as

simple to construct directly the general counterterm ICTi,k ( , L). Let us lex-

icographically order the set Z'0 " Z'0, so that (i, k) < (j, l) if i < j or ifi = j and k < l. Let us write

W<(i,k) (P, I) ='

(j,l)<(i,k)

!jWj,l (P, I) .



Figure 6. This diagram, which is just the infinitesimal ver-sion of the renormalization group equation, explains whyICTi,k ( , L) is independent of L. In this diagram, W+,+ is short-

hand for

W+,+

5

6P ( , L), I !'

(a,b)<(i,k)

ICTa,b ( )

7

8 .

We can write this expression in terms of stable graphs, as follows. Let '<(i,k)

denote the set of stable graphs with genus smaller than i, or with genus equalto i and fewer than k external edges. Then,

W<(i,k) (P, I) ='

##<(i,k)

!g( )

|Aut |w (P, I).

Let us assume, by induction, we have constructed counterterms ICTj,l ( )

for all (j, l) < (i, k), with the following properties.(1) For all L, the limit

lim&0

W<(i,k)

5

6P ( , L), I !'

(j,l)<(i,k)

!jICTj,l ( )

7

8

exists.(2) The counterterms ICT

j,l ( ) are local for (j, k) < (i, k).


10. CONSTRUCTING LOCAL COUNTERTERMS 65

Then, we define the counterterm

ICTi,k ( , L) = Sing Wi,k

5

6P ( , L), I !'

(j,l)<(i,k)

!jICTj,l ( )

7

8 .

As before, it is immediate that

Wi,k

5

6P ( , L), I !'

(j,l)<(i,k)

!jICTj,l ( )! !iICT

i,k ( , L)

7

8

= Wi,k

5

6P ( , L), I !'

(j,l)<(i,k)

!jICTj,l ( )

7

8! ICTi,k ( , L)

is non-singular.What we need to show is that(1) ICT

i,k ( , L) is independent of L.(2) ICT

i,k ( , L) is local.As before, the second statement follows from the first one. To show

independence of L it su#ces to show that

(†) ddL

Wi,k

5

6P ( , L), I !'

(j,l)<(i,k)

!jICTj,l ( )

7

8

is non-singular, that is, the limit of this expression as $ 0 exists. A proofof this is illustrated in figure 6. In this diagram, an expression for (†) isgiven as a sum over graphs whose vertices are labeled by

Wj,l

5

6P ( , L), I !'

(r,s)%(j,l)

!rICTr,s ( )

7

8

for various (j, l) < (i, k). We know by induction that these are non-singular.On the unique edge of these graphs we put the heat kernel KL, which issmooth. Thus, the expression resulting from each graph is non-singular.

10.4. In fact, the use of Feynman graphs is not at all necessary in thisproof; I first came up with the argument by thinking of W (P, I) as

W (P, I) = ! log (exp(!"P ) exp(I/!)) .

From this expression, one can see that

W-P (L,L!),W (P ( , L), I)

.= W

-P ( , L!), I

.

Wi,k (P ( , L), I) = Wi,k

-P ( , L), I<(i,k)

.+ Ii,k.

The first identity is obvious, and the second identity can be seen (for in-stance) using the expression of W (P, I) in terms of Feynman graphs.



These two identities are all that is really needed for the argument. In-deed, suppose we have constructed counterterms ICT

r,s ( ) for all (r, s) < (i, k),such that, for all L, the limit

lim&0

W<(i.k)

5

6P ( , L), I !'

(r,s)<(i,k)

!rICTr,s ( )

7

8

exists. Let us suppose, by induction, that these counterterms are local andindependent of L.

Then, we define the next counterterm by

ICTi,k (L, ) = Sing Wi,k

5

6P ( , L), I !'

(r,s)<(i,k)

!rICTr,s ( )

7

8 .

The identity

Wi,k

5

6P ( , L), I !'

(r,s)<(i,k)

!rICTr,s ( )! !iICT

i,k (L, )

7

8 =

Wi,k

5

6P ( , L), I !'

(r,s)<(i,k)

!rICTr,s ( )

7

8! ICTi,k (L, )

shows that the limit

lim&0

W%(i.k)

5

6P ( , L), I !'

(r,s)<(i,k)

!rICTr,s ( )! !iICT

i,k (L, )

7

8

exists.


11. PROOF OF THE MAIN THEOREM. 67

To show locality of the counterterm ICTi,k (L, ), it su#ces, as before, to

show that it is independent of L. If L! > L, we have

ICTi,k (L!, ) = Sing Wi,k

5

6P ( , L!), I !'

(r,s)<(i,k)

!rICTr,s ( )

7

8

=Sing Wi,k

5

6P (L,L!),W

5

6P ( , L), I !'

(r,s)<(i,k)

!rICTr,s ( )

7

8

7

8

=Sing Wi,k

5

6P (L,L!),W<(i,k)

5

6P ( , L), I !'

(r,s)<(i,k)

!rICTr,s ( )

7

8

+ !iWi,k

5

6P ( , L), I !'

(r,s)<(i,k)

!rICTr,s ( )

7

8

7

8

=Sing Wi,k

5

6P (L, L!),W<(i,k)

5

6P ( , L), I !'

(r,s)<(i,k)

!rICTr,s ( )

7

8

7

8

+ Sing Wi,k

5

6P ( , L), I !'

(r,s)<(i,k)

!rICTr,s ( )

7

8

Since

Wi,k

5

6P (L,L!),W<(i,k)

5

6P ( , L), I !'

(r,s)<(i,k)

!rICTr,s ( )

7

8

7

8

is non-singular, the last equation reduces to

ICTi,k (L!, ) = Sing Wi,k

5

6P ( , L), I !'

(r,s)<(i,k)

!rICTr,s ( )

7

8

= ICTi,k (L, )

as desired.Thus, ICT

i,k (L, ) is independent of L, and can be written as ICTi,k ( ); and

we can continue the induction.

11. Proof of the main theorem.

Let I & O+loc(C

"(M))[[!]] be a local action functional. What we haveshown so far allows us to construct the corresponding theory. Let

WR (P (0, L), I) = lim&0

W-P ( , L), I ! ICT ( )

.



be the renormalized version of the renormalization group flow from scale0 to scale L applied to I. The theory associated to I has scale L e!ectiveinteraction WR (P (0, L), I). It is easy to see that this satisfies all the axiomsof a perturbative quantum field theory, as given in Definition 7. The localityaxiom of the definition of perturbative quantum field theory follows fromTheorem 9.5.1.

We need to show the converse: that to each theory there is a correspond-ing local action functional. This is a simple induction. Let {I[L]} denotea collection of e!ective interactions defining a theory. Let us assume, byinduction on the lexicographic ordering as before, that we have constructedlocal action functionals Ir,s for (r, s) < (i, k) such that

WRa,b

5

6P (0, L),'

(r,s)<(i,k)

!rIr,s

7

8 = Ia,b[L]

for all (a, b) < (i, k).Then, the infinitesimal renormalization group equation implies that

WRi,k

5

6P (0, L),'

(r,s)<(i,k)

!rIr,s

7

8! Ii,k[L]

is independent of L. The locality axiom for the theory {I[L]} – which saysthat each Ii,k[L] has a small L asymptotic expansion in terms of local actionfunctionals – implies that this quantity is local. Thus, let

Ii,k = Ii,k[L]!WRi,k

5

6P (0, L),'

(r,s)<(i,k)

!rIr,s

7

8 .

Then,

WRa,b

5

6P (0, L),'

(r,s)%(i,k)

!rIr,s

7

8 = Ia,b[L]

for all (a, b) * (i, k), and we can continue the induction.

11.1. Recall that we defined T (n) to be the space of theories definedmodulo !n+1, and T (") to be the space of theories. Thus, we have shownthat the choice of renormalization scheme sets up a bijection between T (")

and functionals I & O+loc(C

"(M))[[!]], and between T (n) and functionalsI & O+

loc(C"(M))[!]/!n+1.

The more fundamental statement of the theorem on the bijection be-tween theories and local action functionals is that the map T (n+1) $ T (n)

makes T (n+1) into a principal bundle for the group Oloc(C"(M)), in acanonical way (independent of any arbitrary choices, such as that of a renor-malization scheme). In this subsection we will use the bijection constructedabove to prove this statement.


12. PROOF OF THE PARAMETRIX FORMULATION 69

The bijection between theories and Lagrangians shows that the mapT (n+1) $ T (n) is surjective; this is the only place the bijection is used.

To show that T (n+1) $ T (n) is a principal bundle, suppose that {I[L]},{J [L]} are two theories which are defined modulo !n+2 and which agreemodulo !n+1.

Let I0[L] & T (0) be the classical theory corresponding to both I[L] andJ [L]. Let us consider the tangent space to T (0) at I[L], which includesinfinitesimal deformations of classical theories which do not have to be atleast cubic. More precisely, let TI0[L]T

(0) be the set of H[L] & O(C"(M))such that

I0[L] + H[L]satisfies the classical renormalization group equation modulo 2,

I0,i[L] + Hi[L] = W0,i

1P ( , L),

'I0,j [ ] + Hj [ ]

2modulo 2,

and which satisfy the usual locality axiom, that H[L] has a small L asymp-totic expansion in terms of local action functionals.

The bijection between classical theories and local action functionals iscanonical, independent of the choice of renormalization scheme. This is trueeven if we include non-cubic terms in our e!ective interaction, as long asthese non-cubic terms are accompanied by nilpotent parameters.

Thus, we have a canonical isomorphism of vector spaces

TI0[L]T(0) %= Oloc(C"(M)).

The following lemma now shows that T (n+1) $ T (n) is a torsor forOloc(C"(M)).

Lemma 11.1.1. Let I, J & T (n+1) be theories which agree in T (n). Then,the functional

I0[L] + !$(n+1)(I[L]! J [L]) & O(C"(M))

satisfies the classical renormalization group equation modulo 2, and so de-fines an element of

TI0[L](T (0)) %= Oloc(C"(M)).

Note that !$(n+1)(I[L] ! J [L]) is well-defined as I[L] and J [L] agreemodulo !n+1.

Proof. This is a simple calculation. !

12. Proof of the parametrix formulation of the main theorem

In this page we will prove the equivalence of the definition of theorybased on arbitrary parametrices, explained in Section 8 with the definitionbased on the heat kernel. Since this result is not used elsewhere in the book,I will not give all the details.



Thus, suppose we have a theory in the heat kernel sense, given by a fam-ily I[L] of e!ective interactions satisfying the renormalization group equationand the locality axiom. If P is a parametrix, let us define a functional I[P ]by

I[P ] = W (P ! P (0, L), I[L]) & O(C"(M))[[!]].Since P (0, L) and P are both parametrices for the operator D+m2, thedi!erence between them is smooth. Thus, W (P ! P (0, L), I[L]) is well-defined.

Lemma 12.0.2. The collection of e!ective interactions {I[P ]}, definedfor each parametrix P , defines a theory using the parametrix definition oftheory.

Proof. To prove this, we need to verify the following.(1) If P ! is another parametrix, then

I[P !] = W-P ! ! P, I[P ]

.

(this is the version of the renormalization group equation for thedefinition of theory based on parametrices.

(2) By choosing a parametrix P with support close to the diagonal, wecan make the distribution

Ii,k[P ] & D(Mk)Sk

on Mk supported as close as we like to the small diagonal.(3) The functional I[P ] has smooth first derivative. Recall, as ex-

plained in Section 8, that this means the following. There is acontinuous linear map

C"(M)$ O(C"(M))[[!]]

/$ dI[P ]d

.

Saying that I[$ has smooth first derivative means that this mapextends to a continuous linear map

D(M)$ O(C"(M))[[!]]

where D(M) is the space of distributions on M .In order to verify these properties, it is convenient to choose a renormaliza-tion scheme, so that we can write

I[L] = lim&0

W-P ( , L), I ! Ict( )

..

Now let us choose a cut-o! function ( & C"(M " M) which is 1 in aneighbourhood of the diagonal, and 0 outside a small neighbourhood of thediagonal. Then, (P (0, L) is a parametrix, which agrees with P (0, L) nearthe diagonal. Thus, we have

I[(P (0, L)] = lim&0

W-(P ( , L), I ! Ict( )

..


13. VECTOR-BUNDLE VALUED FIELD THEORIES 71

Since I and Ict( ) are local, if we choose the function ( to be supportedin a very small neighbourhood of the diagonal then we can ensure thatIi,k[(P (0, L)] is supported within an arbitrarily small neighbourhood of thesmall diagonal in Mk.

From this and from the identity

I[P ] = W (P !(P (0, L), I[(P (0, L)])

we can check that, by choosing the parametrix P to have support very closeto the diagonal, we can ensure that Ii,k[P ] has support arbitrarily close tothe small diagonal in Mk. The point is that the combinatorial formulae forW ($, J) (where $ & C"(M2) and J & O(C"(M2))+[[!]]) allows one tocontrol the support of Wi,k ($, J) in terms of the support of J and that of$.

This shows that I[P ] satisfies the first two properties we want to ver-ify. It remains to check that I[P ] has smooth first derivative. This followsfrom the fact that all local functionals have smooth first derivative, andthat the property of having smooth first derivative is preserved under therenormalization group flow.

!Now, we need to prove the converse. This follows from the following

lemma.

Lemma 12.0.3. Let {I(r,s)[P ]} and {I !(r,s)[P ]} be two parametrix theories,defined for all (r, s) * (I,K). Suppose that I(r,s)[P ] = I !(r,s)[P ] if (r, s) <

(I,K). Then,J(I,K) = I(I,K)[P ]! I !(I,K)[P ]

is a local functional, that is, an element of Oloc(C"(M)).

Proof. Note that the renormalization group equation implies that thefunctional J(I,K) is independent of P . The locality axiom implies that J(I,K)

is supported on the small diagonal of MK . Further, J(I,K) has smooth firstderivative. Any distribution J & D(MK)SK which is supported on the smalldiagonal and which has smooth first derivative is a local functional. !

13. Vector-bundle valued field theories

We would like to have a bijection between theories and Lagrangians fora more general class of field theories. The most general set-up we will needis when the fields are sections of some vector bundle on a manifold; andthe interactions depend smoothly on some additional supermanifold. In thissection we will explain how to do this on a compact manifold.

Definition 13.0.4. A nilpotent graded manifold is the following data:(1) A smooth manifold with corners X,(2) A sheaf A of commutative superalgebras over the sheaf of algebras

C"X ,



satisfying the following properties:(1) A is locally free of finite rank as a C"

X -module. In other words, Ais the sheaf of sections of some super vector bundle on X.

(2) A is equipped with an ideal I such that A/I = C"X , and Ik = 0 for

some k > 0. The ideal I, its powers I l, and the quotient sheavesA/I l, are all required to be locally free sheaves of C"

X -modules.

The algebra '(X,A) of C" global sections of A will be denoted by A .Everything in this section will come in families, parameterized by a

nilpotent graded manifold (X,A).

13.1. We are interested in vector-valued theories on a compact manifoldM . As in the case of scalar field theories, we will fix the data of the freetheory, which gives us our propagator; and then consider possible interactingtheories which deform this.

The following definition aims to be broad enough to capture all of the freefield theories used in this book, and in future applications. Unfortunately,it is not particularly transparent.

Definition 13.1.1. A free theory on a manifold M consists of the fol-lowing data.

(1) A super vector bundle E over the field R or C on M , equipped with adirect sum decomposition E = E14E2 into the spaces of propagatingand non-propagating fields, respectively. We will denote the spaceof smooth global sections of E or Ei by E , Ei respectively.

We will let

E !1 = '(M,E)

1 ,Dens(M)).

There is an inclusion

E !1 ) E )

1 .

(2) An even, A -linear, order two di!erential operator

DE1 : E1 ,A $ E1 ,A

(where the tensor product is the completed projective tensor prod-uct).

DE1 must be a generalized Laplacian, which means that the sym-bol

(DE1) & '(T +M, Hom(E,E)),A

must be the identity on E times a smooth family of Riemannianmetrics

g & C"(T +M), C"(X).

(Recall that C"(X) ) A is a subalgebra, as A is the global sectionsof a bundle of algebras on X).



(3) A di!erential operator

D! : E !1 $ E1.

This operator is required to be symmetric: the formal adjoint

(D!)+ : E !1 $ E1

is required to be equal to D!.(4) Let

D+E1

: E !1 $ E !

1

be the formal adjoint of DE1. We require that

D! D+E1

= DE1 D!.

We will abuse notation and refer to the entirely of the data of a freetheory on M as E .

The most basic example of this definition is the free scalar field the-ory, as considered in chapter 2. There, the space E1 of fields is C"(M).The space E2 of non-interacting fields is 0. The operator DE1 is the usualpositive-definite Laplacian operator C"(M) $ C"(M).The operator D! isthe identity, where we have used the Riemannian volume element to trivi-alize the bundle of densities on M , and so to identify E1 with E !

1.More interesting examples will be presented in chapter 5, when we con-

sider the Batalin-Vilkovisky formalism. The use of graded vector bundleswill be essential in the BV formalism.

13.2. The space E2 of non-propagating fields is introduced into thisdefinition with an eye to future applications: none of the examples treatedin this book will have non-propagating fields. Thus, the reader will losenothing by ignoring the space of non-propagating fields.

For those who are interested, however, let me briefly explain the reasonfor considering non-propagating fields. Let us consider a free scalar fieldtheory on a Riemannian manifold M with metric g0. Let us consider per-turbing the metric to g0 +h. The action of the scalar field theory is given byS( ) =

&6g0+h , as usual. Notice that the action is not just local as a

function of the field , but also local as a function of the perturbation h ofthe metric. However, h is not treated as a quantum field, only as a classicalfield: we do not consider integration over the space of metrics.

In this situation, the tensor h is said to be a background field, or (in theterminology adopted here) a non-propagating field.

In this example, the space of fields is

E = C"(M)4 '(M, Sym2 TM)

where the space E1 of propagating fields is C"(M), and the space E2 ofnon-propagating fields is '(M, Sym2 TM).



The action of the theory is S( ) =&M 6g0+h . This action can, as

usual, be split into quadratic and interacting parts:

S( ) =!

M6g0 +

!

M(6g0+h !6g0) .

The quadratic part of the action is the only part relevant to the definitionof a free field theory as presented above. As we see, the quadratic part onlydepends on the propagating field , and not on h. However, the interactionterm depends both on and h. It is a general feature of the interactionterms that they must have some dependence on ; they canot be functionsjust of the non-propagating field h. We will build this into our definition ofinteractions in the presence of non-propagating fields shortly.

If we take the free theory associated to this example – given by discardingthe interacting terms in the action S – we fit it into the general definition asfollows. The operator DE1 is the Laplacian 6g0 : C"(M) $ C"(M). Theoperator D! is the identity map C"(M)$ C"(M).

13.3. Now that we have the general definition of a free field theory, wecan start to define the concept of e!ective interaction in this context. First,we have to define the heat kernel.

If the manifold M is compact, there is a unique heat kernel

Kt & E !1 , E1 , C"(R>0),A

for the operator DE1 .Composing with the operator D! gives an element

D!Kt & E1 , E1 , C"(R>0),A .

We will view this as an element of E , E , C"(R>0).The adjointness properties of the di!erential operators D!, DE1 imply

that D!Kt is symmetric.The propagator for the theory is

P ( , L) =! L

D!Kt & E (2 ,A

which is again symmetric.Note that unless we impose additional positivity conditions on the oper-

ator DE1 , the heat kernel Kt may not exist at t = #; thus, the propagatorP ( ,#) may not exist. In almost all examples, however, the operator DE1

is positive, and so the heat kernel K" does exist.If we specialize the case of the free scalar field theory on a Riemannian

manifold (M, g0), then, as we have seen, E1 = C"(M), DE1 = 6g0 is thenon-negative Laplacian for the metric g0. In this example the operator D!

is the identity. Thus, the propagator prescribed by this general definitioncoincides with the propagator presented in our earlier analysis of the freefield theory.



13.4. As before, we can define the algebra

O(E , A ) =/

Hom(E (n, A )Sn

of all functionals on E with values in A . Here Hom denotes the space ofcontinuous linear maps. The properties of the symmetric monoidal categoryof nuclear spaces, as detailed in Appendix 2, show that

O(E , A ) =

"/

n

Symn E )

%,A .

There is a subspace

Ol(E , A ) ) O(E , A )

of A -valued local action functionals, defined as follows.

Definition 13.4.1. A functional $ & O(E , A ) is a local action func-tional if, when we expand $ as a sum $ =

*$n of its homogeneous com-

ponents, each$n : E (n $ A

can be written in the form

$n(e1, . . . , en) =k'

j=1

!

M(D1,je1) · · · (Dn,jen)d

whered & Densities(M)

is some volume element on M , and each

Di,j : E ,A $ C"(M),A

is an A -linear di!erential operator.

Note that Ol(E , A ) is not a closed subspace. However, as we will see inAppendix 2, Oloc(E , A ) has a natural topology making it into a completenuclear space, and a module over A in the symmetric monoidal category ofnuclear spaces.

Our interactions will be elements of

Ol(E , A )[[!]].

We would like to allow our interactions to have quadratic and linear termsmodulo !. However, we require that these quadratic terms are accompaniedby elements of the nilpotent ideal

I = '(X, I) ) A

(recall that A /I = C"(X)). If we don’t impose this condition, we willencounter infinite sums.

Thus, let us denote by

O+(E , A )[[!]] ) O(E , A )[[!]]



the subset of those functionals which are at least cubic modulo the idealgenerated by I and !.

Then, the renormalization group operator

W (P ( , L), I)

= ! log-exp(!"P ( ,L)) exp(I/!)

.: O+(E , A )[[!]]$ O+(E , A )[[!]]

is well-defined.Because we now allow quadratic and linear interaction terms modulo

!, the Feynman graph expansion of this expression involves univalent andbivalent genus 0 vertices. However, each such vertex is accompanied by anelement of the ideal I of A . Since this ideal is nilpotent, there is a uniformbound on the number of such vertices that can occur, so there are no infinitesums.

Definition 13.4.2. A theory is given by a collection of even elements

I[L] & O+(E , C"((0,#)L),A )[[!]],

such that(1) The renormalization group equation

I[L!] = W-P (L,L!), I[L]

.

holds.(2) Each I(i,k)[L] has a small L asymptotic expansion

I(i,k)[L](e) .'

(r(e)fr(L)

where (r & Ol(E ) are local action functionals.Let T (")(E ) denote the space of such theories, and let T (n)(E ) denote

the space of theories defined modulo !n+1, so that T (")(E ) = lim2!T (n)(E ).

Let me explain more precisely what I mean by saying there is a small Lasymptotic expansion

Ii,k[L] .'

j#Z%0

gr(L)$r.

Without loss of generality, we can require that the local action functionals$r appearing here are homogeneous of degree k in the field e & E .

Recall that A is the global sections of some bundle of algebras A on amanifold with corners X. Let Ax denote the fibre of A at x & X. For everyelement & A , let x & Ax denote the value of at x.

The statement that there is such an asymptotic expansion means thatthere is a non-decreasing sequence dR & Z, tending to infinity, such that forall R, for all fields e & E , for all x & X,

limL&0

L$dRx

"Ii,k[L](e)!

R'

r=0

gr(L)$r(e)

%= 0



in the finite dimensional vector space Ax.Then, as before, the theorem is:

Theorem 13.4.3. The space T (n+1)(E ) has the structure of a principalOloc(E , A ) bundle over T (n)(E ), in a canonical way. Further, T (0)(E )is canonically isomorphic to the space O+

loc(E , A ) of A -valued local actionfunctionals on E which are at least cubic modulo the ideal I ) A .

Further, the choice of renormalization scheme gives rise to a sectionT (n)(E ) $ T (n+1)(E ) of each torsor, and so a bijection between T (")(E )and the space

O+loc(E , A )[[!]]

of local action functionals with values in A , which are at least cubic modulo! and modulo the ideal I ) A .

Proof. The proof is essentially the same as before. The extra di#cul-ties are of two kinds: working with an auxiliary parameter space X intro-duces extra analytical di#culties, and working with quadratic terms in ourinteraction forces us to use Artinian induction with respect to the powers ofthe ideal I ) A .

For simplicity, I will only give the proof when the e!ective interactionsI[L] are all at least cubic modulo !. The argument in the general case is thesame, except that we also must perform Artinian induction with respect tothe powers of the ideal I ) A .

As before, we will prove the renormalization scheme dependent ver-sion of the theorem, saying that there is a bijection between T (")(E ) andO+

loc(E , A )[[!]]. The renormalization scheme independent formulation is aneasy corollary.

Let us start by showing how to construct a theory associated to a localinteraction

I ='

!iI(i,k) & Oloc(E ,A )[[!]].

We will assume that I(0,k) = 0 if k < 3.The argument is essentially the same as the argument we gave earlier.

We will perform induction on the set Z'0 " Z'0 with the lexicographicalorder: (i, k) < (r, s) if i < r or if i = r and k < s.

Suppose, by induction, we have constructed counterterms

ICT(i,k)( ) & Oloc(E , C"((0,#) ),A )

for all (i, k) < (I,K). The ICT(i,k) are supposed, by induction, to have the

following properties:(1) Each ICT

(i,k)( ) is homogeneous of degree k as a function of the fielde & E .

(2) Each ICT(i,k)( ) is required to be a finite sum

ICT(i,k)( ) =

'gr( )$r



where gr( ) & C"((0,#) ) and $r & Oloc(E , A ). Each gr( ) isrequired to have a finite order pole at 0; that is, lim &0

kgr( ) = 0for some k > 0.

(3) Recall that A is the space of global sections of a vector bundle Aon X. For any element & A, let x & Ax denote its value atx & X.

We require that, for all L & (0,#) and all x & X, the limit

lim&0

W(r,s)

5

6P ( , L), I !'

(i,k)%(r,s)

!iICT(i,k)( )

7

8

x

exists in the topological vector space Hom(E (r, R) , Ax. Here,Hom(E (r, R) is given the strong topology (i.e. the topology of uni-form convergence on bounded subsets).

Now we need to construct the next counterterm ICT(I,K)( ). We would like

to define

ICT(I,K)( ) = Sing W(I,K)

5

6P ( , L), I !'

(r,s)<(I,K)

!rI(r,s)CT ( )

7

8 .

In order to be able to define the singular part like this, we need to knowthat

W(I,K)

5

6P ( , L), I !'

(r,s)<(I,K)

!rI(r,s)CT ( )

7

8

has a nice small asymptotic expansion. The required asymptotic expansionis provided by the following theorem, proved in Appendix 1.

Theorem 13.4.4. For all graphs , and all I & Oloc(E , A )[[!]], thereexist local action functionals $r & Oloc(E , A ,C"((0,#)L) and functions gr

in the space P ((0,#) ) ) C"((0,#) ) of periods, such that for all l & Z'0,there is a small asymptotic expansion

"l

"lLw (P ( , L), I) . "l

"lL

'

r'0

gr( )$r.

This means that for each l and each x & X, there exists a non-decreasingsequence dR & Z, tending to #, such that

lim&0

$dR"l

"lL

5

6w (P ( , L), I)!'

r'0

gr( )$r

7

8

x

= 0

in the topological vector space Hom(E (T ( ), R),Ax. Here, Hom(E (T ( ), R)is given the strong topology (i.e. the topology of uniform convergence onbounded subsets).

Also, each gr has a finite order pole at = 0. This means that the limitlim &0

kgr( ) is 0 for some k > 0.



Further, each

$r(L, e) & Hom(E (T ( ), C"((0,#)L))

has a small L asymptotic expansion

$r .'

hs(L)(r,s

where each (r,s & Oloc(E , A ). The definition of small L asymptotic expan-sion is in the same sense as before: there exists a non-decreasing sequencedS & Z'0, tending to #, such that, for all x & X, and all S & Z'0,

limL&0

L$dS

"$r(e)!

S'

s=0

hs(L)(r,s(e)

%

x

= 0

in the topological vector space Hom(E (T ( ), R),Ax.

It follows from this theorem that it makes sense to define the next coun-terterm ICT

(I,K) simply by

ICT(I,K)( , L) = Sing

5

6W(I,K)

5

6P ( , L), I !'

(r,s)<(I,K)

!rICT(r,s)( )

7

8

7

8 .

We would like to show the following properties of ICT(I,K)( , L).

(1) ICT(I,K)( , L) is independent of L.

(2) ICT(I,K)( ) is local, that is, it is an element of

Oloc(E ,A ),alg C"((0,#) ).

(3) For all L & (0,#) and all x & X, the limit

lim&0

W(I,K)

5

6P ( , L), I !'

(i,k)%(I,K)

!iICT(i,k)( )

7

8

x

exists in the topological vector space Hom(E (K , R),Ax.If we can prove these three properties, we can continue the induction.

The third property is immediate: it follows from the small asymptoticexpansion of Theorem 13.4.4.

For the first property, observe that

"

"LSing W(I,K)

5

6P ( , L), I !'

(r,s)<(I,K)

!rICT(r,s)( )

7

8

= Sing"

"LW(I,K)

5

6P ( , L), I !'

(r,s)<(I,K)

!rICT(r,s)( )

7

8 .

This follows from the fact that the small asymptotic expansion proved inTheorem 13.4.4 commutes with taking L derivatives.



Thus, to show that ICT(I,K) is independent of L, it su#ces to show that,

for all L, all x & X, and all e & E ,

"

"LW(I,K)

5

6P ( , L), I !'

(r,s)<(I,K)

!rICT(r,s)( )

7

8

x

(e) & Ax

has an $ 0 limit. This is immediate by induction, using the renormaliza-tion group equation.

The small L asymptotic expansion in Theorem 13.4.4 now implies thatICT(I,K)( ) is local.

Thus, ICT(I,K)( ) satisfies all the required properties, and we can continue

our induction, to construct all the counterterms.So far, we have shown how to construct the e!ective interactions

I[L] = lim&0

W

5

6P ( , L), I !'

(r,s)

!rICT(r,s)( )

7

8 .

These e!ective interactions satisfy the renormalization group equation. Itis immediate from Theorem 13.4.4 that the I[L] satisfy the locality axiom,so that they define a theory.

Now, we need to prove the converse. This is again an inductive argument.Suppose we have a theory, given by e!ective interactions

I[L] & O+(E , A )[[!]].

Suppose that we have a local action functional

J ='

(i,k)<(I,K)

!iJ(i,k) & O+loc(E , A )[[!]]

with associated e!ective interactions J [L].Suppose, by induction, that

J(i,k)[L] = I(i,k)[L]

for all (i, k) < (I,K).We need to find some J !

(I,K) such that, if we set J ! = J + !IJ !(I,K), then

J !(I,K)[L] = I(I,K)[L].

We simply letJ !

(I,K) = I(I,K)[L]! J(I,K)[L].

The renormalization group equation implies that J !(I,K) is independent of L.

It is automatic thatJ !

(I,K)[L] = I(I,K)[L].

Finally, the fact that both J(I,K)[L] and J(I,K)[L] satisfy the small L asymp-totics axiom of a theory implies that J !

(I,K) is local.!


14. FIELD THEORIES ON NON-COMPACT MANIFOLDS 81

14. Field theories on non-compact manifolds

A second generalization is to non-compact manifolds. On non-compactmanifolds, we don’t just have ultraviolet divergences (arising from smallscales) but infrared divergences, which arise when we try to integrate overthe non-compact manifold.

However, by imposing a suitable infrared cut-o!, we will find a notionof theory on a non-compact manifold; and a bijection between theories andLagrangians.

The infrared cut-o! we impose is rather brutal: we multiply the propaga-tor by a cut-o! function so that it becomes supported on a small neighbour-hood of the diagonal in M2. However, the notion of theory is independentof the cut-o! chosen.

We will also show that there are restriction maps, allowing one to restricta theory on a non-compact manifold M to any open subset U . This allowsone to define a sheaf of theories on any manifold. If we are on a compactmanifold, global sections of this sheaf are theories in the sense we definedbefore.

The sheaf-theoretic statement of our main theorem asserts that thissheaf is isomorphic to the sheaf of local action functionals. As always, thisisomorphism depends on the choice of a renormalization scheme. The renor-malization scheme independent statement of the theorem is that the sheafof theories defined modulo !n+1 is a torsor over the sheaf of theories definedmodulo !n, for the sheaf of local action functionals on M .

14.1. Now let us start defining the notion of theory on a possibly non-compact manifold M .

As in Section 13, we will fix a nilpotent graded manifold (X,A), and afamily of free field theories on M parameterized by (X,A). The free fieldtheory is given by a super vector bundle E on M , whose space of globalsections will be denoted by E ; together with various auxiliary data detailedin Definition 13.1.1.

We will use the following notation. We will let E denote the space of allsmooth sections of E, Ec the space of compactly supported sections, E thespace of distributional sections and E c the space of compactly supporteddistributional sections. The bundle E ,DensM will be denoted E!. We willuse the notation E !, E !

c , E! and E

!c to denote spaces of smooth, compactly

supported, distributional and compactly supported distributional sectionsof the bundle E!. sections of We will let E denote the space of distribu-tional sections of E , and E c denote the compactly supported distributionalsections. With this notation, E ) = E

!c, E v

c ee = E!, and so on.

14.2.Definition 14.2.1. Let M,X be topological spaces. A subset C ) Mn"

X is called proper if each of the projection maps i : Mn "X $ M "X isproper when restricted to C.



We say that a section of a vector bundle on Mn"X has proper supportif its support is a proper subset of Mn "X.

Recall that we can identify the space O(Ec, A ) of A -valued functionson Ec (modulo constants) with the completed symmetric algebra

O(Ec, A ) =/

n>0

Symn(E !),A ,

where we have identified E! – the space of distributional sections of the

bundle E! on M – with (Ec)). We will let

Op(Ec, A ) ) O(Ec, A )

be the subset consisting of those functionals $ each of whose Taylor com-ponents

$n & Symn E!

have proper support. We are only interested in functions on Ec moduloconstants.

Note that Op(E , A ) is not an algebra; the direct product of two properlysupported distributions does not necessarily have proper support.

Note also that every A -valued local action functional I & Ol(E , A ) isan element of Op(E , A ).

14.3. Recall that the super vector bundle E on M has additional struc-ture, as described in Definition 13.1.1. This data includes a decompositionE = E1 4 E2 and a generalized Laplacian DE1 : E1 $ E1.

We are interested in the heat kernel for the Laplacian)E1 . On a compactmanifold M , this is unique, and is an element of

Kt & E !1 , E1 , C"(R>0),A .

On a non-compact manifold, there are many heat kernels, corresponding tovarious boundary conditions. In addition, such heat kernels may grow onthe boundary of the non-compact manifold in ways which are di#cult tocontrol.

To remedy this, we will introduce the concept of fake heat kernel. A fakeheat kernel is something which solves the heat equation but only up to theaddition of a smooth kernel.

Definition 14.3.1. A fake heat kernel is a smooth section

Kt & E !1 , E1 , C"(R>0),A

with the following properties.(1) Kt extends, at t = 0 to a distribution. Thus, Kt extends to an

element ofE

!1 , E 1 , C"(R'0),A

Further, K0 is the kernel for the identity map E1 $ E1.



(2) The support

SuppKt ) M "M " R>0 "X

is proper. Recall that this means that both projection maps fromSuppKt to M " R>0 "X are proper.

(3) The heat kernel Kt satisfies the heat equation up to exponentiallysmall terms in t. More precisely, d

dtKt+DE1 Kt extends to a smoothsection

ddt

Kt + DE1 Kt & E !1 , E !

1 , C"(R'0),A

which vanishes at t = 0, with all derivatives in t and on M , fasterthan any power of t.

(4) The heat kernel Kt admits a small t asymptotic expansion whichcan be written, in normal coordinates x, y near the diagonal of M ,in the form

Kt . t$dim M/2e$.x$y.2/t'

i'0

ti$i(x, y).

Let me explain more carefully about what I mean by a small t asymptoticexpansion. We will let

KNt = (x, y)t$ dim M/2e$.x$y.2/t

N'

i=0

ti$i(x, y)

be the N th partial sum of this asymptotic expansion (where we have intro-duced a cut-o! (x, y) so that KN

t is zero outside of a small neighbourhoodof the diagonal).

Then, we require that for all compact subsets C )M "M "X,999"k

t

-Kt !KN

t

.999C,l,m

= O(tN$dim M/2$l/2$k).

where +!+l,m refers to the norm on '(M,E) , Dens(M)) , E , A wherewe di!erentiate l times on M "M , m times on X, and take the supremumover the compact subset C.

These estimates are the same as the ones satisfied by the actual heatkernel on a compact manifold M , as detailed in (BGV92).

The fake heat kernel Kt satisfies the heat equation up to a functionwhich vanishes faster than any power of t. This implies that the asymptoticexpansion

t$ dim M/2e$.x$y.2/t'

i'0

ti$i(x, y).

of Kt must be a formal solution to the heat equation (in the sense describedin (BGV92), Section 2.5. This characterizes the functions $i(x, y) (definedin a neighbourhood of the diagonal) uniquely.



Lemma 14.3.2. Let Kt, :Kt be two fake heat kernels. Then Kt ! :Kt

extends across t = 0 to a smooth kernel, that is,

Kt ! :Kt & E ! , E ,A , C"(R'0).

Further, Kt ! :Kt vanishes to all orders at t = 0.

Proof. This is clear from the existence and uniqueness of the small tasymptotic expansion. !

Lemma 14.3.3. A fake heat kernel always exists.

Proof. The techniques of (BGV92) allow one to construct a fake heatkernel by approximating it with the partial sums of the asymptotic expan-sion. !

14.4. Let us suppose that M is an open subset of a compact manifoldN , and that the free field theory E on M is restricted from one, say F , onN .

Then, the restriction of the heat kernel for F to M is an element:Kt & E !

1 , E1 , C"(R>0),A ,

which satisfies all the axioms of a fake heat kernel except that of requiringproper support.

If ( & C"(M "M) is a smooth function, with proper support, whichtakes value 1 on a neighbourhood of the diagonal, then

( :Kt & E !1 , E1 , C"(R>0),A

is a fake heat kernel.

14.5. The bundle E1 of propagating fields is a direct summand of thebundle E of all fields. Thus, the fake heat kernel Kt can be viewed as anelement of the space

E ! , E , C"(R>0),A .

The operator D! : E !1 $ E1 extends to an operator E ! $ E . For 0 < <

L <#, let us define the fake propagator by

P ( , L) =! L -

(D! , 1)Kt.dt & Sym2 E ,A ,

where : E (2 $ Sym2 E is the symmetrization map.We would like to define a theory, for the fake heat kernel Kt, to be a

collection of e!ective interactions

I[L] & O+p (Ec, A )[[!]]

satisfying the renormalization group equation defined using the fake prop-agator P ( , L). Recall that the subscript p in the expression Op(Ec, A )indicates that we are looking at functionals which are distributions with



proper support. A function in O(Ec, A ) is in Op(Ec, A ) if its Taylor com-ponents, which are A -valued distributions on Mn, have support which is aproper subset of Mn "X.

In order to do this, we need to know that the renormalization groupflow is well defined. Thus, we need to check that if we construct theweights attached to graphs using the propagators P ( , L) and interactionsI & Op(Ec, A )+[[!]].

Lemma 14.5.1. Let be a connected graph with at least one tail. LetP & E ,E ,A have proper support. Suppose for each vertex v of we havea continuous linear map

Iv : E (H(v)c $ A

which has proper support.Then,

w (P, {Iv}) : E (T ( )c $ A

is well defined, and is a continuous linear map with proper support.

Proof. Let f & E (T ( )c . The expression w (P, {Iv})(f) is defined by

contracting the tensor

f ,e#E( ) Pe & E (H( )

given by putting a propagator P on each edge of and f at the tails of ,with the distribution

,v#V ( )Iv : E (H( )c $ A .

Neither quantity has compact support. However, the restrictions we placedon the supports of f , P and each Iv means that the intersection of thesupport of ,Iv with that of f ,Pe is a compact subset of MH( ). Thus, wecan contract ,Iv with f , Pe to get an element of A .

The resulting linear map

E (T ( ) $ A

f /$ w (P, {Iv})(f)

is easily seen to have proper support. !Note that this lemma is false if the graph has no tails. This is the

reason why we only consider functionals on E modulo constants, or equiva-lently, without a constant term.

Corollary 14.5.2. The renormalization group operator

O+p (Ec, A )[[!]] $ O+

p (Ec,A )[[!]]I $W (P ( , L), I)

is well-defined.

As always, O+p (Ec,A )[[!]] refers to the subspace of O+

p (Ec, A )[[!]] con-sisting of elements which are at least cubic modulo ! and the ideal '(X,m) )A .



Proof. The renormalization group operator is defined by

W (P ( , L), I) =' 1

+Aut) )+!g( )w (P ( , L), I).

The sum is over connected stable graphs; and, as we are working with func-tionals on E modulo constants, we only consider graphs with at least onetail. Lemma 14.5.1 shows that each w (P ( , L), I) is well-defined. !

14.6. Now we can define the notion of theory on the manifold M , usingthe fake propagator P ( , L).

Definition 14.6.1. A theory is a collection {I[L] | L & R>0} of ele-ments of O+

p (Ec, A )[[!]] which satisfy(1) The renormalization group equation,

I[L] = W (P ( , L), I[ ])

(2) The asymptotic locality axiom: there a small L asymptotic expan-sion

I[L] .'

fi(L)(i

in terms of local action functionals (i & O+loc(Ec, A )[[!]]. We will

assume that the functions fi(L) appearing in this expansion haveat most a finite order pole at L = 0; that is, we can find some nsuch that limL&0 Lnfi(L) = 0.

Let T (n)(E ,A ) denote the set of theories defined modulo !n+1, and letT (")(E , A ) denote the set of theories defined to all orders in !.

14.7. One can ask how the definition of theory depends on the choiceof fake heat kernel. It turns out that there is no dependence. Let Kt, :Kt

be two heat fake heat kernels, with associated propagators P ( , L), :P ( , L).We have seen that Kt ! :Kt vanishes to all orders at t = 0. It followsthat P (0, L) ! :P (0, L) is smooth, that is, an element of E , E ,A . Also,P (0, L)! :P (0, L) has proper support. Further, as L$ 0, P (0, L) and all ofits derivatives vanish faster than any power of L.

Thus, the renormalization group operator

W1P (0, L)! :P (0, L),!

2: O+

p (Ec,A )[[!]]$ O+p (Ec, A )[[!]]

is well-defined.

Lemma 14.7.1. Let {:I[L]} be a collection of e!ective interactions defin-ing a theory for the propagator :P ( , L). Then,

I[L] = W1P (0, L)! :P (0, L), :I[L]

2

defines a theory for the propagator P ( , L).Further, the small L asymptotic expansion of I[L] is the same as that of

:I[L].



Proof. The renormalization group equation

W (P ( , L), I[ ]) = I[L]

is a corollary of the general identity,

W-P,W

-P !, I

..= W

-P + P !, I

.

for any P, P ! & E , E ,A of proper support, and any I & O+p (E ,A )[[!]].

The statement about the small L asymptotics of I[L] – and hence the localityaxiom which says that I[L] defines a theory – follows from the locality axiomfor :I[L] and the fact that P (0, L) and all its derivatives tend to zero fasterthan any power of L, as L $ 0. !

14.8. Now let U ) M be any open subset. The free field theory on M –defined by the vector bundle E, together with certain di!erential operatorson it – restricts to define a free field theory on U ) M . One can ask if atheory on M will restrict to one on U as well.

The following proposition, which will be proved later, shows that onecan do this.

Proposition 14.8.1. Let U )M be an open subset.Given any theory

{I[L] & O+p (E , A )[[!]]}

on M (defined using any fake heat kernel on M), there is a unique theory

{IU [L] & O+p ('(U,E |U ),A )[[!]]}

again defined using any fake heat kernel on U , with the property that thesmall L asymptotic expansion of IU [L] is the restriction to U of the small Lasymptotic expansion of I[L].

The existence of this restriction map shows that there is a presheaf onM which assigns to an open subset U ) M the set T (n)('(U,EU )), A ) oftheories on U . We will denote this presheaf by T (n)(E , A ).

14.9. Now we are ready to state the main theorem.

Theorem 14.9.1. (1) The presheaves T (n)(E , A ) (of theories de-fined modulo !n+1) and T (")(E , A ) (of theories defined to all orderin !) on M are sheaves.

(2) There is a canonical isomorphism between the sheaf T (0)(E , A )and the sheaf of local action functionals I & Oloc(Ec, A ) which areat least cubic modulo the ideal '(X,m) ) A .

(3) For n > 0, T (n)(E , A ) is, in a canonical way, a torsor overT (n$1)(E , A ) for the sheaf of abelian groups Oloc(Ec, A ), in acanonical way.



(4) Choosing a renormalization scheme leads to a map

T (n)(E , A ) $ T (n+1)(E ,A )

of sheaves for each n, which is a section. Thus, the choice of arenormalization scheme leads to an isomorphism of sheaves

T (")(E ,A ) %= O+loc(Ec, A )[[!]]

on M .

14.10. The proof of the theorem, and of Proposition 14.8.1, is along thesame lines as before, using counterterms.

Proposition 14.10.1. Let us fix a renormalization scheme, and a fakeheat kernel Kt on M .

LetI =

'!iIi,k & O+

loc(Ec, A )[[!]].

Then:(1) there is a unique series of counterterms

ICT ( ) ='

!iICTi,k ( )

where

ICTi,k ( ) & Oloc(E , A ),alg P((0,#))<0

is purely singular as a function of , with the property that the limit

lim&0

W-P ( , L), I ! ICT ( )

.

exists, for all L.(2) This limit defines a collection of elements I[L] & O+

p (E , A )[[!]],satisfying the renormalization group equation and locality axiom,and so defines a theory.

(3) The counterterms ICTi,k ( ) do not depend on the choice of a fake heat

kernel.(4) If U ) M is an open subset, then the counterterms for I restricted

to U are the restrictions to U of the counterterms ICTi,k ( ) for I.

Thus, counterterms define a map of sheaves

O+loc(Ec, A )[[!]]$ O+

loc(Ec, A ,alg P((0,#) )<0)[[!]].

Proof. The proof of the first two statements is identical to the proofof the corresponding statement on a compact manifold, and so is mostlyomitted. One point is worth mentioning briefly, though: the countertermsare defined to be the singular parts of the small asymptotic expansion ofthe weight w (P ( , L), I) attached to a graph . One can ask whether suchasymptotic expansions exist when we use a fake heat kernel rather than anactual heat kernel.



The asymptotic expansion, as constructed in Appendix 1, only relies onthe small t expansion of the heat kernel Kt, and thus exists whenever thepropagator is constructed from a kernel with such a small t expansion.

The third clause is proved using the uniqueness of the counterterms, asfollows. Suppose that Kt, :Kt are two fake heat kernels, with associated fakepropagators P ( , L), :P ( , L). Let ICT ( ), :ICT ( ) denote the countertermsassociated to the two di!erent fake heat kernels. We need to show that theyare the same.

Note that

lim&0

W1:P (0, L)! P (0, L),W

-P ( , L), I ! ICT ( )

.2

exists. We can write the expression inside the limit as

W1P (0, )! :P (0, ),W

1:P ( , L), I ! ICT ( )

22.

Note that P (0, ) ! :P (0, ) tends to zero, with all derivatives, faster thanany power of . Also, W

1:P ( , L), I ! ICT ( )

2has a small asymptotic

expansion in terms of functions of of polynomial growth at the origin.From these two facts it follows that

lim&0

W1:P ( , L), I ! ICT ( )

2

exists.Uniqueness of the counterterms implies that ICT ( ) = :ICT ( ).The fourth clause can be proved easily using independence of the coun-

terterms of the fake heat kernel. !14.11. Now we can prove Proposition 14.8.1 and Theorem 14.9.1. The

proof is easy once we know that counterterms exist.Let us start by regarding the set of theories T (")(E , A ) as just a set,

and not as arising from a sheaf on M . (After all, we have not yet provedProposition 14.8.1, so we do not know that we have a presheaf of theories).

Lemma 14.11.1. Let us choose a renormalization scheme, and a fakeheat kernel. Then the map of sets

O+loc(Ec, A )[[!]]$ T (")(E , A )

I /$ {I[L] = lim&0

W-P ( , L), I ! ICT ( )

.}

is a bijection.

Proof. This is proved by the usual inductive argument. !Now we can prove Proposition 14.8.1, which we restate here for conve-

nience.

Proposition 14.11.2. Let U ) M be an open subset.Given any theory

{I[L] & O+p (E , A )[[!]]}



on M , for any fake heat kernel, U , there is a unique theory

{IU [L] & O+p ('(U,E |U ),A )[[!]]}

with the property that the small L asymptotics of IU [L] is the restriction toU of the small L asymptotics of I[L].

Proof. Uniqueness is obvious, as any two theories on U with the samesmall L asymptotic expansions must coincide.

For existence, we will use the bijection between theories and Lagrangianswhich arises from the choice of a renormalization scheme. We can as-sume that the theory {I[L]} on M arises from a local action functionalI & O+

loc(Ec, A )[[!]]. Then, we define IU [L] to be the theory on U associatedto the restriction of I to U .

It is straightforward to check that, with this definition, I[L] and IU [L]have the same small L asymptotics. !

It follows from the proof of this lemma that the map

O+loc(Ec, A )[[!]]$ T (")(E , A )

of sets actually arises from a map of presheaves on M . Since O+loc(Ec,A )[[!]]

is actually a sheaf on M , it follows that T (")(E , A ) is also a sheaf, andsimilarly for T (n)(E , A ).

The remaining statements of Theorem 14.9.1 are proved in the same wayas the corresponding statements on a compact manifold.


CHAPTER 3

Field theories on Rn

This chapter shows how the main theorem of Chapter 2 works on Rn.However, we will only deal with translation-invariant theories on Rn.

Working on Rn presents di#culties not present when dealing with theo-ries on compact manifolds. In this situation, the finite dimensional integralsone attaches to Feynman graphs are now over products of copies of Rn,and as such may not converge. Divergences of this form are called infrareddivergences.

We have seen in Section 14 how to deal with field theory on non-compactmanifolds in general. This involved choosing a “fake heat kernel”, which isa kernel only satisfying the heat equation in an approximate way, and whichhas certain conditions on its support. In this section we will verify that,on Rn, there is no need to choose a fake heat kernel. As long as we onlyconsider e!ective interactions I[L] when L < #, we can use the ordinaryheat kernel on Rn. It will take a certain amount of work, however, to provethis statement, and to prove the bijection between theories and Lagrangiansin this context.

The sheaf-theoretic nature of quantum field theories, as stated in 14.9.1,implies that any theory on Rn which is invariant under Rn " SO(n) definesa theory on any manifold with a flat metric.

0.12. We would like to say that a theory on Rn is given by a collectionof e!ective interactions I[ ] satisfying the renormalization group equation

I[L] = W (P ( , L), I[ ]) .

The possible presence of infrared divergences makes it di#cult to definethis renormalization group flow. The renormalization group operator I /$W (P ( , L), I) is only defined for functionals I which satisfy certain growthconditions: roughly, the component distributions I(i,k) of I must be tem-pered distributions on Rnk which are of rapid decay away from the smalldiagonal Rn ) Rnk.

In Section 1 we give a definition of the space of distributions on Rnk ofrapid decay away from the small diagonal. The condition is, roughly, thatthe distribution D decays at # as fast as e$b.x.2

for some b > 0.

0.13. In Section 2, we prove the main theorem for scalar field theorieson Rn. We define a theory to be a collection of e!ective interactions I[L],satisfying the growth conditions defined in section 1, the renormalization

91


92 3. FIELD THEORIES ON Rn

group equation, and a locality axiom. The main theorem, as before, is asfollows.

Theorem 0.13.1. Let T (m) be the space of theories defined modulo !n+1.Then T (m+1) $ T (m) is a torsor for the abelian group of translation in-variant local action functionals on Rn.

Moreover, T (0) is canonically isomorphic to the space of translationinvariant local action functionals on Rn which are at least cubic modulo !.

The choice of a renormalization scheme yields a section of each torsorT (m+1) $ T (m), and so an isomorphism between the space T (") of the-ories and the space O+

loc(S (Rn))[[!]] of series in ! whose coe"cients aretranslation invariant local action functionals on Rn, and which are at leastcubic modulo !.

The proof is along the same lines as the proof of the corresponding state-ment on compact manifolds. We first prove the renormalization scheme in-dependent version of the statement, that there is a bijection between theoriesand local action functionals. Given a local action functional I, we constructthe e!ective interactions I[L] by introducing local counterterms ICT ( ) anddefining

I[L] = lim&0

W-P ( , L), I ! ICT ( )

..

Some work is required to show that the functionals I[L] satisfy the requisitegrowth conditions; this is the main point where the proof of the theorem onRn di!ers from that on a compact manifold.

The more canonical renormalization scheme independent version of thestatement is a straightforward corollary.

0.14. Section 3 proves the main result for a more general class of theorieson Rn, where the fields are sections of some vector bundle E , and whereeverything may depend on an auxiliary ring A . As with scalar field theories,we only treat the case where everything is translation invariant. The proofof the bijection between theories and local action functionals in this moregeneral context is essentially the same as the proof for scalar field theories.

1. Some functional analysis

Before we discuss theories on Rn, we need to develop a little functionalanalysis. We need to construct a space of distributions on Rnk which are ofrapid decay away from the small diagonal Rn ) Rnk. The e!ective interac-tions which appear in the definition of a theory will lie in these spaces ofdistributions of rapid decay.

This section should probably be skipped on first reading, and referredback to as necessary.


1. SOME FUNCTIONAL ANALYSIS 93

1.1. Let us fix throughout this section an auxiliary manifold with cor-ners X, and a (possibly graded) vector bundle A on X. We will assume thatA has the structure of super-commutative algebra in the category of vectorbundles on X. Let A = '(X,A) denote the space of global sections of A.

The first thing to do is introduce some of the basic spaces of functionson Rn. Let S (Rn) be the space of Schwartz functions on Rn. This is anuclear Frechet space. Let S (Rn) , A denote the completed projectivetensor product of S (Rn) and A . We can think of S (Rn),A as the spaceof Schwartz functions on Rn with values in A .

Let D(Rn,A ) denote the space of continuous linear maps S (Rn)$ A .Thus, D(Rn, A ) is the space of A -valued tempered distributions on Rn.

There is an A -bilinear direct product map

D(Rn, A )"D(Rk, A )$ D(Rn+k, A )((,$) /$ (" $.

The direct product ( " $ is uniquely determined by the property that forall Schwartz functions f & S (Rn), g & S (Rk),

((" $)(f " g) = ((f)$(g).

Here the product on the right hand side is taken in the algebra A , andf " g & S (Rn+k) is the usual exterior product of functions.

1.2. We are interested in spaces of distributions of rapid decay on Rn.By “rapid decay” we will mean, roughly, that they decay as fast as e$b.x.2

for some b > 0.Such distributions will be continuous linear maps on spaces of functions

whose growth is bounded by all eb.x.2. We will first introduce these function

spaces.

Definition 1.2.1. Let V,W be finite dimensional vector spaces over R.For all a & Z'0, b & R>0 and l & Z'0, let us define a norm +!+a,b,l onS (V 4W ) by the formula

+f+a,b,l ='

|I|%l

Sup(v,w)#V /W

;;;(1 + +v+2)ae$b.w.2

"If;;; .

Let us extend this to a map C"(V 4W )$ [0,#] by the same formula.Let

T (V,W ) ) C"(V 4W )be the subspace of those functions such that, for all a, b and l,

+f+a,b,l < #.

Let us give T (V,W ) the topology induced by the norms +!+a,b,l. This isthe coarsest linear topology containing, as open neighbourhoods of zero, thesets {f | +f+a,b,l < 1}. In this topology, a sequence fi $ 0 if and only if+fi+a,b,l $ 0 for all a, b, l.



If the space V = 0, we will use the notation

T (W ) = T (0, W ).

Definition 1.2.2. A continuous linear map

$ : S (V 4W )$ A

is of rapid decay along W if it extends to a continuous linear map

$ : T (V,W )$ A .

Let K ) X be a compact subset, and let D : A $ A be a di!erentialoperator. Let +a+K,D denote the norm on A given by taken the supremumover K of Da. The topology on A is defined by these norms, as K and Dvary.

Lemma 1.2.3. (1) S (V 4W ) ) T (V,W ) is dense.(2) A continuous linear map $ : S (V 4W ) $ A extends to a con-

tinuous linear map T (V,W ) $ A if and only if, for all compactsubsets K ) X and all di!erential operators D : A $ A , thereexist some a, b, l and C such that

+$(f)+K,D * C +f+a,b,l .

(If this extension exists, it is of course unique.)(3) Let $ : T (V,W ) $ A and ( : T (V !,W !) $ A be continuous

linear maps. Then the direct product

$"( : S (V 4 V ! 4W 4W !)$ A

extends to a continuous linear map

T (V 4 V !,W 4W !)$ A .

Proof. The proof is straightforward and omitted. !

1.3. Now we are ready to introduce our main objects, which are gooddistributions. These are certain distributions on Rnk which are of rapiddecay away from the small diagonal.

We will view Rnk as a configuration space of k points on Rn; in this way,it inherits an Rn action. When we refer to translation invariant objects onRnk we will always mean objects invariant under this Rn action.

Definition 1.3.1. A tempered distribution $ : S (Rnk) $ A is good ifit has the following two properties.

(1) $ is translation invariant.(2) If we write Rnk as an orthogonal direct sum Rn 4 Rn(k$1), where

Rn ) Rnk is the small diagonal, then $ has rapid decay alongRn(k$1). That is, $ extends to a continuous linear map

$ : T (Rn, Rn(k$1))$ A .



1.4. The main result of this section is that we can use Feynman graphsto contract good distributions, and that the result is again a good distribu-tion.

Let be a connected graph. Let H( ), T ( ), V ( ) and E( ) refer tothe sets of half-edges, tails, vertices and internal edges of . For everyv & V ( ), let H(v) ) H( ) refer to the set of half-edges adjoining v. Forevery e & E( ), let H(e) ) H( ) be the pair of half-edges forming e.

Suppose that we have the following data:(1) For every v & V ( ), we have a good distribution

Iv & Dg(RnH(v), A ).

In particular, Iv is an A -valued tempered distribution on RnH(v).(2) Suppose that for each edge e & E( ), we have a function

Pe & C"(RnH(e)).

Let h1, h2 denote the two half edges of e, and let xhi : RnH(e) $Rn be the corresponding linear maps. Let us assume that Pe isinvariant under the Rn action on RnH(e); this amounts to sayingthat Pe is independent of xh1 +xh2 . Let us further assume that forany multi-index I, there exists b such that

|"IPe| * e$b+xh1$xh2+

2

.

Then we can attempt to form the distribution w ({Iv}, {Pe}) on RnT ( )

by contracting the distributions Iv with the functions Pe, according to acombinatorial rule given by , as before.

The result is that this procedure works.

Theorem 1.4.1. The distribution w ({Iv}, {Pe}) is well-defined, and isa good distribution on RnT ( ).

“Well-defined” means that w ({Iv}, {Pe}) is defined by contracting tem-pered distributions which are of rapid decrease in some directions with asmooth function which is of bounded growth in the corresponding direc-tions. A more precise statement is given in the proof.

1.5. Proof of the theorem. In fact we will prove something a littlemore general. We will relax the assumption that each Iv is a good distri-bution. Instead, we will simply assume that Iv is an A -valued tempereddistribution on RnH(v), of rapid decay along RnH(v)/Rn.

More precisely: let us write

RnH(v) = Rn 41RnH(v)/Rn

2,

where Rn ) RnH(v) is the small diagonal, and this direct sum is orthogonal.Then, we will assume that Iv is a continuous linear map

Iv : T (Rn, RnH(v)/Rn)$ A .



In particular, Iv is an A -valued tempered distribution on RnH(v).Observe that we can write

RnH( ) = 4v#V ( )RnH(v)

= RnV ( ) 414v#V ( )

1RnH(v)/Rn

22.

The direct product "v#V ( )Iv is a continuous linear map

"v#V ( )Iv : T1RnV ( ),4v#V ( )RnH(v)/Rn

2$ A .

Thus, "Iv is an A -valued tempered distribution on RnH( ), of rapid decreasein certain directions.

Recall that for each edge e & E( ), we have a function

Pe & C"(RnH(e)).

with the property that, for any multi-index I, there exists b such that

|"IPe| * e$b+xh1$xh2+

2

.

Let us write RnT ( ) as an orthogonal direct sum

RnT ( ) = Rn 41RnT ( )/Rn

2.

Let& T (Rn, RnT ( )/Rn).

We will view as a function on RnH( ) via the map RnH( ) $ RnT ( ).Then, /

e#E( )

Pe & C"(RnH( )).

Lemma 1.5.1. (1) The function<

e#E( ) Pe is an element of thespace

T1RnV ( ),4v#V ( )RnH(v)/Rn

2.

Thus, we can pair<

e#E( ) Pe with "Iv to yield an element

w ({Iv}, {Pe})( ) & A .

(2) The map$ w ({Iv}, {Pe})( )

is a continuous linear map

w ({Iv}, {Pe}) : T (Rn, RnT ( )/Rn)$ A .

(3) If we further assume that each Iv is invariant under the Rn action,so that each Iv is a good distribution, then w ({Iv}, {Pe}) is alsoinvariant under the Rn action, and so is a good distribution.



Proof. The third clause is obvious from the first two.For the first two clauses, it su#ces to show that the map

T (Rn, RnT ( )/Rn)$ C"(RnH( ))

/$/

e#E( )

Pe

is a continuous linear map

T (Rn, RnT ( )/Rn)$ T1RnV ( ),4v#V ( )RnH(v)/Rn

2.

Let +!+H( )a,b,l refer to the norm on T

-RnV ( ),4v#V ( )RnH(v)/Rn

., and let

+!+T ( )c,d,m refer to the norm on T (Rn, RnT ( )/Rn).It su#ces to show that for all a, b, l, there exist c, d,m and C such that

for all ,99999

/

e

Pe

99999

H( )

a,b,l

* C + +T ( )c,d,m .

Let us introduce some notation which will allow us to express this con-dition more explicitly.

For h & H( ), let xh : RnH( ) $ Rn be the coordinate function.For every vertex v & V ( ), let

cv ='

h#H(v)

xh : RnH( ) $ Rn.

Letnv : RnH( ) $ RnH(v) $ RnH(v)/Rn

be the composition of the natural map RnH( ) $ RnH(v) with the projectiononto the subspace orthogonal to Rn ) RnH(v).

For every edge e & E( ), corresponding to half-edges h1, h2, let

de = xh1 ! xh2 .

(Of course, de depends on choosing an orientation of the edge e, but thiswill play no role).

Let us view T ( ) as a subset of H( ), and let

cT ='

t#T ( )

xt : RnH( ) $ Rn.

Finally, letnT : RnH( ) $ RnT ( ) $ RnT ( )/Rn

be the composition of the natural map RnH( ) $ RnT ( ) with the projectiononto the orthogonal complement to the subspace Rn ) RnT ( ).



To show the inequality we want, it su#ces to show the following: for alla & Z'0, b & R>0, and l & Z'0, and all multi-indices I and J with |I| * l,there exist c, d and C such that for all ,

SupRnH( )

;;;;;;

5

61 +'

v#V ( )

+cv+27

8a

e$P

v&V ( ) b.nv.2

("I )

5

6"J

/

e#E( )

Pe

7

8

;;;;;;

* C + +T ( )c,d,l .

Now,|"I | * C ! + +T ( )

c,d,l (1 + +cT +2)$ced.nT .2

,

for some constant C !; and we can find some b! and C !! such that

"J

/

e#E( )

Pe * C !e$b#P

e&E( ).de.2

.

Putting these inequalities together, we see that it su#ces to show that, forall a & Z'0, b, b! & R>0, there exists c & Z'0 and d & R>0 such that

SupRnH( )

;;;;;;

5

61 +'

v#V ( )

+cv+27

8a

e$P

v&V ( ) b.nv.2

e$b#P

e&E( ).de.2

(1 + +cT +2)$ced.nT .2;;; < #

Let us write RnH( ) as an orthogonal direct sum

RnH( ) = Rn 4 RnH( )/Rn,

where Rn ) RnH( ) is the small diagonal. Let

nH : RnH( ) $ RnH( )/Rn

be the projection.All of the maps nv, for v & V ( ), de, for e & E( ), and nT factor through

the projection RnH( ) $ RnH( )/Rn.Further, connectedness of the graph implies that the quadratic form

b'

v#V ( )

+nv+2 + b!'

e#E( )

+de+2

is positive definite on RnH( )/Rn. Thus, we can find some > 0 so that

b'

v#V ( )

+nv+2 + b!'

e#E( )

+de+2 > +nH+2 .

Also, since nT : RnH( ) $ RnT ( )/Rn factors through RnH( )/Rn, wehave

ed.nT .2

* ed.nH.2

.

Thus, by choosing d su#ciently small, we can assume that

e$P

v&V ( ) b.nv.2

e$b#P

e&E( ).de.2

ed.nT .2

* e$ .nH.2


2. THE MAIN THEOREM ON Rn 99

for some > 0.It remains to show that for all a and all > 0, there exists c such that

SupRnH( )

;;;;;;

5

61 +'

v#V ( )

+cv+27

8a

(1 + +cT +2)$ce$ .nH.2

;;;;;;<#.

For all c and all , there exists a constant C such that

(1 + +cT +2)$ce$ .nH.2

< C(1 + +cT +2 + +nH+2)$c.

Thus, it su#ces to show that for all a, we can find some c such that

SupRnH( )

;;;;;;

5

61 +'

v#V ( )

+cv+27

8a

(1 + +cT +2 + +nH+2)$c

;;;;;;<#.

But this follows immediately from the fact that the quadratic form +cT +2 ++nH+2 on RnH( ) is positive definite. !

This completes the proof of the theorem.

2. The main theorem on Rn

In this section we state and prove the main theorem for scalar fieldtheories on Rn.

We would like to give a definition of theory along the same lines as thedefinition we gave on compact manifolds. To do this, we need to have adefinition of the renormalization group flow. The results of Section 1 allowus to construct the renormalization group flow on Rn.

In Section 1, we defined the space Dg(Rnk) of good distributions on Rnk.These distributions are invariant under the diagonal Rn action, and of rapiddecay away from the small diagonal Rn ) Rnk.

We will letO(S (Rn)) =

/

k'1

Dg(Rnk)Sn

be the space of formal power series on S (Rn) whose Taylor componentsare good distributions. Note that these power series do not have a constantterm.

Note that O(S (Rn)) is not an algebra; the direct product of good dis-tributions is no longer good.

A good distribution$ & Dg(Rnk)

will be called local if it is supported on the small diagonal Rn ) Rnk. Trans-lation invariance of good distributions implies that any such local $ can bewritten as a finite sum

f(x1, . . . , xn)$'!

x#Rn("If)(x, . . . , x)



where "I : S (Rnk)$ S (Rnk) are constant-coe#cient di!erential operatorscorresponding to multi-indices I & (Z'0)nk. Thus, these distributions arelocal in the sense used earlier.

We will letOloc(S (Rn)) ) O(S (Rn))

denote the subspace of functionals on S (Rn) whose Taylor components arelocal elements of Dg(Rnk).

Finally, we will let

O+(S (Rn))[[!]] ) O(S (Rn))[[!]]

O+loc(S (Rn))[[!]] ) Oloc(S (Rn))[[!]]

denote the subspaces of O(S (Rn)[[!]] and Oloc(S )[[!]] of functionals whichare at least cubic modulo !.

2.1. Now we are ready to define translation invariant theories on Rn.As before, we will always assume that the kinetic part of our actions is!1

2

+, (D+m2)

,where D is the non-negative Laplacian, and m 5 0 is the

mass.We will let

P ( , L) =! L

e$tm2Ktdt =

! L

t$n/2e$tm2e$.x$y.2/tdt & C"(Rn " Rn).

be the propagator.As on a compact manifold, for any I & O+(S (Rn))[[!]], we will define

W (P ( , L), I) ='

!g( ) 1|Aut( )|w (P ( , L), I)

where the sum, as before, is over stable graphs . Theorem 1.4.1 shows thateach w (P ( , L), I) is well-defined.

The definition of quantum field theory on Rn is essentially the sameas that on a compact manifold. The only essential di!erence is that thee!ective interactions I[L] are required to be elements of O+(S (Rn))[[!]], sothat the component distributions I(i,k)[L] are good distributions on Rnk. Inparticular, the e!ective interactions I[L] are translation invariant; we willonly discuss translation invariant theories on Rn.

A second di!erence between the definitions on Rn and a compact man-ifold is more technical: the locality axiom on Rn can be made somewhatweaker. Instead of requiring that I[L] has a small L asymptotic expansionin terms of local action functionals, we instead simply require that I[L] tendsto zero away from the diagonals in Rnk. Translation invariance guaranteesthat this weaker axiom is su#cient to prove the bijection between theoriesand Lagrangians.

Let us now give the definition more formally.



Definition 2.1.1. A scalar theory on Rn, with mass m, is given by acollection of functionals

I[L] = O+(S (Rn), C"((0,#)))[[!]]

where L is the coordinate on (0,#), such that:(1) The renormalization group equation

W-P (L,L!), I[L]

.= I[L!]

is satisfied, where

P (L, L!) =! L#

Le$tm2

Ktdt

is the propagator.(2) The following locality axiom holds. Let us regard Ii,k[L] as a dis-

tribution on Rnk, and let C ) Rnk be a compact subset in the com-plement of the small diagonal. Then, for all functions f & S (Rnk)with compact support on f ,

limL&0

Ii,k[L](f) = 0

As before, we will let T (") denote the set of theories, and let T (n) definethe set of theories defined modulo !n+1.

The construction of counterterms in the previous section applies, mutatismutandis, to yield the following theorem.

Theorem 2.1.2. The space T (m+1) is a principal bundle over T (m)

for the group Oloc(S (Rn)), in a canonical way, and T (0) is canonicallyisomorphic to the subset of Oloc(S (Rn)) of functionals which are at leastcubic.

The choice of a renormalization scheme yields a section of each torsorT (m+1) $ T (m), and so a bijection between the set T (") of theories andthe set O+

loc(S (Rn))[[!]] of !-dependent translation invariant functionals onS (Rn) which are at least cubic modulo !.

Proof. The proof is along the same lines as the proof of the sametheorem on compact manifolds. Thus, I will sketch only what needs to bechanged.

First, we will show how to construct a theory from a translation-invariantlocal action functional

I ='

!iIi,k & O+l (S (Rn))[[!]].

We will do this in two steps. The first step will show the result with amodified propagator, and the second step will show how to get a theory forthe actual propagator from a theory for this modified propagator.

Let f be any compactly supported function on Rn which is 1 in a neigh-bourhood of 0. Let:P ( , L)(x, y) = f(x! y)P ( , L)(x, y) = f(x! y)t$n/2e$.x$y.2/t & C"(R2n).



This will be our modified propagator. Note that :P ( , L)(x, y) is zero if x!yis su#ciently large, and that :P ( , L) = P ( , L) if x! y is su#ciently small.

Suppose, by induction, we have constructed translation invariant localcounterterms

ICTi,k ( ) & Ol(S (Rn)),alg P((0,#)eps),

for all (i, k) < (I,K), such that

lim&0

Wi,k

5

6:P ( , L), I !'

(r,s)%(i,k)

!rICTr,s ( )

7

8

exists for all (i, k) < (I,K).The results proved in Appendix 1 imply that

WI,K

5

6:P ( , L), I !'

(r,s)%(i,k)

!rICTr,s ( )

7

8 ( , L) .'

fi( )(i( , L)

where . means there is a small asymptotic expansion of this form; thefunctions fi( ) & P((0,#)) are smooth functions of , and are periods, asdefined in Section 9 of Chapter 2.

The distributions

(i( , L) : S (Rnk)$ C"((0,#)L)

are translation invariant, and are also supported on the product of the smalldiagonal Rn ) Rnk with a compact subset K ) Rn(k$1). Thus, they are gooddistributions, that is

(i & Dg(Rnk, C"((0,#)L)).

Then, as before, we let

ICT(I,K)( ) = Sing

5

6W(I,K)

5

6:P ( , L), I !'

(r,s)%(i,k)

!rICT(r,s)( )

7

8

7

8 .

The existence of the small asymptotic expansion mentioned above impliesthat it makes sense to take the singular part. This singular part is an element

ICT(I,K)( ) & Dg(Rnk, C"((0,#)L)),alg C"((0,#) ).

As before, this singular part is independent of L; this implies that the coun-terterm ICT

(I,K) is local.This allows us to define

:I[L] = lim&0

W1:P ( , L), I ! ICT ( ))

2

as before. Each distribution:I(I,K)[L] : S (Rnk) $ C"((0,#)L)



is translation invariant, and is supported on the product of the small diago-nal Rn ) Rnk with a compact neighbourhood K ) Rn(k$1). Therefore each:I(I,K)[L] is a good distribution.

Thus, the collection of e!ective interactions:I[L] & O(S (Rn, C"((0,#)L)))

define a theory for the modified propagator :P ( , L). It is easy to see thatthe locality axiom holds: if we view :I(i,k)[L] as a distribution on Rnk, and fixany f & S (Rnk) which has compact support way from the small diagonal,the limit

limL&0

:I(i,k)[L](f) = 0.

Now we need to show how to get a theory for the actual propagatorP ( , L) from a theory for the modified propagator :P ( , L).

Note that the distribution P (0, L) ! :P (0, L) is actually a smooth func-tion, because P (0, L) and :P (0, L) agree in a neighbourhood of the diagonal.In fact,1P (0, L)! :P (0, L)

2(x, y) = (1! f(x! y))

! L

t=0t$n/2e$m2te$.x$y.2/tdt

= F (x! y)

where

F (x) = (1! f(x))! L

t=0t$n/2e$m2te$.x.2/tdt.

Since f(x) = 1 in a neighbourhood of x = 0, the function F (x) is Schwartz.Further, the function F and all of its derivatives decay faster than

e$.x.2/2L at #.It follows from this and from the fact that the functionals :I[L] are good

that we can apply Theorem 1.4.1, so that the expression

W1P (0, L)! :P (0, L), :I[L]

2

is well-defined. This expression defines I[L]; and, Theorem 1.4.1 impliesthat I[L] is a good distribution.

It is immediate that I[L] satisfies the renormalization group equation

I[L] = W (P ( , L), I[ ])

and the locality axiom. Thus, {I[L]} describes the theory associated to thelocal functional I.

It remains to show that every theory arises in this way. We will show thisby induction, as before. Suppose we have a theory described by a collectionof e!ective interactions {I[L]}. Suppose we have a local action functional

J ='

(i,k)<(I,K)

!iJ(i,k)



with associated e!ective interactions J [L]. Suppose that

J(i,k)[L] = I(i,k)[L]

for all (i, k) < (I,K). We need to find some J !(I,K) such that if we let

J ! = J + !IJ(I,K)

thenJ !

(I,K)[L] = I(I,K)[L].Let

J !(I,K) = I(I,K)[L]! J(I,K)[L].

The renormalization group equation implies that J(I,K) as so defined is in-dependent of L; and it is immediate that, if we define J ! = J + !IJ(I,K),then

J !(I,K)[L] = I(I,K)[L].

It remains to check that J(I,K) is local.The locality axiom satisfied by I(I,K)[L] and J(I,K)[L] implies that J(I,K)

is a distribution on Rnk supported on the small diagonal. Any such distri-bution which is translation invariant is a local action functional.

!

3. Vector-bundle valued field theories on Rn

We would also like to work theories on Rn where the space of fields isthe space of sections of some vector bundle. As with scalar field theories,we are only interested in translation invariant theories.

As Section 13 of Chapter 2, everything will depend on some auxiliarymanifold with corners X, equipped with a sheaf A of commutative gradedalgebras over the sheaf of algebras C"

X . The space of global sections of Awill be denoted, as before, by A .

The data we need to state our main theorem in this case is the following.(1) A finite dimensional graded vector space E. We let

E = E ,S (Rn).

Thus, E is the space of Schwartz sections of the trivial vector bundleE " Rn.

(2) A degree 0 symmetric element

Kl & C"(Rn " Rn), E(2 , C"((0,#)l),A ,

such that, in some basis ei of E, Kl can be written

Kl ='

Pi,j(x! y, l12 , l$

12 )e$.x$y.2/lei , ej

where eachPi,j & A [xk ! yk, l

12 , l$

12 ]


3. VECTOR-BUNDLE VALUED FIELD THEORIES ON Rn 105

is a polynomial in the variables xk ! yk and l±12 , with coe#cients

in A .The kernel Kl will, in all examples, be a kernel obtained from the heat kernelfor some elliptic operator by di!erentiating in the variables xk ! yk somenumber of times.

The propagator will be of the form

P ( , L) =! L

Kldl.

3.1. Space of functions. As for scalar field theories, there are variousspaces of functionals which are relevant. Recall that

Dg(Rnk, A ) ) D(Rnk, A )

as defined in Section 1 is the space of good distributions, which, heuristically,are those A -valued translation invariant distributions of rapid decay awayfrom the diagonal.

We will let

O(E , A ) =/

k>0

1Dg(Rnk, A ), (E))(k

2

Sk

.

This is the space of A -valued functionals on E whose Taylor componentsare given by good distributions. Since good distributions are defined to betranslation invariant, every element of O(E ,A ) is translation invariant.

A good distribution $ & Dg(Rnk, A ) is called local if it is supported onthe small diagonal Rn ) Rnk. Any local good distribution can be writtenas a finite sum of distributions of the form

f(x1, . . . , xk) /$'

I

!

x#RnaI("If)(x, . . . , x)

where aI & A and I & (Z'0)nk are multi-indices.We let

Oloc(E , A ) ) O(E , A )denote the subspace of functionals whose Taylor components are all local.

We will letO+(E , A )[[!]] ) O(E , A )[[!]]

be the subspace of functionals which are at least cubic modulo the ideal inA [[!]] generated by I and !. The subspace

O+loc(E ,A )[[!]] ) Oloc(E , A )[[!]]

is defined in the same way.Theorem 1.4.1 allows us to define the renormalization group flow

O+loc(E , A )[[!]]$ O+

loc(E , A )[[!]]I /$ W (P ( , L), I) .

Now we are ready to define the notion of theory in this context.



Definition 3.1.1. A family of theories on E , over the ring A , is acollection

{I[L] & O+(E , A )[[!]]}of translation invariant e!ective interactions such that

(1) Each I[L] & O+(E , A )[[!]] is of degree 0.(2) The renormalization group equation

I[L] = W (P ( , L), I[ ])

is satisfied.(3) The following locality axiom holds. Let us expand, as usual,

I[L] ='

!iI(i,k)[L]

where each I(i,k)[L] is an Sk-invariant map E (k $ A . Then, werequire that for all elements f & E (k which are compactly supportedaway from the small diagonal in Rnk, and for all x & X,

limL&0

I(i,k)[L](f)x = 0.

This limit is taken in the finite dimensional graded vector space Ax.The subscript x denotes restricting an element of A = '(X,A) toits value at the fibre Ax of A above x & X.

We let T (")(E ) denote the set of theories, and T (n)(E ) denote the set oftheories defined modulo !n+1.

As before, the main theorem is:

Theorem 3.1.2. The space T (m+1) $ T (m) is a torsor for the spaceO0

loc(E , A )[[!]] of local action functionals of degree 0 on E . Further, T (0)

is canonically isomorphic to the space of degree 0 local action functionals onE which are at least cubic.

If we choose a renormalization scheme, then we find a section of eachtorsor T (m+1) $ T (m). Thus, the choice of renormalization scheme yieldsa bijection

T (")(E , A ) %= O+,0loc (E , A )[[!]]

between the set of theories and the set of (translation-invariant) local actionfunctionals on E which are of degree 0, and which are at least cubic modulothe ideal in A [[!]] generated by I and !.

Proof. As before, we will prove the renormalization scheme dependentversion of the theorem; the renormalization scheme independent version isa simple corollary.

There are two steps to the proof. First, we have to construct the theoryassociated to a local action functional I & Oloc(E , A ). The second step isto show that every theory arises in this way.

The proof of the first step is essentially the same as that of the corre-sponding statement for scalar field theories, Theorem 2.1.2. The fact that weare dealing with sections of a vector bundle rather than functions presents


4. HOLOMORPHIC ASPECTS OF THEORIES ON Rn 107

no extra di#culties. The fact that we are dealing with a family of theories,parameterized by the algebra A , makes the analysis slightly harder. How-ever, the results in Appendix 1, as well as those in Section 1, are all statedand proved in a context where everything depends on an auxiliary ring A .Thus, the argument of Theorem 2.1.2 applies with basically no changes.

The converse requires slightly more work. Suppose we have a the-ory, given by a collection of e!ective interactions I(i,k)[L] & O+(E , A ,C"((0,#)L))[[!]]. Suppose, by induction, that we have a local action func-tional

J ='

(i,k)<(I,K)

!iJ(i,k) & O+loc(E , A )[[!]]

with associated e!ective interactions J [L], such that

J(i,k)[L] = I(i,k)[L] for all (i, k) < (I,K).

As usual, we need to find some J !(I,K) & Oloc(E , A ), homogeneous of

degree K, such that if we set

J ! = J + !IJ !(I,K)

thenJ !

(I,K)[L] = I(I,K)[L].Since

J !(I,K)[L] = J(I,K)[L] + J(I,K),

we must haveJ(I,K) = I(I,K)[L]! J(I,K)[L].

The right hand side of this equation is independent of L. Thus, the distri-bution J(I,K) is an element

J(I,K) & (E))(K ,Hom(S (RnK),A ).

We need to show that it is local.Since both I[L] and J [L] satisfy the locality axiom defining a theory,

the distribution J(I,K) is supported on the small diagonal Rn ) RnK . Inaddition, it is translation invariant.

Every element of Hom(S (RnK), A ) which is supported on the smalldiagonal, and which is translation invariant, is local. !

4. Holomorphic aspects of theories on Rn

In this section we will show that the Fourier transforms of the e!ectiveinteractions describing a theory are entire holomorphic functions on prod-ucts of complexified momentum space. The results in this section will notbe used in the rest of this book. However, the holomorphic nature of theFourier transform of the e!ective interactions is worth noting, as it lendssome insight into the problem of analytically continuing to Lorentzian sig-nature.



4.1. Recall that Fourier transform is a continuous linear isomorphismF : S (Rl) $ S (Rl) for any l. Continuity of Fourier transform meansthat it acts on spaces associated to S (Rl), such as spaces of tempereddistributions.

Suppose we have a translation invariant vector-valued field theory onRn, described by a family of e!ective interactions

I[L] & O+(E , A , C"((0,#)L))[[!]].

The Taylor components I(i,k)[L] of I[L] are good distributions

I(i,k)[L] & Dg(Rnk, A , C"((0,#)L)).

In particular, they are tempered distributions

I(i,k)[L] : S (Rnk)$ A , C"((0,#)L).

Thus, we can take the Fourier transform

F (I(i,k)[L]) : S (Rnk)$ A , C"((0,#)L),

defined by composing I(i,k)[L] with the Fourier transform on S (Rnk). Con-tinuity of the Fourier transform implies that F (I(i,k)[L]) is continuous.

We would like to describe the form taken by the Fourier transform ofI(i,k)[L]. It turns out that it is a holomorphic function of a certain kind.

We will let Rn(k$1) ) Rnk denote the orthogonal complement to thesmall diagonal Rn ) Rnk. We will let Hol(Cl) denote the space of entireholomorphic functions on any Cl. This is a nuclear Frechet space; we willlet Hol(Cl),A denote the projective tensor product. Note that Hol(Cl),Ais the subspace of C"(Cl) ,A of those elements (z1, . . . , zl, x) such that""zi

= 0 for i = 1, . . . , l.The following proposition describes the form taken by the Fourier trans-

form of I(i,k)[L].

Proposition 4.1.1. There exists some

((i,k)[L] & Hol(Cn(k$1)),A , C"((0,#)L)

such that, for all f & S (Rnk),

I(i,k)[L](F (f)) =!

x1,...,xn#RnPxi=0

((i,k)[L](x1, . . . , xn$1)f(x1, . . . , xn).

In other words, the Fourier transform of I(i,k)[L] is the operation whichtakes a function f on Rnk, restricts it to the subspace Rn(k$1), and thenintegrates against ((i,k)[L].

A corollary of this proposition, and of Proposition 1.3.2 in Chapter 4, isthe following.



Corollary 4.1.2. The functions ((i,k)[L] appearing above can be writ-ten as finite sums

((i,k)[L](x1, . . . , xn$1) ='

'r,si,k(7

Lx1, . . . ,7

Lxn$1)Lr/2(log L)s

where r & Z, s & Z'0, and 'r,si,k & Hol(Cn(k$1)).

This corollary shows that the ((i,k)[L] have very tightly constrainedbehaviour as functions of L.

4.2. Proposition 4.1.1 follows immediately from a general lemma aboutthe holomorphic nature of the Fourier transform of a distribution of rapiddecay.

Recall from Section 1 that

T (Rm) ) C"(Rm)

refers to the space of functions f all of whose derivatives are bounded byeb.x.2

, for all b. The space T (Rm) has a sequence of norms +!+b,l, forb & R>0 and l & Z'0, defined by

+f+b,l = Supx#Rm

'

|I|%l

;;;eb.x.2

"If;;;

where the sum is over multi-indices I = (I1, . . . , Il) & (Z'0)m, and |I| =*Ii.The space T (Rm) is given the topology defined by the family of norms

+!+b,l. A continuous linear function

$ : T (Rm)$ A

is an A -valued distribution of rapid decay.Since the inclusion S (Rm) !$ T (Rm) is continuous, any A -valued dis-

tribution of rapid decay is in particular an A -valued tempered distribution.Thus, we can take its Fourier transform.

Lemma 4.2.1. Let $ : T (Rm)$ A be an A -valued distribution of rapiddecay. Then there exists an entire holomorphic function

=$ & Hol(Cm),A

such that, for all f & S (Rm),

$(F (f)) =!

x#Rm

=$(x)f(x).

Proof. Observe that the map

Cm $ T (Rm)

x /$ ei0y,x1

is holomorphic (meaning that it is smooth and satisfies the Cauchy-Riemannequation).



Define a function=$ : Cm $ A

by=$(x) = $y(ei0y,x1)

(where the subscript in $y means that $ is acting on the y variable).Since =$ is the composition of a holomorphic map Cm $ T (Rm) with a

continuous linear map T (Rm)$ A , =$ is holomorphic. Thus,=$ & Hol(Cm),A .

Further, the map

Hom(T (Rm),A ) $ Hol(Cm),A

$ /$ =$

is continuous (this follows from the fact that ei0y,x1, and any number of itsx and y derivatives, can be bounded by some eb.y.2

uniformly for x in acompact set).

It remains to show that =$ is indeed the Fourier transform of $. This istrue for all $ in

C"c (Rm),A ) Hom(T (Rm),A ),

where C"c (Rm) denotes the space of smooth functions with compact support.

The former subspace is dense in the latter; continuity of $ /$ =$ impliesthat =$ must be the Fourier transform for all $. !

Corollary 4.2.2. Let $ & Dg(Rnk, A ) be a good distribution. Thenthere exists some =$ & Hol(Cn(k$1)),A such that, for all f & S (Rnk),

$(F (f)) =!

x1,...,xk#RnPxi=0

=$(x1, . . . , xk$1)f(x1, . . . , xk).

Proof. Let us write Rnk = Rn"Rn(k$1), where Rn ) Rnk) is the smalldiagonal, and Rn(k$1) is its orthogonal complement.

Translation invariance of $ implies that it can be written as a directproduct

$ = 1 " $!

where 1 is the distribution on Rn given by integrating, and $! is an A -valuedtempered distribution on Rn(k$1). The rapid decay conditions satisfied bya good distribution mean that $! gives a continuous linear map

$! : T (Rn(k$1))$ A .

Fourier transform commutes with direct product of distributions, andthe Fourier transform of the distribution

&is the function at the origin.

Thus,F ($) = x=0 " F ($!).



The previous lemma implies that F ($!) is an entire holomorphic functionon Cn(k$1), as desired. !

This completes the proof of the proposition.



CHAPTER 4

Renormalizability

1. The local renormalization group flow

1.1. One point of view on renormalizability says that a theory is renor-malizable if there are finitely many counterterms. The usual justificationfor this condition is that it selects, in the infinite dimensional moduli spaceof theories, a finite dimensional space of well-behaved theories. Finite di-mensionality means that renormalizable theories are predictive: they haveonly finitely many free parameters, which can be fixed by performing finitelymany experiments.

Any natural condition which selects a finite dimensional subspace of thespace of theories can be regarded as a criterion for renormalizability. Themore natural the condition, of course, the better.

The philosophy of this book is that the counterterms themselves haveno intrinsic importance; they are simply a tool in the construction of thebijection between theories and local action functionals. The choice of adi!erent renormalization scheme will lead to a di!erent set of countertermsassociated to a theory. Thus, we must reject the criterion asking for finitelymany counterterms as unnatural.

We will use a renormalization criterion arising from the Wilsonian pointof view. This criterion selects, in a natural way, a finite dimensional spaceof renormalizable theories in the infinite dimensional space of all theories.

The idea is that a theory given by a set of e!ective interactions {I[L]}is well-behaved if I[L] doesn’t grow too fast as L $ 0, when measured inunits appropriate to the length scale L. We will take “not too large” tomean they grow at most logarithmically as L $ 0. A theory will be calledrenormalizable if it satisfies this growth condition and if, in addition, thespace of deformations of the theory which also satisfy this growth condition isfinite dimensional. This means that one can always specify a renormalizabletheory by a finite number of parameters.

1.2. Notation. Throughout this chapter, we will only consider trans-lation invariant theories on Rn. We will use the notation from Chapter 3;thus, O(S (Rn)) will refer to the space of translation invariant functionalson Rn, satisfying the technical conditions described in Chapter 3.

Local action functionals will also be translation invariant. We will let

Oloc(S (Rn)) ) O(S (Rn))

113


114 4. RENORMALIZABILITY

be the space of translation invariant local action functionals on S (Rn).Theories on Rn which are invariant under Rn " SO(n) yield theories on

any manifold with a flat metric.

1.3. Now we will start to make the definition of renormalizability pre-cise. First, we need to introduce the local renormalization group flow. Thisis a flow on the space of theories which is a combination of the renormaliza-tion group flow we already know, a rescaling on Rn, and a rescaling of thefield . These rescalings are the change of units mentioned above.

Let us define an operation

Rl : S (Rn)$ S (Rn)

Rl( )(x) = ln/2$1 (lx)

for x & Rn. If I & O(S (Rn)) is a functional on S (Rn), define

R+l (I)( ) = I(Rl$1 ).

ThusR+

l (I)( (x)) = I(l1$n/2 (l$1x))The reason for this definition is that then the pairing between functionalsand fields is invariant:

I( ) = (R+l I)(Rl ).

Definition-Lemma 1.3.1. The local renormalization group flow

RGl : T (") $ T (")

on the space of theories is defined as follows. If {I[L]} is a collection ofe!ective interactions defining a theory, then

RGl ({I[L]}) = {RGl(I[L])}is defined by

RGl(I[L]) = R+l

-I[l2L]

..

This collection of e!ective interactions I[L] & O(S (Rn), C"((0,#)l) ,C"((0,#)L) defines a smooth family of theories parameterized by l, in thesense of Definition 3.1.1.

Before we proceed, I should comment on the choice of the power ln/2$1

which appears in the definition of the local renormalization group flow. Letus suppose we have a classical field theory described by a local action func-tional S. If c & R,, then we can define a new action functional Sc bySc( ) = S(c ). The action functionals S and Sc describe equivalent classi-cal field theories: they are related by an automorphism of the vector bundleof fields covering the identity on the space-time manifold.

The local renormalization group flow is defined by a certain change ofcoordinates on the space of fields, of the form (x) /$ c(l) (lx), wherec(l) & R,. Because changes of coordinates of the form (x) $ c (x) takea field theory to an equivalent field theory, it doesn’t matter what function


1. THE LOCAL RENORMALIZATION GROUP FLOW 115

c(l) we choose (as long as we consider how these changes of coordinates acton actions instead of interactions).

We have been dealing throughout with interactions, and suppressingthe quadratic term in the action. Thus, it is convenient to choose c(l) sothat the action for the massless free field is preserved under this change ofcoordinates. Taking c(l) = ln/2$1 is the unique way to do this.

Now let us turn to the proof of the definition-lemma stated above.

Proof. In order to check that RGl(I[L]) defines a theory, we need tocheck that the locality axiom and the renormalization group equation aresatisfied.

The locality axiom of is immediate. We need to check that RGl (I[L])satisfies the renormalization group equation. Now,

P ( , L)(x, y) =! L

t$n/2e$.x$y.2/tdt

so that

RlP ( , L) = ln$2! L

t$n/2e$l2.x$y.2/tdt

=! l$2L

l$2u$n/2e$.x$y.2/udu where u = l$2t

= P (l$2 , l$2L).

The way we defined the action R+l on functionals O(S (Rn)) means that, for

all P & C"(M2), and all I & O(S (Rn)),

R+l ("P I) = "RlP R+

l I.

The renormalization group equation for I[L] says that

I[l2L] = ! log>exp

-!"P (l2 ,l2L)

.exp

-!$1I[l2 ]

.?.

Applying R+l to both sides, we find

RGl (I[L]) = ! log>exp

-!"P ( ,L)

.exp

-!$1R+

l I[l2 ].?

= ! log>exp

-!"P ( ,L)

.exp

-!$1RGl (I[ ])

.?

= W (P ( , L),RGl (I[ ]))

as desired.!

Proposition 1.3.2. Let {I[L]} be a theory on Rn (as always, translationinvariant). Then,

RGl(I[L]) & O+(S (Rn))[[!]], C[log l, l, l$1].

In other words, each RGl(Ii,k[L]), as a function of l, is a polynomial in l,l$1 and log l.



We will prove this later.A term in RGl (I[L]) is called(1) relevant if it varies as lk(log l)r for some k 5 0 and some r & Z'0.(2) irrelevant if it varies as lk(log l)r for some k < 0 and some r & Z'0.(3) marginal if it varies as (log l)r for some r & Z'0.

Thus, marginal terms are a subset of relevant terms. Logarithmic termsonly appear because of quantum e!ects; at the classical level (modulo !) allterms scale by lk for some k & 1

2Z.

Definition 1.3.3. A theory {I[L]} is relevant if, for each L, RGl (I[L])consists entirely of relevant terms; or, in other words, if

RGl (I[L]) & O+(S (Rn))[[!]], C[log l, l].

A theory is marginal if, for each L, each RGl (I[L]) consists entirely ofmarginal terms; that is,

RGl (I[L]) & O+(S (Rn))[[!]], C[log l].

Let

R(n) ) T (n)

(respectively,

M (n) ) T (n)2

denote the subset of the space T (n) of theories consisting of relevant (respec-tively, marginal) theories defined modulo !n. Let R(") = lim2!R(n) and letM (") = lim2!M (n) be the spaces of relevant and marginal theories defined toall orders in !.

In this definition, and throughout this chapter, we consider only trans-lation invariant theories.

Passing to small length scales (that is, high energy) corresponds to send-ing l $ 0. A theory which is well-behaved at small length scales will haveonly relevant terms, and a theory which is well-behaved at large length scaleswill have only marginal and irrelevant terms. A theory which is well-behavedat all length scales will have only marginal terms.

Definition. A theory on Rn is renormalizable if it is relevant and, termby term in !, has only finitely many relevant deformations.

In other words, a theory {I[L]} & T (") is renormalizable if {I[L]} &R(") and, for all finite n, T{I[L]}R

(n) is finite dimensional.A theory is strictly renormalizable if it is renormalizable and it is mar-

ginal.A theory is strongly renormalizable it it is strictly renormalizable, and all

of its relevant deformations are marginal; in other words, if {I[L]} & M (")

and T{I[L]}R(") = T{I[L]}M

(").



1.4. The choice of a renormalization scheme leads to a bijection

T (") %= O+loc(S (Rn))[[!]]

between the space of theories and the space of local action functionals. Boththeories and functionals are, as always in this chapter, translation invariant.

The local renormalization group flow translates, under this bijection, toan R>0 action on the space of local action functionals. If IO+

loc(S (Rn))[[!]]we will let RGl(I) be the family of local action functionals arising by theaction of the local renormalization group. It follows from the fact thatRGl({I[L]}) is a smooth family of theories, and from Theorem 3, that RGl(I)is a smooth family of local action functionals, that is, an element

RGl(I) & O+loc(S (Rn), C"((0,#)l))[[!]].

Modulo !, the action of the local renormalization group on local actionfunctionals is simple. If

I & O+loc(S (Rn))[[!]],

thenRGl(I) = R+

l (I) mod !.

At the quantum level more subtle things happen, as we will see shortly.

1.5. Every local action functional I & Oloc(S (Rn)) which is homoge-neous of degree k with respect to the field & S (Rn) can be written as afinite sum of functionals of the form

(†) /$!

Rn("I2 ) · · · ("Ik )

where the Ij are multi-indices, and

"Ij =#"

"x1

$Ij,1

· · ·#"

"xn

$Ij,n

are the corresponding translation invariant di!erential operators.The operator R+

l acts diagonally on the space of functionals, with eigen-values lk for k & 1

2Z. We say a functional I has dimension k if

R+l I = lkI.

For example, if I is the functional in equation (†), then

R+l (I) = l

n+k“1$1

2n”$

P|Ij |I

so that I is of dimension n + k-1! 1

2n.!*

|Ij |.Let

Oloc,k(S (Rn)) ) Oloc(S (Rn))be the space of local action functionals of dimension k.

LetOloc,'0(S (Rn)) ) Oloc(S (Rn))



be the subspace of functionals which are sums of functionals of non-negativedimension. There are projection maps

Oloc(S (Rn)) $ Oloc,'0(S (Rn))

andOloc(S (Rn)) $ Oloc,0(S (Rn)).

The main theorem proved in this chapter is the following.

Theorem 1.5.1. The space R(m+1) is, in a canonical way, a torsor overR(m) for the abelian group Oloc,'0(S (Rn)) of local action functionals of non-negative dimensions. Also, R(0) is canonically isomorphic to the subspaceof Oloc,'0(S (Rn)) of functionals which are at least cubic.

In a similar way, M (m+1) is a torsor over M (m) for the abelian groupOloc,0(S (Rn)) of local action functionals of dimension zero. Also, M (0) iscanonically isomorphic to the subspace of Oloc, 0(S (Rn)) of functionals ofdegree 0.

The choice of a renormalization scheme leads to sections of the torsorsM (m+1) $ M (m) and R(m+1) $ R(m), and thus to bijections

R(") %= O+loc,'0(S (Rn))[[!]]

andM (") $ O+

loc,0(S (Rn))[[!]].

The following is an immediate corollary.Corollary 1.5.2. Let us choose a renormalization scheme.Then, we find a bijection between (translation invariant) strictly renor-

malizable scalar field theories on Rn and Lagrangians of the form:(1) The 3 theory on R6.(2) The 4 theory on R4.(3) The 6 theory on R3.(4) The free field theory on Rn when n = 5 or n > 6.

More precisely: there is a bijection between strictly renormalizable the-ories on R3 and interactions of the form

f(!)!

R3D + g(!)

!

R3

6

where f & !R[[!]] and g & R[[!]]; and similarly for R4 and R6. The pointis that, on R3, the space of local action functionals of dimension zero isspanned by

&R3 D and

&R3

6; and similarly on R4 and R6.On Rn where n = 5 or n > 6, the only local action functional of dimen-

sion zero is&

Rn D . Thus, renormalizable theories in these dimensions cor-respond to purely quadratic interactions of the form f(!)

&R3 D . These

are free theories.There are no renormalizable scalar field theories on R or R2, as any rel-

evant theory in these dimensions has infinitely many relevant deformations.



1.6. Now let us consider the quantum e!ects which appear in the localrenormalization group.

Let us fix a renormalization scheme RS0. Recall that P((0, 1)) )C"((0, 1)) is the algebra of smooth functions on (0, 1) which are periods. Arenormalization scheme is defined by a subspace

RS0 = P((0, 1))<0 )P((0, 1))

of purely singular periods; this subspace is required to be complementary tothe subspace P((0, 1))'0 of periods which admit an $ 0 limit.

LetRSl = {f &P((0, 1)) | f(l$2 ) & RS0}.

In other words: the action of R>0 on itself induces an action of R>0 on thespace of renormalization schemes, and we will denote by RSl the renormal-ization scheme obtained by applying l$2 & R>0 to the original renormaliza-tion scheme RS0.

There is a change of renormalization scheme map

$l,0 = $RSl&RS0 : O+loc(S (Rn))[[!]]$ O+

loc(S (Rn))[[!]].

By definition, a theory associated to the action I and the renormalizationscheme RSl is equivalent to the theory associated to the action $l,0(I) andthe renormalization scheme RS0.

Throughout, we will use the renormalization scheme RS0 to identify thespace T (") with the space O+

loc(S (Rn))[[!]] of local action functionals usingthe renormalization scheme RS0. Once we have made this identification, thelocal renormalization group flow RGl is a map

RGl : O+loc(S (Rn))[[!]]$ O+

loc(S (Rn))[[!]].

Lemma 1.6.1. The local renormalization group flow RGl is the composi-tion

RGl = $l,0 -R+l : O+

loc(S (Rn))[[!]]$ O+loc(S (Rn))[[!]].

Proof. The theory associated to a local action functional I is definedby the e!ective interactions

WR (P (0, L), I) = lim&0

W-P ( , L), I ! ICT ( )

..

By definition of the local renormalization group flow,

WR (P (0, L),RGl(I)) = R+l W

R-P (0, l2L), I

.

= lim&0

R+l W

-P ( , l2L), I ! ICT ( )

.

= lim&0

W-RlP ( , l2L), R+

l I !R+l I

CT ( ).

= lim&0

W-P (l$2 , L), R+

l I !R+l I

CT ( ).

= lim&0

W-P ( , L), R+

l I !R+l I

CT (l2 ).



where in the last step we reparameterize the dummy variable . Note thatR+

l ICT (l2 ) is purely singular for the renormalization scheme RSl. Since the

limit exists, it follows that R+l I

CT (l2 ) is the counterterm for R+l I with this

renormalization scheme.Thus, the e!ective interaction

lim&0

W-P ( , L), R+

l I !R+l I

CT (l2 ).

defines the theory associated to R+l I and the renormalization scheme RSl.

By definition of the change of renormalization scheme map, this is the sameas the theory associated to $l,0R+

l I and the renormalization scheme RS0.Thus, we find that

WR (P (0, L),RGl(I)) = WR (P (0, L),$l,0R+l I)

so that$l,0R

+l I = RGl(I)

as desired. !1.7. As we have seen, the choice of renormalization scheme leads to

a bijection between R(") and the space O+loc,'0(S (Rn))[[!]] of local action

functionals of non-negative dimension. Since the local renormalization groupflow acts on R("), via this bijection, it also acts on O+

loc,'0(S (Rn))[[!]]. Wewill continue to use the notation RGl to refer to this action:

RGl : O+loc,'0(S (Rn))[[!]]$ O+

loc,'0(S (Rn))[[!]].

In a similar way, the bijection between the space M (") of marginal theoriesand the space O+

loc,0(S (Rn))[[!]] of local action functionals of dimension zeroleads to an action of RGl on the latter space.

We have to be careful here; the diagram

O+loc,'0(S (Rn))[[!]] RGl ! !

! "

""

O+loc,'0(S (Rn))[[!]]

! "

""

O+loc(S (Rn))[[!]]

RGl !! O+loc(S (Rn))[[!]]

does not commute, where the vertical arrow is the naive inclusion. Nei-ther does the corresponding diagram where the inclusion is replaced by theprojection

O+loc(S (Rn))[[!]]$ O+

loc,'0(S (Rn))[[!]].However, there is a modified inclusion map

O+loc,'0(S (Rn))[[!]] !$ O+

loc(S (Rn))[[!]]

which corresponds to the inclusion

R(") !$ T (");

this inclusion is, of course, equivariant with respect to the local renormal-ization group action.



Classically, that is, modulo !, the local renormalization group action istrivial on a local action functional of dimension zero. However, there arequantum e!ects. We can see this, for instance, in the 4 theory on R4. Letus use the notation

+4,

=!

R4

4.

Lemma 1.7.1.

RGl

-c 1

4!

+4,.

= 14!

+4,#

c + !c2 316 2

log l

$modulo !2

(This agrees with a standard result in the physics literature).

Proof. We will fix a renormalization scheme RS0 with the propertythat the functions $1 and log are purely singular.

In Section 4 of this chapter, the counterterms for the action

I =14!

c+

4,

are calculated to one loop. We find (Proposition 4.0.5) that

ICT1,2 ( ) = 2$6 $2c $1

+2,

ICT1,4 ( ) = ! $22$8c2 log

+4,

ICT1,i ( ) = 0 if i 8= 2, 4.

Lemma 1.6.1 says that the local renormalization group flow RGl is ob-tained as a composition

RGl = R+l - $l,0

of the rescaling operator R+l and a change of renormalization scheme map

$l,0. As discussed earlier, $l,0 is the change of renormalization scheme mapfrom the renormalization scheme RSl, whose purely singular functions are ofthe form f(l2 ) where f( ) is purely singular for RS0, to the renormalizationscheme RS0.

By definition of $l,0, the theory associated to action I and renormaliza-tion scheme RSl is equivalent to the theory associated to action $l,0(I) andrenormalization scheme RS0.

With I = 14!c

+4,, the one-loop counterterms with renormalization

scheme RSl are

ICT,RSl1,2 ( ) = 2$6 $2c $1

+2,

ICT,RSl1,4 ( ) = ! $22$8c2

-log + log l2

. +4,

ICT,RSl1,i ( ) = 0 if i 8= 2, 4

To see this, observe that the functions $1 and log l2 = log + log l2 arepurely singular for the renormalization scheme RSl.



It follows that

$l,0(I) = I + ! $22$8c2 log l2+

4,

modulo !2.

Since R+l

+4,

=+

4,, this yields the desired result.

!

2. The Kadano!-Wilson picture and asymptotic freedom

2.1. The definition of renormalizability given above is closely related tothe Kadano!-Wilson picture.

The Kadano!-Wilson renormalization criterion says that a theory isrenormalizable if

(1) It converges to a fixed point under the local renormalization groupflow as l $ 0.

(2) The unstable manifold for the local renormalization group at thisfixed point is finite dimensional; in other words, all but finitelymany directions are attractive.

“Unstable” means that as the local RG parameter l increases, and we tendto low energy, points in the unstable manifold move away from the fixedpoint. To avoid using the term “renormalizable” for several slightly di!er-ent concepts, we will refer to a theory satisfying these two conditions asKadano!-Wilson (KW) renormalizable. A fixed point satisfying condition(2) will be called a Kadano!-Wilson (KW) fixed point.

Any real-world measurements one makes of a quantum field theory oc-cur at low energy, that is, as the local renormalization group parameter lbecomes large. Suppose we have a theory which is a Kadano!-Wilson fixedpoint, and we make a small deformation in the stable direction. Then, asl $ #, this deformation is drawn back to the fixed point we started with.Thus, deformations in the stable direction make no di!erence to real worldmeasurements; they are called irrelevant.

If the Kadano!-Wilson criterion holds, there are only a finite number ofdeformations which make any di!erence to low-energy measurements; thesedeformations occur in the unstable manifold. Thus, to specify a theory up toterms which are irrelevant for experimental purposes amounts to specifyinga point in the unstable manifold. The KW criterion says that this unstablemanifold is finite dimensional, and so a theory is specified by only a finitenumber of parameters.

2.2. The Kadano!-Wilson criterion for renormalizability is an ideal non-perturbative definition. The definition we give is a perturbative approxima-tion to this ideal.

Our perturbative definition of renormalizability contains two conditions:firstly, that the coupling constants in the theory grow at most logarithmicallyas l $ 0; and secondly, that the theory has only finite many deformationssatisfying this condition.


2. THE KADANOFF-WILSON PICTURE AND ASYMPTOTIC FREEDOM 123

The first condition says that there are no obvious perturbative obstruc-tions to convergence to a fixed point. If a term in the theory exhibits poly-nomial growth as l $ 0, then one can reasonably conclude that the theorydoesn’t converge to a fixed point. The second condition says that if thetheory does converge to a fixed point, then this fixed point must satisfy theKadano!-Wilson criterion.

The presence of terms with logarithmic growth does not preclude conver-gence to a fixed point at the non-perturbative level. For instance, supposewe had a theory with a single coupling constant c, and that at c = 0 the the-ory is a fixed point (for example, a free theory). Suppose that the couplingconstant c changes as

c /$ cl! = c + !c log l + · · · .

Since the formal parameter ! should be viewed as being greater than zero,if c changes in this way then the theory does converge to a fixed point.

On the other hand, if the coupling constant c changes as

c /$ cl$! = c! !c log l + · · ·

we would not expect convergence to a fixed point.Thus, our definition of perturbative renormalizability is not as strong

as the ideal non-perturbative definition given above: our definition onlyexcludes the theories which obviously don’t converge to a fixed point, butincludes theories which don’t converge to a fixed point for more subtle rea-sons.

2.3. A more refined definition of renormalizability would require ourtheory to be asymptotically free; this idea is explained in this subsection.

Let us consider a theory with a single coupling constant c. Since we arealways making the ! dependence explicit, our coupling constant c dependson !,

c(!) ='

i'0

!ici.

Geometrically, we will think of the coupling constant c as being a map offormal schemes

Spec R[[!]]$ Spec R[c]or equivalently, a section

Spec R[[!]]$ Spec R[c]" Spec R[[!]].

The renormalization group flow then acts on the space of coupling con-stants:

c(!) /$ RGl(c(!)).This action is ! multilinear, in the sense that if we Taylor expand RGl as afunction of c(!), the terms in the Taylor series are ! multilinear maps

R[[!]](k $ R[[!]].



(This multilinearity property follows from the fact that the change of renor-malization scheme maps are !-multilinear.) Geometrically, this means thatRGl is an automorphism

RGl : Spec R[c]" Spec R[[!]] $ Spec R[c]" Spec R[[!]]

compatible with the projection to Spec R[[!]].Since RGlm = RGl -RGm, and since RGl(c(!)) is smooth as a function of

l, there exists a vector field

X = f(c, !)"

"c& R[[!]], R[c]

"

"c

which generates the renormalization group flow RGl. This vector field Xis an infinitesimal automorphism of Spec R[[!]]" Spec R[c] over Spec R[[!]],or, in other words, an !-dependent family of vector fields on the a#ne lineSpec R[c]. The statement that X generates RGl means that

RGl = e(log l)X .

We will say that a theory is asymptotically free, if, to leading order in cand !, the vector field X moves the coupling constant c closer to the originin the a#ne line. This means that if we write

X = f(c, !)"

"c= ckfk(!)

"

"c+ O(ck+1),

then

fk(!) = !l + O(!l+1)

where > 0.What this means is that if we start with a theory with some very small

coupling constant c, then the renormalization group flow will make thiscoupling constant smaller.

In practise, theories are very rarely asymptotically free. For example,Lemma 1.7.1 shows that the renormalization group vector field X for the

4 interaction is, to leading order,

X = c2! ""c

.

Thus, the renormalization group flow takes a small positive coupling con-stant and moves it towards the origin, and a small negative coupling constantand moves it away from the origin. However, our sign conventions mean thatthe physically relevant coupling constant is negative.

Of course, the famous result of Gross-Wilczek (GW73) and Politzer(Pol73) says that Yang-Mills theory is asymptotically free. Although thisbook proves renormalizability of Yang-Mills theory, proof of asymptotic free-dom in our context will have to wait for future work.


3. UNIVERSALITY 125

3. Universality

A key feature of the Kadano!-Wilson picture is the idea of universality.If we are at a KW fixed point, then all small deformations of the theorybecome equivalent to small deformations lying on the unstable manifoldin the large length (low-energy) limit. The theories lying on the unstablemanifold are said to be universal among all theories near the fixed point.

One can ask whether is a similar picture in the perturbative version ofthe Kadano!-Wilson philosophy we are using here. Let us start by makinga precise definition of universality.

Definition 3.0.1. Two theories, given by collections of e!ective inter-actions {I[L]}, {I ![L]}, are in the same universality class if for all L,

RGl (I[L])! RGl

-I ![L]

.$ 0 as l $#.

Thus, given a renormalizable theory, one can ask whether all small de-formations of it are in the same universality class as a renormalizable defor-mation. The strongest results along these lines can be obtained for theorieswhich satisfy an extra condition.

Definition 3.0.2. We say a theory is strongly renormalizable if(1) It is strictly renormalizable, that is, it only contains marginal terms.(2) All renormalizable deformations of the theory are also strictly renor-

malizable. In other words, there are no deformations which intro-duce relevant terms.

The following theorem is very easy to prove1.

Theorem 3.0.3. Any infinitesimal deformation of a strongly renormal-izable theory (to any order) is in the same universality class as a renormal-izable deformation.

Unfortunately, scalar field theories are never strongly renormalizable.However, as we will see later, pure Yang-Mills theory in dimension 4 isstrongly renormalizable. Thus, the quantization constructed later for pureYang-Mills theory is universal.

A weaker result holds for theories which are just strictly renormalizable.

Theorem 3.0.4. Any first-order deformation of a strictly renormaliz-able theory is in the same universality class as a renormalizable first-orderdeformation.

Unfortunately, this is not true for higher order deformations. If we havea first order deformation which involves both relevant and irrelevant terms,then these can combine at higher order to create extra marginal or relevantterms which are not present in any renormalizable deformation.

1When we discuss Yang-Mills theory, we will give a detailed proof of a stronger versionof this theorem.



4. Calculations in 4 theory

In this section, we will calculate explicitly the one-loop counterterms for4 theory, with interaction

I = I0,4 = c14!

!4.

This allows explicit calculations of the local renormalization group flow tothe first non-trivial order (Lemma 1.7.1).

Let us fix a renormalization scheme with the property that the functions$1 and log are purely singular.

Proposition 4.0.5. The one-loop counterterms for the action I are

ICT1,2 ( )( ) = 2$6 $2c $1

!

x#R4(x)2

ICT1,4 ( )( ) = ! $22$8c2 log

!

x#R4(x)4

ICT1,i ( )( ) = 0 if i 8= 2, 4.

The rest of this section will be devoted to the proof of this proposition.

4.1. The first counterterm arises from the graph

It is easy to see that the weight is

w 1(P ( , L), I)( ) = c

! L

l=

!

R4Kl(x, x) (x)2

= c

! L

l=

!

R4(4 l)$2 (x)2

= c1

16 2

- $1 ! L$1. !

R4

2

Recall that we are using a renormalization scheme where $1 is purely sin-gular. Then, the first counterterm is

ICT1,2 ( ) = c

164 2

$1!

x#R4(x)2.

(the extra factor of 14 arises from the automorphism of the graph).


4. CALCULATIONS IN 4 THEORY 127

Because ICT1,2 (l ) is purely singular, this graph does not contribute to the

local renormalization group flow.The next graphs we need to consider are

These graphs contribute with opposite signs. The singularity in the graph2 arises from the loop on the second vertex; this singularity is counteracted

by the counterterm ICT1,2 ( ) in the graph 3. Thus, the sum of contributions

of these two graphs is non-singular, so they do not contribute any newcounterterms.

4.2. The next graph that needs to be considered is

The weight for this graph is

w 4( ) =c2!

l1,l2#[ ,L]

!

x1,x2#R4Kl1(x1, x2)Kl2(x1, x2) (x1)2 (x2)2

=c22$8 $4!

l1,l2#[ ,L]

!

x1,x2#R4

@l$21 l$2

2

e$.x1$x2.2((4l1)$1+(4l2)$1) (x1)2 (x2)2A



Let us perform a change of variables on R4 " R4, by letting

y = 12(x1 + x2)

z = 12(x1 ! x2).

This introduces a Jacobian factor of 24, so that this becomes

w 4( ) = c22$4 $4!

l1,l2#[ ,L]

!

y,z#R4l$21 l$2

2 e$.z.2(l$11 +l$1

2 ) (y!z)2 (y+z)2.

We can get an asymptotic expansion for the integral over z with thehelp of Wick’s lemma. Since we’re ultimately only interested in the singularpart, this asymptotic expansion contains all the information we need.

We find that!

z#R4l$21 l$2

2 e$.z.2(l$11 +l$1

2 ) (y ! z)2 (y + z)2

= 2l$21 l$2

2 (l$11 + l$1

2 )$2

#(y)4 + O

#l1l2

l1 + l2

$$

= 2(l1 + l2)$2

#(y)4 + O

#l1l2

l1 + l2

$$

where O (l1l2/(l1 + l2)) indicates some polynomial in l1l2/(l1 + l2), whereeach term in the polynomial is multiplied by a local action functional of .

The terms like (l1 + l2)$2$klk1 lk2 do not contribute to the singular part,because the integral

!

l1,l2#[0,L](l1 + l2)$2$klk1 lk2dl1dl2

converges absolutely for any k > 0.Thus, we find that

Sing w 4(P ( , L), I)( ) = Sing (f( ))!

y#R4(y)4

where

f( ) = $22$4c2!

l1,l2#[ ,L](l1 + l2)$2dl1dl2

= $22$4c2 (! log 2 + 2 log(L + )! log 2L) .

We will choose a renormalization scheme such that log is purely singular.Then,

Sing f( ) = $22$4c2 log .

The counterterm is

ICT1,4 =

1|Aut 4|

Sing w 4(P ( , L), I)( )

= ! 1|Aut 4|

$22$4c2 log!

y#R4(y)4.


4. CALCULATIONS IN 4 THEORY 129

Now,

|Aut 4| = 2$4

so that

ICT1,4 ( )( ) = !2$8 $2c2 log

!

y#R4(y)4.

4.3. It turns out that all other one-loop counterterms vanish. The cal-culation which proves this is similar to the calculations we have alreadyseen.

We say that a connected graph is one-particle reducible, or 1PR, if it hasa separating edge, that is, an edge which when cut leaves a disconnectedgraph. A graph which is not one-particle reducible is one-particle irreducible,or 1PI. If we cut a one-particle reducible graph along all of its separatingedges, we are left with a disjoint union of one-particle irreducible graphs;these are called the irreducible components of the graph.

One-particle reducible graphs do not contribute to the counterterms,because they are already made non-singular by the counterterms for itsirreducible components. We have already seen this for the graphs 2 and

3 in the calculation above; although the contribution of the graph 2 issingular, this singularity is cancelled out by the contribution of the graph

3. The graph 3 contains the counterterms for the irreducible componentsof 2.

Thus, when calculating the remaining one-loop counterterms, we onlyhave to worry about 1PI graphs. Every one-loop 1PI graph is given by asequence of vertices arranged in a circle, like

The integral attached to each such graph converges absolutely, and so thesegraphs do not contribute to the counterterms. We will see this explicitly for

5; the calculation for other circular graphs is similar.



Note that

w 5(P ( , L), I)( )

= c3!

l1,l2,l3#[ ,L]

!

x1,x2,x3#R4

>Kl1(x2, x3)Kl2(x3, x1)

Kl3(x1, x2) (x1)2 (x2)2 (x3)2?

Since we are only interested in showing that the limit as $ 0 exists, wewill not worry about constants appearing in the formulae.

Plugging the expression

Kl(x, y) = (4 l)$2e$.x$y.2/4l

into the above formula, we see that we need to show that the integral!

l1,l2,l3#[0,L]

!

x1,x2,x3#R4

@(l1l2l3)$2e$.x2$x3.2/4l1$.x3$x1.2/4l2$.x1$x2.2/4l3

(x1)2 (x2)2 (x3)2A

is absolutely convergent.Let us perform the change of variables

z1 = 12 (x2 ! x3)

z2 = 12 (x3 ! x1)

y = 12 (x1 + x2 + x3) .

The integral we need to show is absolutely convergent is!

l1,l2,l3#[0,L]

!

y,z1,z2#R4(l1l2l3)$2e$.z1.2/l1$.z2.2/l2$.z1+z2.2/l3F (y, z1, z2)

where F is a Schwartz function on R4 " R4 " R4. Since F is Schwartz, forall n > 0 there exists a constant Cn such that

F (y, z1, z2) * Cn(1 + +y+)$n

for all y, z1, z2. Thus, it su#ces to show that the integral!

l1,l2#[0,L]

!

z1,z2#R4(l1l2l3)$2e$.z1.2/l1$.z2.2/l2$.z1+z2.2/l3

converges absolutely.Note that!

z1,z2#R4e$.z1.2/l1$.z2.2/l2$.z1+z2.2/l3 = 4(detA)$1/2

where A is the matrix of the quadratic form

+z1+2 /l1 + +z2+2 /l2 + +z1 + z2+2 /l3.

Sincedet A = (l$1

1 l$12 + l$1

1 l$13 + l$1

2 l$13 )4,


5. PROOFS OF THE MAIN THEOREMS 131

we are left with the integral!

l1,l2,l3#[0,L](l1l2l3)$2(l$1

1 l$12 +l$1

1 l$13 +l$1

2 l$13 )$2 =

!

l1,l2,l3#[0,L](l1+l2+l3)$2.

This integral is easily seen to be absolutely convergent.

5. Proofs of the main theorems

5.1. Before we prove the main results of this chapter, we will need alemma about the behaviour of the change of renormalization scheme map.

Let RS0, RS1 be two renormalization schemes. Let

$RS0&RS1 : O+loc(S (Rn))[[!]]$ O+

loc(S (Rn))[[!]]

be the change of renormalization scheme map, defined so that for all I &O+

loc(S (Rn))[[!]] the theory associated to I and RS0 is the same as thetheory associated to $RS0&RS1(I) and RS1.

Lemma 5.1.1. $RS0&RS1(I) is a series of the form

$RS0&RS1(I) = I +'

a>0, b'0

!aPa,b(I)

where:(1) Each Pa,b(I) & Oloc(S (Rn)) is homogeneous of degree b as a func-

tion on S (Rn);(2) Pa,b(I) & Oloc(S (Rn)) is a polynomial in the Ir,s where 2r ! s *

2a! b.Note that the space Oloc(S (Rn)) of translation invariant local

action functionals on S (Rn), homogeneous of degree k, is isomor-phic, as a topological vector space, to R" = 4i#Z%0

R. A polynomialmap R" $ R" is simply an element of

R"[t1, t2, . . .] = 4i#Z%0R[t1, t2, . . .].

Proof. The proof is by induction. As usual, let us write $RS0&RS1(I)as a sum

$RS0&RS1(I) ='

!a$RS0&RS1(I)a,b

where $RS0&RS1(I)a,b is homogeneous of degree b as a function of the field.Let us assume, by induction, that $RS0&RS1(I)a,b is of the desired form

for all (a, b) < (A,B).Let us write SingRS0 and SingRS1 to denote the operator which picks

out the singular part of a function of using the renormalization schemesRS0 and RS1 respectively. In order to keep the notation short, let us write:Ia,b for $RS0&RS1(I)a,b. Let :ICT

a,b ( ) refer to the counterterms for :Ia,b, takenwith respect to the renormalization scheme RS1. Similarly, let ICT

a,b ( ) referto the counterterms for Ia,b taken with respect to the renormalization schemeRS1.



Let us assume, by induction, that each :ICTa,b ( ) and ICT

a,b ( ) is a polynomialin I(r,s) for 2r ! s * 2a! b.

Then, observe that

:ICTA,B( ) = SingRS1 WA,B

5

6P ( , L),'

(a,b)<(A,B)

!a:Ia,b ! !a:ICTa,b ( )

7

8

Note that, in a stable graph of genus A with B tails, only vertices of type(r, s) can appear if 2r ! s * 2A!B. It follows that we can write

:ICTA,B( ) = SingRS1 WA,B

5

6P ( , L),'

2a$b%2A$B


7

8 .

Since everything on the right hand side of this expression is a polynomial inIa,b where 2a! b * 2A!B, it follows that :ICT

A,B( ) is such a polynomial.In a similar way, ICT

A,B( ) is a polynomial in Ia,b for 2a! b * 2A!B.By definition, we have

:IA,B + lim&0

5

6WA,B

5

6P ( , L),'

(a,b)<(A,B)


7

8! :ICTA,B( )

7

8

= IA,B+lim&0

5

6WA,B

5

6P ( , L),'

(a,b)<(A,B)

!aIa,b ! !aICTa,b ( )

7

8! ICTA,B( )

7

8 .

Thus,

:IA,B = IA,B

! lim&0

5

6WA,B

5

6P ( , L),'

(a,b)<(A,B)


7

8! :ICTA,B( )

7

8

+ lim&0

5

6WA,B

5

6P ( , L),'

(a,b)<(A,B)

!aIa,b ! !aICTa,b ( )

7

8! ICTA,B( )

7

8 .

Everything on the right hand side is a polynomial in Ia,b for 2a!b * 2A!B;thus, :IA,B is such a polynomial, as desired.

!

Next, we will prove Proposition 1.3.2.If I & O+

loc(S (Rn))[[!]] is a local action functional, then the countertermsICT ( ) depend in a polynomial way on I. Thus, the e!ective interactionsWR (P (0, L), I) depend in a polynomial way on I. It follows that to proveProposition 1.3.2, it su#ces to prove the following.


5. PROOFS OF THE MAIN THEOREMS 133

Lemma 5.1.2. For any I & O+loc(S (Rn))[[!]],

RGl(I) & O+loc(S (Rn))[[!]], C[l, l$1, log l].

In other words, for each (i, k), RGl(Ii,k) is a polynomial in l, l$1 and log lwhen considered as a function of l.

Proof. Recall that the family of theories RGl({I[L]}) is a smooth familyof theories parameterized by l, in the sense of Definition 3.1.1 of Chapter 3.It follows from this and from Theorem 3 of Chapter 3 that the local actionfunctionals RGl(I) are a smooth family parameterized by l, that is,

RGl(I) & Oloc(S (Rn), C"((0,#)l)).

Recall thatRGl = $l,0 -R+

l

where $l,0 is the change of renormalization scheme map.We will prove the result by the usual inductive argument. Fix some

I & O+loc(S (Rn))[[!]]. Let us fix (a, b) & Z>0 " Z'0, and let us assume, by

induction, that for all (r, s) < (a, b).

RGl(I)r,s & Oloc(S (Rn)), C[l, l$1, log l].

The operator $l,0 is “upper triangular”, as proved in Theorem 5.1.1 inChapter 2. We have

RGml(I) = RGm (RGl(I)) = $m,0R+m (RGl(I)) .

It follows that, for all m, l & R>0, we have a di!erence equation of the form

RGml(I)a,b = R+mRGl(I)a,b + J

whereJ & Oloc(S (Rn)), C[l, l$1, log l].

We are using the induction assumption to specify the form of the dependenceof J on l. Of course, J is homogeneous of degree a in the field . Also, Jdepends on m, but this dependence is suppressed.

Let us writeRGl(I)a,b =

'

c

RGl(I)ca,b

where RGl(I)ca,b is of dimension c & 1

2Z. Then, we find that

(‡) RGml(I)ca,b = mcRGl(I)c

a,b + Jc

for some Jc & Oloc,c(S (Rn)), C[l, l$1, log l] of dimension c.This is the point at which we use the fact that RGl(I) is smooth as

a function of l. Because of this, the di!erence equation (‡) for RGl(I)ca,b

implies that RGl(I)ca,b satisfies a di!erential equation of the form

l"

"lRGl(I)c

a,b = cRGl(I)ca,b + :Jc

for some :Jc & Oloc,c(S (Rn)), C[l, l$1, log l] of dimension c.



Any solution of this di!erential equation is an element Oloc,c(S (Rn)),C[l, l$1, log l].

!

Next, we can prove Theorem 1.5.1. The following is an equivalent state-ment of the theorem.

Theorem 5.1.3. Let

I & O+loc,0(S (Rn))[[!]]

be any local action functional of dimension 0. Then, there exists a unique

:I & !Oloc,2=0(S (Rn))[[!]]

which is a sum of terms of non-zero dimension, such that the theory associ-ated to I + :I is marginal, that is, an element of M (").

Similarly, for each local action functional

I & O+loc,'0(S (Rn))[[!]]

which is a sum of terms of non-negative dimension, there exists a unique

:I & !Oloc,2=0(S (Rn))[[!]]

which is a sum of terms of negative dimension, such that the theory associ-ated to I + :I is relevant, that is, an element of R(").

Proof. We will prove the first statement; the proof of the second state-ment is essentially the same.

We need to show that there is a unique :I which is a sum of terms ofnon-zero dimension such that

RGl(I + :I) & O+loc(S (Rn))[[!]], C[log l].

The proof is similar to the proof of Lemma 5.1.2. Let us assume induc-tively that there are unique :Ir,s for all (r, s) < (a, b) such that:

(1) Each :Ir,s is translation invariant, homogeneous of degree r in thefield & S (Rn), and is a sum of terms of non-zero dimension.

(2) Let:I<(a,b) =

'

c

'

(r,s)<(a,b)

!r:Ir,s.

Then,

RGl

1I + :I<(a,b)

2

r,s& O+

loc(S (Rn))[[!]], C[log l]

for all (r, s) < (a, b).


6. GENERALIZATIONS OF THE MAIN THEOREMS 135

Let RGl

1I + :I<(a,b)

2c

a,bdenote the part of dimension c & 1

2Z. As in the

proof of the previous lemma, we can see inductively that RGl

1I + :I<(a,b)

2c

a,bsatisfies a di!erential equation of the form

l"

"lRGl

1I + :I<(a,b)

2c

a,b= cRGl

1I + :I<(a,b)

2c

a,b+ :Jc

for some :Jc & Oloc,c(S (Rn)), C[log l] of dimension c.If c = 0, we see immediately that

RGl

1I + :I<(a,b)

2c

a,b& Oloc,0(S (Rn)), C[log l]

varies only logarithmically with l, as desired.If c 8= 0, this di!erential equation implies that

RGl

1I + :I<(a,b)

2c

a,b= lcH + H !

for some H & Oloc,c(S (Rn)) and H ! & Oloc,c(S (Rn)) , C[log l], both ofconformal weight c.

Now set:Ica,b = !H.

Then,

RGl

1I + :I<(a,b) + !a:Ic

a,b

2c

a,b= RGl

1I + :I<(a,b)

2c

a,b+ !alc:Ic

a,b

= RGl

1I + :I<(a,b)

2c

a,b! !alcH

= H !

& Oloc,c(S (Rn)), C[log l]

as desired.We do this for all c; for all but finitely many c, we find :Ic

a,b = 0. Thus,we can set

:Ia,b ='

c2=0

:Ica,b

and we can continue the induction. !

6. Generalizations of the main theorems

The main results of this chapter hold in a more general context. Thisgeneralization will be needed when we consider Yang-Mills theory in Chapter6.

In this section, everything will depend on some auxiliary manifold withcorners X, equipped with a sheaf A of commutative superalgebras over thesheaf of algebras C"

X . The space of global sections of A will be denoted, asbefore, by A .

Let us assume we are in the situation of Chapter 3, Subsection 3. Thus,we have



(1) A finite dimensional super vector space E. Let

E = E ,S (Rn).

Thinking of E as a trivial vector bundle on Rn, E is the space ofSchwartz sections of E.

(2) An even symmetric element

Kl & C"(Rn " Rn), E(2 , C"((0,#)l),A ,

playing the role of the heat kernel. We assume that in some basisei of E, Kl can be written

Kl ='

Pi,j(x! y, l$1/2)e$.x$y.2/lei , ej

where thePi,j & A

Bx! y, l±1/2

C

are polynomials in the variables xk!yk and l±1/2, with coe#cientsin A .

Then, we can write the propagator as

P ( , L) =! L

l=Kldl

as an integral of the kernel Kl.Let us further assume that E is equipped with a direct sum decomposi-

tionE =

D

i#12Z

Ei

into super vector spaces. Ei is the space of elements of E of dimension i.The direct sum decomposition on E induces an R>0 action on E =

S (Rn),E, byRl (f(x)ei) = f(lx)liei

where f(x) & S (Rn) and ei & Ei.We require that the propagator scales as

Rl(P ( , L)) = P (m$2 ,m$2L).

As explained in Chapter 3, Subsection 3, we have a notion of theorydefined using this propagator; let T (")(E , A ) denote the set of theories,and let T (n)(E , A ) denote the set of theories defined modulo !n+1. Thechoice of renormalization scheme leads to a bijection

T (")(E , A ) %= O+,evloc (E , A )[[!]]

between theories and local action functionals I & Oloc(E )[[!]] which areeven and at least cubic modulo !. As always, both theories and local actionfunctionals are translation invariant.

There is an action

RGl : T (")(E , A )$ T (")(E , A )


6. GENERALIZATIONS OF THE MAIN THEOREMS 137

of the local renormalization group on the space of theories. And, just as inthe case of scalar field theories, we have:

Proposition 6.0.4. For any theory {I[L]} &-T (")(E ,A )

.,

RGl(I[L]) & O+(E , A )[[!]], C[l, l$1, log l].

Then, we define the spaces R(")(E , A ) of relevant theories to be thespace of those theories such that

RGl(I[L]) & O+(E , A )[[!]], C[l, log l].

Similarly, the space M (")(E , A ) of marginal theories is defined to be thespace of theories such that

RGl(I[L]) & O+(E , A )[[!]], C[log l].

LetOloc,k(E , A ) ) Oloc(E ,A )

be the space of local action functionals which are of dimension k & 12Z.

Then, there are projection maps

Oloc(E , A )$ Oloc,0(E , A )Oloc(E , A )$ Oloc,'0(E , A )

onto the space of local action functionals of dimension zero and the spaceof local action functionals which are sums of terms of non-zero dimension.

The main theorem is analogous to the previous theorem:

Theorem 6.0.5. The space R(n)(E , A ) is a torsor over R(n$1)(E , A )for the abelian group Ol,'0(E ,A ) of local action functionals of non-negativedimension. The space R(0)(E , A ) is canonically isomorphic to the subspaceof Ol,'0(E ,A ) of functionals which are at least cubic.

In a similar way, M (n)(E , A ) is a torsor over M (n$1)(E , A ) for theabelian group Ol,0(E , A ) of local action functionals of dimension zero, andM (0)(E ,A ) is canonically isomorphic to the subspace of Ol,0(E , A ) whichare sums of terms which are at least cubic.



CHAPTER 5

Gauge symmetry and the Batalin-Vilkoviskyformalism

1. Introduction

1.1. The philosophy of this book is that everything one says about aquantum field theory should be said in terms of its e!ective theories. Thebijection between theories and local action functionals allows one to trans-late statements about e!ective theories into statements about local actionfunctionals.

We would like to apply this philosophy to understand gauge theories.Naively, one could imagine that to give a gauge theory would be to givean e!ective gauge theory at every energy level, in a way related by therenormalization group flow.

One immediate problem with this idea is that the space of low-energygauge symmetries is not a group. The product of low-energy gauge sym-metries is no longer low-energy; and if we project this product onto itslow-energy part, the resulting multiplication on the set of low-energy gaugesymmetries is not associative.

For example, if g is a Lie algebra, then the Lie algebra of infinitesimalgauge symmetries on a manifold M is C"(M), g. The space of low-energyinfinitesimal gauge symmetries is then C"(M)<! , g where we use thetraditional Wilsonian cut-o! – C"(M)<! is the direct sum of all eigenspacesof the Laplacian with eigenvalue less than ". In general, the product of twofunctions in C"(M)<! can have arbitrary energy; so that C"(M)<!, g isnot closed under the Lie bracket.

This problem is solved by a very natural union of the Batalin-Vilkoviskyformalism and the e!ective interaction philosophy. This e!ective BV for-malism is the second main theme of this book.

1.2. The Batalin-Vilkovisky formalism is widely regarded as being themost powerful and general way to quantize gauge theories. The first stepin the BV procedure is to introduce extra fields – ghosts, correspondingto infinitesimal gauge symmetries; anti-fields dual to fields; and anti-ghostsdual to ghosts – and then write down an extended action on this extendedspace of fields. This extended action encodes the original action, the Liebracket on the space of infinitesimal gauge symmetries, and the way thisLie algebra acts on the original space of fields. This action is supposed to

139


140 5. GAUGE SYMMETRY AND THE BATALIN-VILKOVISKY FORMALISM

satisfy the quantum master equation, which is a succinct way of encodingthe following conditions:

(1) The Lie bracket on the space of infinitesimal gauge symmetriessatisfies the Jacobi identity.

(2) This Lie algebra acts in a way preserving the action functional onthe space of fields.

(3) The Lie algebra of infinitesimal gauge symmetries preserves the“Lebesgue measure” on the original space of fields. That is, thevector field on the original space of fields associated to every infin-itesimal gauge symmetry is divergence-free.

(4) The adjoint action of the Lie algebra on itself also preserves the“Lebesgue measure”. Again, this says that the vector field associ-ated to every infinitesimal gauge symmetry is divergence-free.

Unfortunately, the quantum master equation is an ill-defined expression.The 3rd and 4th conditions above are the source of the problem: the diver-gence of a vector field on the space of fields is a singular expression, involvingthe same kind of singularities as those appearing in one-loop Feynman dia-grams.

1.3. This form of the quantum master equation violates the philosophyof this book: we should always express things in terms of the e!ective inter-actions. The QME above is about the original infinite energy / length scalezero action, so we should not be surprised that it doesn’t make sense.

The solution to this problem is to combine the BV formalism with thee!ective interaction philosophy. With Wilson’s sharp energy cut-o!, thisworks as follows. We say that to give an e!ective interaction in the BVformalism is to give a functional S["] on the space of extended fields (ghosts,fields, anti-fields and anti-ghosts) of energy less than ". This energy "e!ective action must satisfy a certain energy " quantum master equation.

The reason that the e!ective action philosophy and the BV formalismwork well together is the following.

Lemma. The renormalization group flow from scale " to scale "! carriessolutions of the energy " QME into solutions of the energy "! QME.

Thus, to give a gauge theory in the e!ective BV formalism is to give acollection of e!ective actions S["] for each ", such that S["] satisfies thescale " QME, and such that S["!] is obtained from S["] by the renormal-ization group flow.

This picture also solves the problem that the low energy gauge symme-tries are not a group. The energy " e!ective action S["], satisfying theenergy " quantum master equation, gives the extended space of low-energyfields a certain homotopical algebraic structure, which has the following in-terpretation:

(1) The space of low-energy infinitesimal gauge symmetries has a Liebracket.


2. A CRASH COURSE IN THE BATALIN-VILKOVISKY FORMALISM 141

(2) This Lie algebra acts on the space of low-energy fields.(3) The space of low-energy fields has a functional, invariant under the

bracket.(4) The action of the Lie algebra on the space of fields, and on itself,

preserves the Lebesgue measure.However, these axioms don’t hold on the nose, but hold up to a sequence ofcoherent higher homotopies.

In this book, of course, we prefer to use the proper-time cut-o!, basedon the heat kernel. Thus, an e!ective BV theory is given by a set of e!ectiveinteractions {I[L]} on the space of extended fields, such that

(1) Each I[L] satisfies the scale L quantum master equation.(2) The various I[L] are related by the renormalization group equation.(3) As L$ 0, I[L] is asymptotically local in the same sense as before.

1.4. Renormalizing gauge theories. It is straightforward to gener-alize the Wilsonian definition of renormalizability given in Chapter 4 toe!ective BV theories. Thus, to give a renormalizable theory on Rn in theBV formalism is to give a collection {I[L]} of e!ective interactions, satisfy-ing the quantum master equation and the renormalization group equationas above, such that

(1) As L$ 0, I[L] grows at most logarithmically in L, when measuredin units appropriate to length scale L.

(2) Term by term in !, the theory has only finitely many deformationssatisfying the first condition. 1

This chapter provides the foundations for the renormalized BV formal-ism, and gives a general homological criterion which is su#cient for renor-malizability. In Chapter 6, we will use these results to prove:

Theorem. Pure Yang-Mills theory on R4, or on any compact four-manifold with a flat metric, with coe"cients in any semi-simple Lie algebrag, is perturbatively renormalizable.

This chapter makes extensive use of homological algebra; an excellentreference for this material is the book-in-progress (KS).

2. A crash course in the Batalin-Vilkovisky formalism

In this section I will give an informal overview of the Batalin-Vilkoviskyapproach to quantizing gauge theories. I will present two di!erent pointsof view on the BV formalism: one where the BV formalism is viewed as amachine for understanding integrals, and a second where the BV formalismis interpreted from the point of view of derived geometry.

1It is more natural to weaken this axiom to require that there are only finitely manydeformations up to Batalin-Vilkovisky equivalence. Roughly, two theories in the BV for-malism are equivalent if they are related by a local symplectic change of coordinates onthe space of fields. A precise definition along these lines is given later.



2.1. The BV construction and integrals. In this section, our vectorspaces and manifolds will always be finite dimensional. Of course, none ofthe di#culties of renormalisation are present in this simple case. Many ofthe expressions we write in the finite dimensional case are ill-defined in theinfinite dimensional case.

Let us suppose that we have a finite dimensional vector space V of fields,which will always be treated as a formal manifold near 0 & V . Let g be aLie group acting on V , in a possibly non-linear way. Since V is formal, thisaction integrates to an action of the formal group G associated to g.

Let O(V ) = 0Sym+V ) denote the algebra of formal power series on V .

Let f & O(V ) be a G-invariant function, with the property that 0 & V is acritical point of f .

We will think of this data as defining a finite-dimensional classical gaugetheory, with gauge group G, space of fields V and action functional f .

One is interested in making sense of functional integrals of the form!

V/Gef/!

over the quotient space V/G. The starting point in the Batalin-Vilkoviskyformalism is the BRST construction, which says one should try to interpretthis quotient in a homological fashion.

Functions on the quotient V/G are the same as G-invariant functions onV , or, equivalently, g-invariant functions on V . In homological algebra, it isalways better to take the derived invariants, rather than the naive invariants.The derived invariants for the action of g on the algebra O(V ) of functionson V is the Chevalley-Eilenberg complex

C+(g, O(V )).

Thus, we can define the derived quotient of V be G to be the “spectrum” ofC+(g,O(V )); that is, the object whose algebra of functions is C+(g, O(V )).

As a graded vector space,

C+(g, O(V )) = 0Sym+(g[1]4 V )

(where symmetric powers are taken, as always, in the graded sense, and [1]refers to a change of degree, so g is in degree !1.).

Thus, we can identify the derived quotient of V by G with a di!erentialgraded manifold, whose underlying graded manifold is

g[1]4 V.

The Chevalley-Eilenberg di!erential can be thought of as a degree 1 vectorfield of square 0 on g[1]4 V . Let us denote this vector field by X.

The BRST construction says one should replace the integral over V/Gby an integral over this derived quotient, that is, an integral the form

!

g[1]/Vef/!.



2.2. This leaves us in a better situation than before, as the derivedquotient g[1] 4 V is a smooth formal dg manifold, and we can attempt tomake sense of the integral perturbatively. However, we still have problems;the quadratic part of the functional f is highly degenerate on g[1] 4 V .Indeed, f is independent of g[1] and is constant on G-orbits on V . Thus, wecannot compute the integral above by a perturbation expansion around thecritical points of the quadratic part of f .

This is where the Batalin-Vilkovisky formalism comes in. Let E denotethe shifted cotangent bundle of T +[!1](g[1]4 V ), so that

E = g[1]4 V 4 V )[!1]4 g)[!2].

The various summands of E are usually given the following names: g[1] is thespace of ghosts, V is the space of fields, V )[!1] is the space of antifields andg)[!2] is the space of antighosts. (We will see shortly that these additionalfields also have a natural interpretation from the point of view of derivedgeometry).

The function f on g[1] 4 V pulls back to a function on E, via the pro-jection E $ g[1]4V ; we continue to call this function f . By naturality, thevector field X on g[1] 4 V induces a vector field on E, which we continueto call X. As [X,X] = 0 on g[1]4 V , the same identity holds on E. As Xpreserves f on g[1]4 V , it continues to preserve f on E.

E is a dg manifold equipped with a symplectic form of degree !1, and Xis a degree 1 vector field preserving the symplectic form. Thus, there existsa unique function hX on E whose Hamiltonian vector field is X, and whichvanishes at the origin in E. As X is of degree 1 and the symplectic pairingis of degree !1, the function hX is of cohomological degree 0.

As E has a symplectic form of cohomological degree !1, the space offunctions on E has a Poisson bracket of cohomological degree 1. The state-ment that [X,X] = 0 translates into the equation {hX , hX} = 0. Thestatement that Xf = 0 becomes {hX , f} = 0. And, as f is pulled back fromg[1]4 V , it automatically satisfies {f, f} = 0. These identities together tellus that the function f + hX satisfies the Batalin-Vilkovisky classical masterequation,

{f + hX , f + hX} = 0.

The function f + hX is the BV action

SBV (e) = f(e) + hX(e).

Let us split SBV into “kinetic” and “interacting” parts, by

SBV (e) =12'e,Qe(+ IBV (e)

where Q : V $ V is a cohomological degree 1 linear map, skew self-adjointfor the degree !1 pairing '!,!(, and IBV is a function which is at leastcubic. The fact that SBV satisfies the classical master equation implies that



Q2 = 0. Also, the identity

QIBV +12{IBV , IBV } = 0

holds as a consequence of the classical master equation for f + hX .Let L ) E be a small, generic, Lagrangian perturbation of the zero

section g[1]4 V ) E. The Batalin-Vilkovisky formalism tells us to considerthe functional integral

!

e#Lexp(SBV (e)/!) =

!

e#Lexp(

12! 'e,Qe(+

1!IBV (e))

As L is generic, the pairing 'e,Qe( will have very little degeneracy on L. Infact, if the complex (E,Q) has zero cohomology, then the pairing 'e,Qe( isnon-degenerate on a generic Lagrangian L. This means we can perform theabove integral perturbatively, around the critical point 0 & L.

2.3. The BV construction is the derived critical scheme. In thissubsection we will describe how the BV complex, as described above, can beinterpreted as the derived critical scheme of the function f on the derivedquotient of V by G. In my opinion, this is the best way to interpret the BVformalism; this point of view is developed extensively in (CG10).

As before, let V be a vector space, viewed as a formal manifold completednear the origin. Let g be a Lie algebra, with a possibly non-linear action onV (defined in a formal neighbourhood of the origin). Let G be the formalgroup associated to g; G also acts on the formal manifold V . Let f be ag-invariant function on V .

The BV construction, as described above, associates to this data a formaldi!erential graded supermanifold

E = g[1]4 V 4 V )[!1]4 g)[!2].

The di!erential on the algebra O(E) of formal power series on E is givenby bracketing with the BV action SBV .

In this subsection we will show how we can interpret this di!erentialgraded supermanifold as the derived critical scheme for the function f onthe derived quotient of V by g.

In general, if we have a scheme M with a function f on M , the criticalscheme of f is simply the zero locus of the one-form df . The derived criticalscheme of f is the di!erential graded manifold obtained by forming a Koszulresolution of the equation df = 0.

Explicitly, the derived critical scheme of f has, as underlying gradedmanifold, the shifted cotangent bundle T +[!1]M of M . The algebra offunctions on this space is

C"(T +[!1]M) = '(M,9+TX).

The di!erential on this algebra of functions which yields the derived criticalscheme of f is given by contracting with the one-form df .



It is clear that H0(C"(T +[!1]M),df:) is the ring of functions on thecritical locus of f (that is, the Jacobi ring of f).

We will see that when we apply this construction to the derived quotientof V by g, we find the BV dg manifold E.

As we have seen above, the derived quotient of V by g is the dg manifoldg[1] 4 V . The algebra of functions on this, O(g[1] 4 V ), has a Chevalley-Eilenberg di!erential which we have been denoting X.

The function f & O(g[1] 4 V ) is X-invariant, that is, closed for thedi!erential. Thus, we can form the derived critical scheme for the functionf ; we find the dg manifold T +[!1](g[1] 4 V ). The di!erential on this dgmanifold has two terms; one arising from the di!erential X on the baseg[1]4 V , and the other from contracting with the one-form df .

It is straightforward to check that

O(T +[!1](g[1]4 V )) = O(E)

where the di!erential on the left hand side comes from the derived criticalscheme construction, and that on the right hand side from bracketing withthe BV action function SBV .

2.4. The BV construction as symplectic reduction. Yet anotherpoint of view on the BV construction is to interpret it as being obtained bya derived graded symplectic reduction from the derived critical scheme ofthe function f on V .

The derived critical scheme of f is the dg graded symplectic manifold(with symplectic form of degree !1)

T +[!1]V = V 4 V )[!1],

with di!erential arising from contracting with the one-form df .The Lie algebra g acts on the dg graded symplectic manifold T +[!1]V ,

preserving the di!erential. The claim is that the dg graded symplectic man-ifold E constructed above is the homotopical symplectic reduction of the dggraded symplectic manifold T +[!1]V by g.

Symplectic reduction has two steps: first, we set the moment map tozero; and then, we form the quotient.

There is a map of dg Lie algebras from g to the dg Lie algebra of sym-plectic vector fields on T +[!1]V . Since symplectic vector fields correspondto Hamiltonian functions, we get a map from g to the Lie algebra of Hamil-tonian functions on T +[!1]V of degree !1 which vanish at the origin. Thismap can be viewed as a map of dg Lie algebras with bracket of degree +1,

g[1]$ O(T +[!1]V ).

This map induces a map of commutative algebras

0Sym+(g[1]) = O(g)[!1]) $ O(T +[!1]V ).



This map is in fact a map of graded Poisson algebras (with Poisson bracketof degree 1). The map of formal dg manifolds

m : T +[!1]V $ g)[!1]

is the moment map.The first step in the homotopical symplectic reduction is to take the

homotopy fibre of this moment map over 0 & g)[!1]. This homotopy fibrem$1(0) can be defined by a Koszul complex:

O(m$1(0)) = O(T +[!1]V ),LO(g'[$1]) C

. O(T +[!1]V 4 g)[!2])

with a certain Koszul di!erential.The next step in the symplectic reduction procedure is to take the quo-

tient of the homotopy fibre m$1(0) of the moment map by the action ofg. The quotient is taken in the homotopical sense, yielding the Chevalley-Eilenberg complex

R HomUg(C, O(T +[!1]V 4 g)[!2]) = O(g[1]4 T +[!1]V 4 g)[!2]).

The formal dg graded symplectic manifold g[1] 4 T +[!1]V 4 g)[!2], withdi!erential arising from the symplectic reduction procedure described above,is the same as the formal dg graded symplectic manifold

E = g[1]4 V 4 V )[!1]4 g)[!2]

with di!erential arising from the action SBV .

2.5. Quantum master equation. Let us now turn to a more generalsituation, where E is a finite dimensional vector space with a symplecticpairing of cohomological degree !1 and Q : E $ E is a operator of squarezero and degree 1, which is skew self-adjoint for the pairing. E is not nec-essarily of the form constructed above.

The symplectic pairing gives an isomorphism

E[!1] %= E).

The symplectic two-form is an element

& 92E)

of cohomological degree !1. Note that

92E) %=-Sym2 E

.[!2].

Under this isomorphism, the symplectic form becomes an element) & Sym2 E

of cohomological degree 1.Let

) : O(E) $ O(E)



be the linear order two di!erential operator corresponding to ). Thus, )is the unique order two di!erential operator which vanishes on Sym%1 E),and which, on Sym2 E), is given by pairing with ).

Let I & O(E)[[!]] be an !-dependent function on E, which modulo ! isat least cubic. The function I satisfies the quantum master equation if

(Q + !))eI/! = 0.

This equation is equivalent to the equation

QI +12{I, I} + !)I = 0.

The key lemma in the Batalin-Vilkovisky formalism is the following.

Lemma 2.5.1. Let L ) E be a Lagrangian on which the pairing 'e,Qe(is non-degenerate. (Such a Lagrangian exists if and only if H+(E,Q) = 0.)Suppose that I satisfies the quantum master equation. Then the integral

!

e#Lexp(

12! 'e, Qe(+

1!I(e))

is unchanged under deformations of L.

The non-degeneracy of the inner product on L, and the fact that I is atleast cubic modulo !, means that one can compute this integral perturba-tively.

Suppose E,Q, ' , ( , I are obtained as before from a gauge theory. ThenI automatically satisfies the classical master equation QI + 1

2{I, I} = 0. If,in addition, )I = 0, then I satisfies the quantum master equation. Thus,we see that we can quantize the gauge theory in a way independent of thechoice of L as long as I satisfies the equation )I = 0. When I does notsatisfy this equation, one looks to replace I by a series I ! = I +

*i>0 !iIi

which does satisfy the quantum master equation QI ! + 12{I

!, I !}+!)I ! = 0.

2.6. Geometric interpretation of the quantum master equation.The quantum master equation has a geometric interpretation, first describedby Albert Schwarz (Sch93). I will give a very brief summary; the readershould refer to this paper for more details.

Let denote the unique up to scale translation invariant “measure” onE, that is, section of the Berezinian. The operator ) can be interpretedas a kind of divergence associated to the measure , as follows. As E hasa symplectic form of degree !1, the algebra O(E) has a Poisson bracketof degree 1. Every function I & O(E) has an associated vector field XI ,defined by the formula XIf = {I, f}.

The operator ) satisfies the identity

LXI = ()I)

where LXI refers to the Lie derivative. In other words, )I is the infinitesimalchange in volume associated to the vector field XI .



Thus, the two equations

{I, I} = 0)I = 0

say that the vector field XI has square zero and is measure preserving.This gives an interpretation of the quantum master equation in the case

when I is independent of !. When I depends on !, the two terms of thequantum master equation do not necessarily hold independently. In thissituation, we can interpret the quantum master equation as follows. Let I

be the measure on E defined by the formula

I = eI/! .

We can define an operator )I on O(E) by the formula

LXf I = ()If) I .

This is the divergence operator associated to the measure I , in the sameway that ) is the divergence operator associated to the translation invariantmeasure .

Then, a slightly weaker version of the quantum master equation is equiv-alent to the statement

)2I = 0.

Indeed, one can compute that

!)If = {I, f} + !)f

so that!2)2

If = 12{{I, I}, f} + !{)I, f}.

Thus, )2I = 0 if and only if 1

2{I, I} + !)I is in the centre of the Poissonbracket, that is, is constant.

This discussion shows that the quantum master equation is the state-ment that the measure eI/! is compatible in a certain sense with the gradedsymplectic structure on E.

Remark. In fact it is better to use half-densities rather than densitiesin this picture. A solution of the quantum master equation is then givenby a half-density which is compatible in a certain sense with the gradedsymplectic form. As all of our graded symplectic manifolds are linear, wecan ignore this subtlety.

2.7. Integrating over isotropic subspaces. As we have describedit, the BV formalism only has a chance to work when H+(E,Q) = 0. Thisis because one cannot make sense of the relevant integrals perturbativelyotherwise. However, there is a generalisation of the BV formalism whichworks when H+(E,Q) 8= 0. In this situation, let L ) E be an isotropicsubspace such that Q : L $ Im Q is an isomorphism. Let Ann(L) ) E be



the set of vectors which pair to zero with any element of L. Then we canidentify

H+(E,Q) = Ann(L) 3Ker Q.

We thus have a direct sum decomposition

E = L4H+(E,Q)4 Im Q.

Note that H+(E,Q) acquires a degree !1 symplectic pairing from that on E.Thus, there is a BV operator )H((E,Q) on functions on H+(E,Q). We say afunction f on H+(E,Q) satisfies the quantum master equation if )ef/! = 0.

The analog of the “key lemma” of the Batalin-Vilkovisky formalism is thefollowing. This lemma is well known to experts in the area (Mne09; KL09).

Lemma 2.7.1. Let I & O(E)[[!]] be an !-dependent function on E whichsatisfies the quantum master equation. Then the function on H+(E,Q) de-fined by

a /$ ! log#!

e#Lexp(

12! 'e,Qe(+

1!I(e + a))

$

(where we think of H+(E,Q) as a subspace of E) satisfies the quantum mas-ter equation.

Further, if we perturb the isotropic subspace L a small amount, then thissolution of the QME on H+(E,Q) is changed to a homotopic solution of theQME.

Note that this integral is the renormalization group flow for the e!ectiveinteraction.

This integral is an explicit way of writing the homological perturbationlemma for BV algebras, which transfers a solution of the quantum masterequation at chain level to a corresponding solution on cohomology. Fromthis observation it’s clear (at least philosophically) why the lemma shouldbe true; the choice of the Lagrangian L is essentially the same as the choiceof symplectic homotopy equivalence between E and its cohomology.

I need to explain what a “homotopy” of a solution of the QME is.There is a general concept of homotopy equivalence of algebraic objects,which I learned from the work of Deligne, Gri#ths, Morgan and Sullivan(DGMS75). Two algebraic objects are homotopic if they are connected bya family of such objects parametrised by the commutative di!erential alge-bra &+([0, 1]). In our context, this means that two solutions f0, f1 to thequantum master equation on H+(E,Q) are homotopic if there exists an ele-ment F & O(H+(E,Q)), &+([0, 1])[[!]] which satisfies the quantum masterequation

(dDR + !)H((E,Q))eF/! = 0

and which restricts to f0 and f1 when we evaluate at 0 and 1. Here dDR

refers to the de Rham di!erential on &+([0, 1]).



The quantum master equation imposed on F is equivalent to the equa-tion

dDRF +12{F, F} + !)F = 0.

If we write F (t,dt) = A(t) + dtB(t), then the QME imposed on F becomesthe system of equations

12{A(t), A(t)} + !)A(t) = 0

ddt

A(t) + {A(t), B(t)} + !)B(t) = 0.

The first equation says that A(t) satisfies the ordinary QME for all t, andthe second says that the family A(t) is tangent at every point to an orbit ofa certain “gauge group” acting on the space of solutions to the QME.

2.8. Operadic interpretation of the quantum master equation.In this subsection we will show how solving the quantum master equation canbe interpreted as solving a certain algebraic quantization problem, expressedin the language of operads. This point of view will not be considered in theremainder of the book; thus, I will not give all the details. However, thisoperadic perspective is developed extensively in (CG10).

Let (E,Q) be a cochain complex with a symplectic pairing of degree!1, as above. The algebra O(E) of formal power series on E has a Pois-son bracket {!,!} of cohomological degree 1, derived from the symplecticpairing.

Definition 2.8.1. A P0 algebra is a commutative di!erential gradedequipped with a Poisson bracket of cohomological degree 1. This Poissonbracket must be a derivation in each factor, satisfy the graded Jacobi identity,and must be compatible with the di!erential.

We will let P0 denote the operad (in the category of cochain complexes)whose algebras are P0 algebras.

Remark. In general, a Pk algebra is a commutative di!erential gradedalgebra with a Poisson bracket of degree 1!k. Thus, P1 algebras are ordinaryPoisson algebras. The reason for the terminology is that Pk algebras areclosely related to Ek algebras.

The algebra O(E), with di!erential Q and Poisson bracket arising fromthe symplectic form, is a P0 algebra.

If I & O(E) solves the quantum master equation, we will let

QI = Q + {I,!} : O(E) $ O(E)

is a derivation for both the commutative and Poisson brackets. The classicalmaster equation implies that Q2

I = 0. Thus, (O(E), QI) defines a P0 algebra.



Definition 2.8.2. A BD algebra2 is a flat R[[!]] module A, equipped witha commutative product, a Poisson bracket of degree 1, and a di!erential d,all of which are linear over !; such that

d(a · b) = (da) · b + (!1)|a|a · db + !{a, b}.A BD algebra is not a di!erential graded algebra, because the di!erential

is not a derivation for the commutative product. However, the failure of thedi!erential to be a derivation is measured by the Poisson bracket.

There is an operad BD, in the category of cochain complexes over thering R[[!]], whose algebras are BD algebras. As a graded operad, BD issimply

BD = P0 , R[[!]].The di!erential on BD is defined by

d (! 0 !) = !{!,!}where (! 0 !) , !{!,!} & P0(2) denote the product and bracket.

Definition 2.8.3. A quantization of a P0 algebra A is a BD algebra :A,flat over R[[!]], with an isomorphism of P0 algebras

:A,R[[!]] R %= A.

The main results of this subsection is that quantizations of the P0 algebraO(E) are the same as solutions to the quantum master equation. Here, asbefore, (E,Q, ) is a cochain complex with a symplectic pairing of degree!1.

We will state this result as an equivalence of groupoids (a more refinedanalysis would yield an equivalence of infinity-groupoids). Given a P0 alge-bra A, let us define a groupoid Quant(A) whose objects are quantizationsof :A of A, and whose morphisms are R[[!]]-linear isomorphisms : :A $ :A!

of BD algebras, which are the identity modulo !.Let I & O(E) be a solution to the classical master equation. Let

O>0(E) = O(E)/Rdenote the quotient of E by R. Note that we can identify O>0(E)[1] withthe vector space of derivations of O(E) which preserve the symplectic form.

Let us define a simplicial set QME(E, I) of solutions to the quantummaster equation by saying that the set QME(E, I)[n] of n-simplices areelements

=I & O>0(E), &+()n)[[!]]which satisfy the quantum master equation and which agree with I modulo!. Here, the quantum master equation includes the di!erential on the al-gebra &+()n) of forms on )n. (We will analyze this simplicial set in moredetail in Section 10).

2BD stands for Beilinson-Drinfeld, who considered such algebras in their book (BD04).I use this notation because, unfortunately, the term Batalin-Vilkovisky algebra has beenused in the mathematical literature to refer to a di!erent, but closely related object.



Let us define a groupoid

*QME(E, I)

as the fundamental groupoid of QME(E, I). One can verify easily thatQME(E, I) is a Kan complex. This means that *QME(E, I) can be de-scribed, in concrete terms, as the groupoid whose objects are solutions tothe quantum master equation =I which restrict to I modulo !; and whosemorphisms are homotopies of such solutions; but these homotopies are them-selves only taken up to homotopy.

Proposition 2.8.4. There is an equivalence of groupoids

*QME(E, I) . Quant(O(E), Q + {I,!}).

This proposition won’t be used in the rest of this book.

Proof. Given any 0-simplex =I & QME(E, I), let us define a di!erential

QbI = Q + {=I,!} + !)on O(E),&+()n)[[!]]. The quantum master equation implies that Q2

bI= 0.

This shows that each object of *QME(E, I) (that is, each 0-simplex inQME(E, I)) gives an object of the category Quant(O(E), Q + {I,!}).

Next, we need to show that homotopies between solutions of the QMEgive rise to isomorphisms of quantizations. Suppose that F (t,dt) & O(E),&+([0, 1])[[!]] is such a homotopy. Then, we can write

F (t,dt) = =I(t) + dt =J(t)

where =I(t) is a family of solutions to the QME, depending on t. Let

X(t) = !{ =J(t),!}

be the derivation given by Poisson bracket with =J(t). The statement thatF (t,dt) satisfies the QME implies

ddt

QbI(t) = [X(t), QdI(t)].

(Here, as before, QbI(t) = Q + {=I(t),!} + !)).Let us define a one-parameter family of automorphisms

$(T ) : O(E)[[!]]$ O(E)[[!]]

for T & [0, 1], by requiring that $(0) is the identity, and that $(T ) satisfiesthe di!erential equation

ddT$(T )(f) = X(T )$(T )(f).

The fact that ! is a formal parameter and that O(E) is the completedsymmetric algebra implies that there is a unique such $(T ).

Note that$(T )QbI(0)$(T )$1 = QbI(T ).



In particular, $(1) defines an isomorphism of quantizations

$(1) : (O(E)[[!]], QbI(0))$ (O(E)[[!]], QbI(1)).

Thus, every one-simplex in QME(E, I) defines a morphism in the groupoidQuant(O(E), QI). It is not hard to check that homotopic one-simplices givethe same morphism, so that we have a map of groupoids

*QME(E, I)$ Quant(O(E), QI).

Next we will check that this functor is essentially surjective. If A is aquantization of (O(E), QI), then A (without the di!erential) can be thoughtof as a family of formal graded symplectic manifolds over Spec R[[!]] which,modulo !, is equipped with an isomorphism to the formal graded symplecticmanifold E. The Darboux lemma (which applies in this graded formalsituation) shows that there is an isomorphism of graded P0 algebras overR[[!]] (without the di!erential)

A %= O(E)[[!]].

We will let dA be the di!erential on A, viewed as an operator on O(E)[[!]].This operator is not a derivation, but satisfies

dA(a · b) = (dAa) · b + (!1)|a|a · dAb + !{a, b}.Then,

dA !Q! !)is a derivation for both the commutative and Poisson structure on O(E)[[!]].

Now, every such derivation of O(E)[[!]] is given by bracketing with aHamiltonian function, defined up to a constant. Thus, let =I & O>0(E)[[!]]be the unique element such that

{=I,!} = dA !Q! !).

That is, dA = QbI for some unique =I & O(E)>0[[!]]. The fact that d2A = 0

implies the that =I satisfies the quantum master equation.Thus, we see that the functor

*QME(E, I)$ Quant(O(E), QI)

is essentially surjective.It remains to check that the functor is full and faithful. Thus, suppose

that =I0, =I1 are solutions to the quantum master equation. Suppose that

$ : O(E)[[!]]$ O(E)[[!]]

is an automorphism of P0 algebra O(E)[[!]] such that

$QbI0$$1 = QbI1 ,

and such that $ is the identity modulo !. Then, we can write $ as theexponential of some derivation X of O(E)[[!]]. Let $(t) = etX . Define =I(t)by

QbI(t) = $(t)QbI0$(t)$1.



Let J & O>0(E)[[!]] be the unique element such that !{J,!} = X. Then,we find a homotopy between the solutions =I0 and =I1 of the quantum masterequation by setting

(†) F (t,dt) = =I(t) + Jdt

The di!erential equationddt=I(t) + QJ + !)J + {J, =I(t)} = 0

implies that F (t, dt) satisfies the quantum master equation, and so definesa homotopy.

Thus we have shown that the functor is full. The final step is to verifythat the functor is faithful. This amounts to checking that any homotopybetween two solutions of the quantum master equation =I0 and =I1 is homo-topic to one of the form (†), that is, one such that the coe#cient of dt isindependent of t.

This can be proved by an inductive argument, working term by term in!. Let QME%n(E, I) denote the simplicial set of solutions to the quantummaster equation defined modulo !n+1. One can check that the natural mapQME%n+1(E, I)$ QME%n(E, I) is a fibration. Let

#QME%n

(E, I) ) QME%n(E, I)

be the sub-simplicial set consisting of solutions to the quantum master equa-tion of the form (†).

Let us fix two solutions =I0, =I1 of the quantum master equation whichagree with the original I modulo !. Let F be a homotopy between them.Let Fn denote the reduction of F modulo !n+1; we assume that F0 = 0.Thus, Fn is a one-simplex in QME%n(E, I). Suppose, by induction, that

we have a one-simplex Gn in the subcomplex #QME%n

(E, I) which is of theform (†). Suppose in addition that Hn is a two-simplex in QME%n(E, I))which is constant with value I0 on one edge; and on the other two edges isFn and Gn.

We have a lift Fn+1 of F to QME%n+1(E, I). The fact that the mapQME%n+1(E, I) $ QME%n(E, I) is a fibration means that the homotopyHn lifts to a two-simplex H !

n+1 of QME%n+1(E, I), which on one edge isFn+1 and on another is constant with value I0.

Let H !!n+1 be another such lift of Hn. Then,

H !n+1 !H !!

n+1 = !n+1A

for some A & O(E),&+()2, Horn) (here Horn ) )2 is the union of the firsttwo edges).

Further, the fact that H !n+1 and H !!

n+1 satisfies the quantum masterequation implies

(QI + ddR)A = 0.


3. THE CLASSICAL BV FORMALISM IN INFINITE DIMENSIONS 155

We would like to construct some Hn+1 whose restriction to the third face of

)2 is in the subcomplex #QME%n+1

(E, I). This condition amounts to sayingthat the restriction of Hn+1 to the third edge of )2 lies in the subcomplexof &+([0, 1]) spanned by 1, t and dt. This subcomplex is quasi-isomorphicto &+([0, 1]). Therefore, standard obstruction theory arguments allow usto find some A & O(E) , &+()2, Horn) such that QIA + ddRA = 0, andthat :Hn+1 + !n+1A restricts, on the third edge of )2, to an element of thissubcomplex. !

3. The classical BV formalism in infinite dimensions

In this section we will explain how the constructions and results of theBV formalism in finite dimensions go through with little change at the clas-sical level in infinite dimensions. This leads to many examples of solutionsof the Batalin-Vilkovisky classical master equation in infinite dimensions.At the quantum level, as we will see later, we will need renormalizationtechniques to even define the quantum master equation.

3.1. Before we start, we need a definition.

Definition 3.1.1. Let E1, . . . , En+1 be vector bundles on a manifoldM ; let E1, . . . , En denote the spaces of global sections. Let Di!(Ei, C"(M))be the space of di!erential operators from the vector bundle Ei to the triv-ial vector bundle. We will view Di!(Ei, C"(M)) as a C"(M)-module bymultiplication on the left: (fD)(e) = f(De) for f & C"(M) and e & Ei.

The space of polydi!erential operators

E1 , · · ·, En $ En+1

is the space

PolyDi!(E1 , · · ·, En, En+1)= Di!(E1, C

"(M)),C!(M) · · ·,C!(M) Di!(En, C"(M)),C!(M) En+1.

There is a natural injective map

PolyDi!(E1 , · · ·, En, En+1)$ Hom(E1 , · · ·, En, En+1)

defined by

(D1 , · · ·,Dn , en+1)(e1 , · · ·, en) = en+1(D1e1)(D2e2) · · · (Dnen)

where Di & Di!(Ei, C"(M)) and ei & Ei.

3.2. We will describe the construction of solutions of the classical masterequation on a compact manifold, for simplicity. The construction on Rn oron a non-compact manifold is similar.

Let M be a compact manifold and E a super vector bundle on M .Let E = '(M,E), and, as usual, let O(E ) =

<n'0 Hom(E (n, R)Sn be the

algebra of functionals on E .



The Lie algebra

Der(O(E )) = O(E ), E =/

n'0

Hom(E (n, E )Sn

is the Lie algebra of symmetries of E . There is a subalgebra

Derloc(O(E )) =/

n'0

Homloc(E (n, E )Sn

of local symmetries of E , where

Homloc(E (n, E )

is the space of polydi!erential operators.

Definition 3.2.1. A degree !1 symplectic structure on E is a map

: E ,E $ Dens(M)

of vector bundles on M , where Dens(M) is the line bundle of densities.This map must be of cohomological degree !1, -symmetric, and fibrewisenon-degenerate. This map yields a degree !1 anti-symmetric pairing

: E , E $ C

e1 , e2 /$!

M(e1 , e2).

Let X & Der(O(E )). This vector field is given by a collection of Sn-invariant maps

Xn : E (n $ E

for n 5 0. Formally, we can say that X is symplectic if the Lie derivativeLX vanishes. It is not completely obvious that this equation is well defined.However, it turns out that LX = 0 if and only if the maps

Xn : : E (n+1 $ Ce1 , · · ·, en+1 /$ (e1, Xn(e2, . . . , en+1))

are invariant under Sn+1.If X is symplectic, then there is a Hamiltonian HX & O(E ) whose ho-

mogeneous componentsHn : E (n $ C

are defined by

Hn =1n

Xn$1 :where Xn$1 : is the map defined above.

Formally, X is the Hamiltonian vector field associated to the functionHX .

Definition 3.2.2. A function f & O(E ) is Hamiltonian if it is theHamiltonian function HX associated to some symplectic vector field X. If fis Hamiltonian, then there is a unique such vector field X; we call this theHamiltonian vector field associated to f .



There is a bijection between Hamiltonian functions up to constants andsymplectic vector fields.

Not all functions f & O(E ) are Hamiltonian. However,

Lemma 3.2.3. Every local action functional f & Oloc(E ) is Hamiltonian.Under the bijection between Hamiltonian functions up to constants and sym-plectic vector fields, local action functionals correspond to local symplecticvector fields.

Let f, g & Oloc(E ). Then, we define the BV bracket

{f, g}by

{f, g} = Xfg.

Note that this bracket is of cohomological degree 1. As usual, {f, g} issymmetric and satisfies the graded Jacobi identity.

Definition 3.2.4. An even functional S & Oloc(E ) satisfies the classicalmaster equation if

{S, S} = 0.

3.3. Now let us turn to constructions of solutions of the classical masterequation, analogous to the constructions we have already seen in the finitedimensional situation.

Let F be a vector bundle on M , whose space of global sections is denotedby F .

Definition 3.3.1. A local Lie algebra structure on F is given by acontinuous linear map

F ,F $ F

which is an anti-symmetric polydi!erential operator, satisfying the Jacobiidentity.

Let V be another vector bundle on M , whose space of global sections isdenoted by V . Let F be a local Lie algebra. A local Lie algebra action ofF on V is given by a Lie algebra map

F $ Der(O(V ))

such that the corresponding maps

F , V (n $ V

are polydi!erential operators.

For example, the tangent bundle TM of M is a local Lie algebra. If g isa finite dimensional Lie algebra, then C"(M), g is a local Lie algebra.

If the local Lie algebra F has local action on V , then the infinite di-mensional supermanifold F [1] 4 V has a Chevalley-Eilenberg di!erential,as usual. This is an element

X & Der O(F [1]4 V )



whose homogeneous components are of two types; firstly, we have the Liebracket

F(2 $ F

and secondly, we have the action maps

F , V (n $ V .

Since all of these maps are polydi!erential operators, the vector field Xis local. As usual, the Jacobi identity, together with the axioms for a Liealgebra action, are equivalent to the identity

[X,X] = 0.

3.4. In this situation, we can apply the Batalin-Vilkovisky machineryas in the finite dimensional setting. We need to define the shifted cotangentbundle of F [1]4 V . For any vector bundle U on M , let

U ! = U) ,Dens(M).

Note that there is a natural pairing between '(M,U !) and '(M,U). Thispairing has no kernel.

Let E be the vector bundle on M defined by

E = F [1]4 V 4 V ![!1]4 F ![!2].

The global sections of E will be denoted E , which we write

E = F [1]4 V 4 V ![!1]4F ![!2].

The graded pieces of the vector bundle E are the bundles of ghosts, fields,anti-fields and anti-ghosts, respectively.

The vector bundle E has a degree !1 symplectic pairing

E , E $ Dens(M),

arising from the natural pairings

F , F ! $ Dens(M)

V , V ! $ Dens(M).

3.5. We would like to extend the local vector field X on F [1] 4 V toa vector bundle on E , which we view as the shifted cotangent bundle toF [1] 4 V . Let us recall what happens in the finite dimensional situation.Let N be a finite dimensional supermanifold, and let X be a vector fieldon N . Then, by naturality, X extends to a vector field X ! on the shiftedcotangent bundle T +N [!1] of N . The Hamiltonian function of the vectorfield X ! is linear on the fibres of the projection T +N [!1]$ N ; this propertyuniquely characterizes the lift X ! of X.

Back in our infinite dimensional situation, we have the following.



Lemma 3.5.1. There is a unique (up to the addition of a constant) localaction functional

SGauge & Oloc(E )such that

(1) SGauge is linear on the fibres of the projection map

E $ F [1], V .

That is,

SGauge & O(F [1], V ),Hom(V ![!1]4F ![!2], R) ) O(E ).

(2) The Hamiltonian vector field X ! associated to SGauge is a lift of thevector field X. This means that X ! preserves the subalgebra

O(F [1]4 V ) ) O(E )

and agrees with X on this sub-algebra.Further, SGauge satisfies the classical master equation

{SGauge, SGauge} = 0.

Proof. Uniqueness of SGauge is easy to see. To see existence, we willconstruct SGauge explicitly.

Let us f & F be a ghost, v & V a field, v) & V ! an anti-field, andf) & F ! an anti-ghost. Then let

SGauge(f, v, v), f)) = 12

+[f, f ], f),+

+f(v), v)

,.

By f(v) we mean the vector field f & Derloc(O(V )) evaluated at the pointv & V ; the resulting element of V = TvV is paired with v).

The first term in SGauge is local because the Lie bracket map

F ,F $ F

is a polydi!erential operator.The second term in SGauge is local because f(v) arises from Sn-invariant

polydi!erential operators

F , V (n $ V .

for all n.The associated local symplectic vector field X ! & Derloc(O(E )) preserves

the sub-algebraO(F [1]4 V ) ) O(E ).

It is easy to see that this vector field X ! agrees with the Chevalley-Eilenbergvector field on the algebra O(F [1]4 V ).

The fact that the Chevalley-Eilenberg vector field X has square zeroimplies that

[X !, X !] = 0.

It follows that the function SGauge satisfies

{SGauge, SGauge} = 0.



!3.6. Let

SV & Oloc(V )be a functional preserved by the action of the Lie algebra F . We can pullback SV to a function

SV & Oloc(E ).The fact that this functional is invariant under the action of F implies

thatX !SV = 0

where X ! is the lift of the Chevalley-Eilenberg vector field X on F [1]4 V .This means that

{SGauge, SV } = 0.

Since SV depends on only half of the variables of E ,

{SV , SV } = 0.

Thus, the BV actionS = SGauge + SV

satisfies the classical master equation.

4. Example: Chern-Simons theory

The main theory we will be concerned with is Yang-Mills theory. How-ever, we will start by discussing a much simpler theory, Chern-Simons theoryon an oriented three-manifold M . Let G be a compact Lie group, with Liealgebra g; let us fix an invariant pairing '!,!(g on g.

4.1. The Chern-Simons field is a g-valued connection on M . For sim-plicity, we will assume we are perturbing around the trivial flat connectionon the trivial g-bundle on M . Then, we can identify this space of fields with&1(M), g. The gauge group is the group

G = Maps(M,G)

of all smooth maps from M to G, a compact Lie group whose Lie algebrais g. We will mainly be interested in the Lie algebra of infinitesimal gaugesymmetries, which we take to be

C"(M), g = &0(M), g.

The action of this Lie algebra on the space of fields is the standard a#ne-linear action,

A /$ [X,A] + dX

where A & &1(M), g and X & &0(M), g.Let '!,!( denote the pairing on &+(M), g defined by

' 1 , E1, 2 , E2( =!

M1 9 2 'E1, E2(g

where '!,!(g is our chosen invariant pairing on g.


4. EXAMPLE: CHERN-SIMONS THEORY 161

The Chern-Simons action SCS on &1(M), g is defined by

SCS(A) = 12

!

M'A,dA(+ 1

6 'A, [A, A](

This is preserved by the Lie algebra &0(M), g of infinitesimal gauge sym-metries.

4.2. The Batalin-Vilkovisky space of fields associated for the Chern-Simons gauge theory is obtained by adding on ghosts, corresponding toelements of the Lie algebra of gauge symmetries; anti-fields, dual to thespace of fields; and anti-ghosts, dual to the space of ghosts. The space weend up with is simply

&0(M), g ghosts; degree ! 1

&1(M), g fields; degree 0

&2(M), g anti-fields; degree 1

&3(M), g anti-ghosts; degree 2

In other words, the space of fields for Chern-Simons theory in the BV for-malism is simply

E = &+(M), g[1].

The degree !1 symplectic pairing on E arises from the degree !3 symmetricpairing on &+(M), g described above.

4.3. The Batalin-Vilkovisky action on E is simply

S(e) = 12 'e,de(+ 1

6 'e, [e, e](

as before.One can see this as follows. The BV action can be written

S = SGauge + SCS

whereSCS : &1(M), g$ R

is the Chern-Simons action as above, and SGauge arises from the gaugeaction.

We will denote fields by A, ghosts by X, anti-fields by A) and anti-ghostsby X). Then, the Lie bracket on ghosts gives the term

12

+[X,X], X), .

The action of ghosts on fields gives the term+dX + [X,A], A), .



4.4. In the BV formalism, we always integrate over an isotropic subspaceof the space of fields E . For Chern-Simons theory, the isotropic subspace is

Im d+ ) E .

This, of course, requires the choice of a metric on M . The philosophy isthat all constructions should be independent of the choice of gauge fixingcondition. A precise statement along these lines is proved in (Cos07). See(CM08a) and (Iac08) for a more thorough discussion.

5. Example : Yang-Mills theory

The material in Section 3 leads to many examples of solutions of theclassical master equation. The main example we will be concerned with inthis book is associated to the Yang-Mills gauge theory in four dimensions.

5.1. Let M be an oriented 4-manifold equipped with a conformal classof metrics. Let g be a semi-simple Lie algebra, and fix an invariant pairing'!,!(g on g. The Yang-Mills field is a g-valued connection on M ; since weare perturbing around a given flat connection, we can identify the space offields with &1(M), g.

As with Chern-Simons theory, the gauge group is the group

G = Maps(M,G)

of all smooth maps from M to G, the compact Lie group associated to G.The Lie algebra of infinitesimal gauge symmetries is

C"(M), g = &0(M), g.

The action of this Lie algebra on the space of fields is the standard a#ne-linear action,

A /$ [X,A] + dX

where A & &1(M), g and X & &0(M), g.Let '!,!( denote the pairing on &+(M), g defined by

' 1 , E1, 2 , E2( =!

M1 9 2 'E1, E2(g

where '!,!(g is our chosen invariant pairing on g.

5.2. The Yang-Mills action SY M on &1(M), g is defined by

SY M (A) = 'F (A)+, F (A)+( .

In this expression,

F (A)+ = 12(1 + 0)F (A) & &2

+(M), g


5. EXAMPLE : YANG-MILLS THEORY 163

denotes the projection of the curvature onto the space of self-dual two-forms.This is not the standard way of writing the Yang-Mills action, but observethat

'F (A)+, F (A)+( =+

12(1 + 0)F (A), 1

2(1 + 0)F (A),

= 12 'F (A), F (A)(+ 1

2 '0F (A), F (A)( .The first term is topological, and so independent of the connection A. Thesecond term is the usual way of writing the Yang-Mills action.

The Yang-Mills functional SY M is invariant under the action of the groupG of gauge symmetries, and under the Lie algebra &0(M),g of infinitesimalgauge symmetries. The functional integral one is interested in making senseof is !

A#$1(M)(g/GeSY M (A)/!.

This is the integral over the space of all connections modulo gauge equiva-lence.

5.3. The Lie algebra &0(M) , g is local, as is the action of this Liealgebra on the space &1(M) , g of fields. Thus, we can apply the Batalin-Vilkovisky machinery, as described in Section 3. The resulting graded sym-plectic vector space of BV fields is

E = &0(M), g[1]4 &1(M), g4 &3(M), g[!1]4 &4(M), g[!2].

As always, V [!i] denotes the vector space V situated in degree i.The degree !1 symplectic pairing on E is the obvious pairing between

&0(M), g and &4(M), g and between &1(M), g and &3(M), g.

5.4. The general BV procedure produces an action

S & Oloc(E )

which satisfies the classical master equation

{S, S} = 0.

Let us describe S explicitly. As in Section 3, S is a sum of two terms

S = SGauge + SY M

whereSY M : &1(M), g $ R

is the Yang-Mills action as above.The other term SGauge of S is constructed from the gauge action, as

follows. Let us denote a ghost by X, a field by A, an anti-field by A) and ananti-ghost by X). Then, the Lie bracket on the space of ghosts contributesthe term

12

+[X,X], X), .

The action of ghosts on fields contributes the term+dX + [X,A], A), .



5.5. The quadratic part of S induces a di!erential

Q : E $ E .

Because S satisfies the classical master equation,

Q2 = 0.

The operator Q is explicitly described by the diagram

&0(M), gd !!&1(M), g

d+d !!!!&3(M), gd !!&4(M), g.

5.6. In the BV formalism, we need to integrate over an isotropic sub-space in the space of fields E . As with Chern-Simons theory, we would liketo take our isotropic subspace to be of the form Im QGF for some di!erentialoperator

QGF : E $ E

which is of degree !1, square zero, and self-adjoint with respect to the sym-metric pairing. These conditions are necessary in order to have a propagatorof the kind that can be treated by the renormalization techniques describedin this book.

The only obvious candidate is to take the operator QGF described bythe diagram

&0(M), g &1(M), gd(

## &3(M), gd(+d(## &4(M), g

d(## .

Such an operator requires the choice of a metric on M in the given conformalclass.

Unfortunately, this operator doesn’t satisfy certain conditions requiredto apply our renormalization techniques to the BV formalism. The problemis that QGF is of second order, so that the operator

D = [Q, QGF ]

is of fourth order. The propagator is constructed from the heat kernel forD, and the heat kernel of a fourth order operator is not so well behaved.

This problem will be resolved in Chapter 6, by using the first orderformulation of Yang-Mills theory.

6. D-modules and the classical BV formalism

In this section we will explain how to interpret some aspects of theclassical BV formalism, as described in Section 3, using the theory of D-modules.

As we will see later, when constructing quantum theories in the BVformalism, obstructions to gauge symmetry can lie in the cohomology of thecomplex Oloc(E ), with the di!erential given by bracketing with the (classical)BV action {S,!}. The classical master equation says that this di!erentialis of square zero.

Thus, it is important to be able to compute the cohomology of thecomplex Oloc(E ) with di!erential {S,!}. In this section we will show how


6. D-MODULES AND THE CLASSICAL BV FORMALISM 165

to interpret this complex in terms of the Gel’fand-Fuchs cohomology of acertain DM L" algebra on M .

6.1. Let M be a C" manifold. Let DM denote the algebra of di!erentialoperators on M . A DM -module is simply a sheaf of left modules over thissheaf of algebras on M . Any DM -module is, in particular, a module overthe sheaf C"

M of smooth functions on M . Conversely, if R is a sheaf of C"M -

modules, the extra data which we need to make R into a DM -module is aflat connection. This is, by definition, a map

# : R $ R,C!M&1

M

satisfying the usual Liebniz rule, whose curvature F (#) & R ,C!M&2

M van-ishes.

Thus, we can think of DM -modules as (possibly infinite dimensional)vector bundles on M with a flat connection.

If E,F are vector bundles on M with flat connections, then the tensorproduct bundle E , F has an induced flat connection. In a similar way, ifR,S are DM -modules, then

R,C!M

S

is a new DM -module.In this way, the category of DM -modules becomes a symmetric monoidal

category. Thus, one can talk about algebraic objects in the category of DM -modules: for instance, commutative algebras, Lie algebras, L" algebras,etc.

6.2. Any finite rank Z graded vector bundle E on M gives rise to a DM -module J(E), the DM -module of jets of sections of E. The fibre of J(E) atx & M is the space of formal germs at x of sections of E. The DM -modulestructure on J(E) is characterized by specifying the infinitesimal paralleltransport, which identifies the fibre J(E)x with the fibre at an infinitesimallynear point y. This parallel transport operation simply takes a formal germof a section of E near x, and thinks of it as a formal germ of a section neary; this makes sense, as x and y are infinitesimally close.

Let us choose local coordinates x1, . . . , xn on an open neighbourhood Uof M , and let us choose an isomorphism between E and a trivial gradedvector bundle with fibre E0, for some finite dimensional graded vector spaceE0. Then, we can identify

'(U, J(E)) = C"(U),R R[[x1, . . . , xn]],R E0.

Thus, as a C"M -module, J(E) is an inverse limit of finite dimensional mod-

ules. We will endow the sheaf J(E) with the topology of the inverse limit;all constructions we perform with DM -modules like J(E) must respect thistopology.



The flat connection on J(E) is given by the map

# : C"(U),R R[[x1, . . . , xn]],R E0 $ &1(U),R R[[x1, . . . , xn]],R E0

#(f , g , v) = df , g , v +'

fdxi ,"

"xig , v.

This map is continuous for the topology on J(E).

Lemma 6.2.1. Let E,F be vector bundles on M . Let E , F denote thespaces of global sections of E and F respectively. Then,

Di!(E , F ) = HomDM (J(E), J(F ))

where HomDM refers to the space of continuous DM -module maps.More generally, if E1, . . . , En, F are vector bundles on M , then

PolyDi!(E1 , · · ·, En, F ) = HomDM

-J(E1),C!

M· · ·,C!

MJ(En), J(F )

..

Here J(E1),C!M· · ·,C!

MJ(En) refers to the completed tensor product of the

pro-finitely generated C"M -modules J(Ei).

These identifications are compatible with composition.

It follows immediately that if E is a local Lie algebra on M , in the senseof Section 3, then J(E) is a Lie algebra in the symmetric monoidal categoryof DM -modules.

6.3. LetJ(E)) = HomC!

M(J(E), C"

M )where HomC!

Mdenotes the sheaf of continuous linear maps of C"

M -modules(in other words, J(E)) is the ind vector bundle dual to the pro vector bundleJ(E)). Note that the usual formula for the flat connection on the dual of avector bundle gives the C"

M -module J(E)) the structure of DM -module.For example, if E is the trivial vector bundle with fibre the vector space

E0, thenJ(E)) = DM ,R E)

0

as a DM -module.Let

O(J(E)) = 0Sym+J(E)) =

/

n'0

Symn J(E))

denote the completed symmetric algebra on J(E)). The symmetric productis taken over C"

M . Note that O(J(E)) is a DM -algebra.

6.4. As in Section 3, let

Derloc(O(E )) =/

n'0

PolyDi!(E (n, E )Sn

be the Lie algebra of local symmetries of E . Lemma 6.2.1 immediatelyimplies the following.

Lemma 6.4.1. The Lie algebra Derloc(O(E )) is canonically isomorphicto the Lie algebra of derivations of the DM -algebra O(J(E)).



Let us suppose we have some

X & Derloc(O(E ))

which is of cohomological degree 1, and which satisfies

[X,X] = 0.

Any classical theory in the BV formalism gives rise to a super vector bundleE with such an X, as explained in Section 3.

Thus, we can think of X as an element

X & Der(O(J(E)),

satisfying the equation [X,X] = 0.There is a standard dictionary relating L"-algebras and square zero

vector fields. Applied in our context, it shows the following. Let us considerthe Taylor components of X as maps

Xn : (J(E)[!1])(n $ J(E)[!1]

of degree 2!n. Then, these give J(E)[!1] the structure of DM L"-algebra.Thus, we have shown the following.

Lemma 6.4.2. Let E be a Z-graded vector bundle, whose sections arethe fields for a classical theory in the BV formalism. Then, the DM -moduleJ(E)[!1] has a canonical structure of a DM L"-algebra.

We can consider O(J(E)) as a cochain complex, with the di!erential[X,!]. This lemma shows that we can view O(J(E)) as the Chevalley-Eilenberg cochain complex, taken in the symmetric monoidal category ofDM -modules, of the DM L"-algebra J(E)[!1]. To be precise, we are tak-ing the topological version of Chevalley-Eilenberg cochains, where we takethe continuous dual of the completed exterior algebras of the topologicalDM L"-algebra J(E)[!1]. This topological version of Chevalley-Eilenbergcochains is known as Gel’fand-Fuchs cohomology.

6.5. The L"-algebra J(E)[!1] is something very familiar in the caseof Chern-Simons theory. Let M be a three-dimensional manifold, and let gbe a Lie algebra with an invariant pairing. The vector bundle of fields forChern-Simons theory on M with coe#cients in g is

E = &+M , g[1].

Thus, E[!1] = &+M , g has the structure of a local di!erential graded Lie

algebra. The di!erential is the de Rham di!erential, and the Lie bracketarises from wedge product of forms and the Lie bracket on g. This Liealgebra structure carries over to one on J(E)[!1].



6.6. Let E and X be as above. The main object we would like tounderstand is the complex Oloc(E ) of local action functionals on E , with thedi!erential arising from the local derivation X of O(E ).

Let us consider Oloc(E ) and O(J(E)) both as cochain complexes, withdi!erential arising from X. Recall that DensM , the line bundle of densitieson M , is a right DM -module in a natural way.

Lemma 6.6.1. There is an isomorphism of cochain complexes

Oloc(E ) = DensM ,DM O(J(E)).

Proof. The vector bundle DensM ,C!M

O(J(E)) is, essentially by defi-nition, the space of Lagrangians on the vector bundle E. Taking the tensorproduct over DM amounts to quotienting by the space of Lagrangians whichare a total derivative. A local action functional on E is defined to be some-thing arising from the integral of a Lagrangian; thus, local action functionalsare Lagrangians up to the addition of a total derivative.

It is straightforward to check that this isomorphism is compatible withdi!erentials. !

I should emphasize that the tensor product appearing in the statement ofthis lemma is not derived. We have a slightly weaker result using the derivedtensor product. Let R ) Oloc(E ) denote the space of constant local actionfunctionals; similarly, let C"

M ) O(J(E)) denote the submodule spanned bythe identity in O(J(E)). Note that O(J(E))/C"

M is a DM -module, in fact,a non-unital DM -algebra.

Lemma 6.6.2. There is a canonical quasi-isomorphism of cochain com-plexes

Oloc(E )/R . DensM ,LDM

(O(J(E))/C"M )

Proof. In light of the previous lemma, it su#ces to check that

(O(J(E))/C"M ),L

DMDensM = (O(J(E))/C"

M ),DM DensM .

Note thatO(J(E))/C"

M =/

n>0

Symn J(E)).

Each DM -module Symn J(E)), for n > 0, is flat as a DM -module. Indeed,Symn J(E)) is a direct summand in (J(E)))(n). If we choose a local iso-morphism between the vector bundle E on M and a trivial vector bundlewith fibre E0, we can identify

J(E)) %= E)0 ,DM

as DM -modules. Thus, J(E)) is free, and therefore flat. More generally,

(J(E)))(n %= (E)0 )(n ,D(n

M

locally. Here, the tensor power of DM is taken over C"M . It is easy to see

that D(nM is flat, and so (J(E)))(n is. !



6.7. Let us suppose that M = Rn. Further, let us assume that thevector bundle E on Rn is equipped with a flat connection; let E0 be thefibre at 0 & Rn. Suppose that the vector field X on O(E ) is translationinvariant, that is, preserved by the flat connection on E.

Translation gives an Rn action on Oloc(E ), and the di!erential on Oloc(E )is preserved by this action.

We would like to be able to compute the cohomology of the Rn-invariantsof Oloc(E ). The following lemma allows us to do this.

Recall that O(J(E)), defined as<

n'0 Symn(J(E))), is a DRn-algebra.Let O(J(E))0 denote its fibre at 0 & Rn. The fact that O(J(E)) is aDRn-algebra implies that O(J(E))0 is acted on by the algebra R["1, . . . , "n].The generators "i of this algebra act by derivations on O(J(E))0; that is,O(J(E))0 has an action of the Lie algebra Rn by derivations.

Lemma 6.7.1. There is a canonical quasi-isomorphism of complexes

(Oloc(E )/R)Rn %= (O(J(E))0/R),LR["1,...,"n] |det|Rn.

(Here |det|Rn is the one dimensional R["i]-module on which each "i actstrivially, and on which GLn acts by the absolute value of the determinant).

This quasi-isomorphism is invariant under GLn.

Note that this theorem can be restated as follows. Recall that J(E)[!1]is a DRn L"-algebra. Thus, J(E)0[!1] is a topological L"-algebra. Thedi!erential graded algebra O(J(E))0 can be interpreted as the Gel’fand-Fuchs Lie algebra cochains C+(J(E)0[!1]). Thus, the lemma asserts thatthere is a quasi-isomorphism

(Oloc(E )/R)Rn . C+red(J(E)0[!1]),R["1,...,"4] R.

Here C+red(J(E)0[!1]) refers to the reduced Gel’fand-Fuchs Lie algebra co-

homology. We will use this form of the lemma when we consider Yang-Millstheory later.

Proof. This is an almost immediate corollary of Lemma 6.6.1. Indeed,there is an isomorphism of complexes

Oloc(E ) %= O(J(E)),DRn DensnR .

We need to identify the translation invariant part of the right hand side.Let us view O(J(E)) as an infinite dimensional vector bundle on Rn.

The Rn action on E, and on Rn, gives rise to an Rn action on O(J(E)).Note that the flat connection on O(J(E)) coming from this action does notcoincide with the flat connection coming from the DRn-module structure.

The Rn action on O(J(E)) allows us to turn an element of O(J(E))0into a translation invariant section of O(J(E)). Thus,

O(J(E))Rn= O(J(E))0.



It is easy to see that-O(J(E)),D(Rn) Dens(Rn)

.Rn

= O(J(E))Rn ,D(Rn)Rn (DensnR)Rn

= O(J(E))0 ,R["1,...,"n] |det|Rn.

To complete the proof of the lemma, it su#ces to observe that O(J(E))0/Ris flat as a module for R["1, . . . , "n]. !

7. BV theories on a compact manifold

In this section we will give a general definition of the notion of a quantumtheory in the BV formalism. We will explain the set-up on a compact mani-fold first, as in that situation things are somewhat technically easier. Later,we will explain what needs to be changed when we work with translationinvariant theories on Rn, or when we work on a non-compact manifold.

Definition 7.0.2. A free theory on in the BV formalism is described bythe following data:

(1) A Z-graded vector bundle E on M , over R or C, whose space ofglobal sections will be denoted E . For simplicity, we will describewhat happens over C.

(2) E is equipped with anti-symmetric map of vector bundles on M , ofcohomological degree !1,

'!,!(loc : E , E $ Dens(M).

This must be non-degenerate on each fibre.This pairing induces an integration pairing on the space of global

sections E ,

'!,!( : E , E $ C

'e1, e2( =!

M'e1, e2(loc .

(3) A di!erential operator Q : E $ E of cohomological degree onewhich is of square zero, and is skew self-adjoint for the pairing.Further, Q must make E into an elliptic complex.

Recall that the statement that (E , Q) is an elliptic complex means thatthe complex of vector bundles ( (Q), +E) on T +M \ M , is exact.

Throughout, we will abuse notation and refer to the totality of data ofa free BV theory by E .

7.1. Recall that, as explained in Section 2, the di!erential graded man-ifold of fields that appear in the BV theory is the derived moduli space ofsolutions to the equations of motion. The equations of motion of a free clas-sical field theory are linear equations. In addition, in a free gauge theory,the gauge group is Abelian and acts in an a#ne-linear way on the space offields.


7. BV THEORIES ON A COMPACT MANIFOLD 171

Thus, if we start with a free classical theory (which may be a gaugetheory) then the derived moduli space of solutions to the equations of motionis linear, and thus described by a cochain complex. This is the cochaincomplex (E , Q).

We will discuss several examples of free BV theories, starting with a freemassless scalar field theory on a manifold M . The equations of motion in thistheory are that D = 0, where D : C"(M) $ C"(M) is the non-negativeLaplacian on M .

Thus, the derived moduli space of solutions to the equations of motionis the two-term cochain complex

E = C"(M) D!$ C"(M)

concentrated in degrees 0 and 1. The degree !1 symplectic pairing on E isdefined by

' , ( =!

MdVolM

where is in the copy of C"(M) in degree 0, and is in the copy in degree1.

7.2. Next let us consider free gauge theory in this setting. Let g bean Abelian Lie algebra with a non-degenerate invariant pairing. Let usconsider the free Chern-Simons theory with gauge Lie algebra g. The fieldsare &1(M) , g. The Lie algebra of gauge symmetries is the Abelian Liealgebra &0(M), g. The action of &0(M), g on &1(M), g is a#ne-linear,defined by

X(A) = A + dX

for A & &1(M), g, X & &0(M), g.Thus, the derived quotient of &1(M), g by &0(M), g is described by

the two-term complex

&0(M), gd!$ &0(M), g,

concentrated in degrees 0 and !1.The action on the space &1(M),g of fields is

&M 'A,dA(g, where '!,!(g

is the invariant pairing on g. Thus, the derived moduli space of solutions tothe equations of motion is defined by the 4-term complex

E = &0(M), g $ &1(M), g $ &2(M), g$ &3(M), g

concentrated in degrees !1, 0, 1, 2. The di!erential is the de Rham di!eren-tial; thus, we can write

E = &+(M), g[1].The symplectic pairing on E is given by

' 1 ,X1, 2 ,X2( =!

M

1 9 2 'X1, X2(g ,

where i & &i(M) and Xi & g[1].



7.3. Let us now consider the free Yang-Mills theory, where the gaugeLie algebra is again an Abelian Lie algebra g with invariant pairing.

The space of fields is &1(M),g, and the gauge Lie algebra is &0(M),g.The action of &0(M) , g on &1(M) , g is, as with Chern-Simons theory,defined by

X(A) = A + dX

for A & &1(M), g and X & &0(M), g.The action S(A) is

S(A) =!

M'dA, 0dA(g

where d is the de Rham di!erential, and we are using the invariant pairingon g.

Thus, the derived moduli space of solutions to the equations of motionis the 4-term complex

&0(M), gd !!&1(M), g

d+d !!!!&3(M), gd !!&4(M), g.

concentrated in degrees !1, 0, 1, 2.

7.4. In the BV formalism, one always integrates over an isotropic sub-space of the space E of fields. The subspaces we integrate over are alwaysof the form Im QGF for a di!erential operator QGF on E .

Definition 7.4.1. A gauge fixing operator on a free BV theory E isgiven by an operator

QGF : E $ E

such that:(1) QGF is of cohomological degree !1, of square zero, and self-adjoint

for the pairing '!,!(.(2) The commutator

D = [Q, QGF ]is required to be a generalized Laplacian, in the sense of (BGV92).

The scalar field theory has a very simple gauge fixing operator. If

E = C"(M)4 C"(M)[!1]

is the graded vector space corresponding to the free scalar field theory onM , then

QGF : E 1 = C"(M)$ E 0 = C"(M)is the identity. Since Q : E 0 $ E 1 is the Laplacian, we see that [QGF , Q] :E i $ E i is again the Laplacian, and thus satisfies the required ellipticityproperties.

In this case, the isotropic subspace Im QGF ) E which we are supposedto integrate over is simply C"(M). Thus, with this gauge fixing condition,functional integrals in the scalar field theory in the BV formalism coincidewith the functional integrals considered in Chapter 2.


8. EFFECTIVE ACTIONS 173

Chern-Simons theory on a 3-manifold M also has a simple gauge fixingoperator satisfying these axioms, namely

QGF = d+ : &+(M), g[1]$ &+(M), g[1].

Again, since [QGF , Q] = [d+, d] is the Laplacian coupled to &i(M), g, it isclear that the required ellipticity conditions are satisfied.

Unfortunately, it seems that, in this formulation, Yang-Mills theory onM has no such gauge fixing operator. In Chapter 6, we will use a slightlydi!erent – but equivalent – formulation of classical Yang-Mills theory, calledthe first-order formulation. In the first-order formulation of Yang-Mills the-ory there is a natural gauge fixing operator.

8. E!ective actions

8.1. A free theory in the BV formalism, together with a choice of gaugefixing condition, gives a free theory in the sense of Chapter 2, Section 13, aswe will now see.

Recall that E ! denotes the space of sections of the bundle E),Dens(M)on M . The pairing on E gives rise to an isomorphism

E ! $ E .

Thus, we can compose this isomorphism with the di!erential operator QGF :E $ E , to give an operator

D! : E ! $ E .

This di!erential operator, together with D : E $ E , define a free theory inthe sense of Chapter 2, Section 13.

8.2. Given any K & E , E , let us define a convolution operator

K# : E $ E

as follows.If e & E , then K , e & E , E , E . The pairing '!,!( on E gives a map

1, '!,!( : E , E , E $ E

of cohomological degree !1, which is compatible with di!erentials.We define

(K # e) = (!1)|K|(1, '!,!()(K , e).The reason for this choice of sign is the following.

Lemma 8.2.1. If K & E , E , then

(QK) # e = [Q,K#]e.

In other words, the map

# : E , E $ End(E )

is a cochain map.



(Here Q is the total di!erential on the complex E , E . This di!erentialis occasionally referred to as Q, 1 + 1,Q).

8.3. There is a heat kernel

Kl & E , E

for the operator e$lD, characterized by the equation

Kl # e = e$lDe.

This heat kernel is symmetric.The propagator of our theory is given by

P ( , L) =! L

l=(QGF , 1)Kldl.

This has the property that

P ( , L) # e = QGF! L

e$lDe,

for any e & E .

8.4. As usual, let

O(E ) = 0Sym+E ) =

/

n'0

Hom(E (n, C)Sn

be the algebra of formal power series on E .We will let

Oloc(E ) ) O(E )

denote the subspace of local action functionals.

Definition 8.4.1. A pre-theory is a collection of e!ective interactions{I[L]} satisfying renormalization group flow and the locality axiom as definedin Chapter 2, Section 13.

More precisely,(1) Each I[L] & O+,ev(E )[[!]] is of degree 0 and at least cubic modulo

!.(2) The renormalization group equation

I[L] = W (P ( , L), I[ ])

is satisfied.(3) Each Ii,k[L] has a small L asymptotic expansion in terms of local

action functionals.

We let ET (")(E ) denote the set of pre-theories, and ET (n)(E ) denote the setof theories defined modulo !n+1.


9. THE QUANTUM MASTER EQUATION 175

We will reserve the word theory for collections of e!ective interactions{I[L]} which satisfy the quantum master equation, as defined below.

The results of Chapter 2, Section 13 show that the space ET (n) is a torsorover ET (n$1) for the abelian group Oloc(E ) of even local action functionals.The choice of renormalization scheme leads to a section of each torsor, andso to a bijection

ET (") %= O+loc(E )[[!]]

between the space of pre-theories and the space of local action functionalswhich are even at least cubic modulo !.

9. The quantum master equation

9.1. As E has a degree !1 symplectic form, one can try to define the BVLaplacian on the space of functionals on E . Unfortunately, this is ill-defined.

Let)l = !"Kl : O(E ) $ O(E )

denote the order two di!erential operator associated to the heat kernel Kl

for l > 0.Then, the BV Laplacian (if it were defined) would be the limit

)0I = ! liml&0"KlI.

If I & Ol(E ) is local, then this expression has the same kind of singularitiesone finds in one-loop Feynman graphs. Thus, the naive quantum masterequation

(Q + !)0)eI/! = 0is ill-defined.

9.2. The philosophy of this book is always that the fundamental objectof a theory is not the local action functional I & O+

loc(E )[[!]] but rather thee!ective interactions {I[L]}. The quantum master equation is an expressionof gauge symmetry. Our philosophy suggests that the correct expression ofgauge symmetry should be an equation imposed on the e!ective interactions{I[L]}.

If we make the unnatural choice of a renormalization scheme, then wefind a bijection between local action functionals I and collections {I[L]} ofe!ective interactions. The quantum master equation we will impose on thee!ective interactions yields a renormalized version of the quantum masterequation on the set of local action functionals.

Definition 9.2.1. A functional

I & O+(E )[[!]]

satisfies the scale L quantum master equation if

(Q + !)L)eI/! = 0.



Figure 1. This diagram illustrates the bracket {I, J}L andthe BV Laplacian )L at scale L. As usual, the subscripts(i, k) indicate the part of the coe#cient of !i homogeneousof degree k in the field variable.

An equivalent expression of the scale L quantum master equation is asfollows. There is a scale L BV bracket

{!,!}L : O(E ),O(E )$ O(E )

defined by

{I, J}L = )L(IJ)!)L(I)J ! (!1)|I|I)L(J).

The scale L quantum master equation can be rewritten as

QI + 12{I, I}L + !)LI = 0.

A diagrammatic expression for {I, J}L and )L is given in figure 1.

Lemma 9.2.2. A functional I & O+(E )[[!]] satisfies the scale quantummaster equation if and only if

W (P ( , L), I)

satisfies the scale L quantum master equation.

Thus, the renormalization group flow takes solutions of the scale QMEto solutions of the scale L QME.

Proof. Note that the propagator P ( , L) is the kernel for the operator! L

QGF e$lDdl.


9. THE QUANTUM MASTER EQUATION 177

This operator has a homological interpretation:FQ,

! L

QGF e$lDdl

G=! L

[Q,QGF ]e$lD

=! L

De$lD

= e$ D ! e$LD.

Thus, the operator& L QGF e$lDdl is a chain homotopy between e$ D and

e$LD.It follows that

(Q, 1 + 1,Q)P ( , L) = K !KL

and that

(†)H"P ( ,L), Q

I= "K ! "KL = )L !)

Thus, the operator "P ( ,L) is a chain homotopy between the BV operator atscale and the BV operator at scale L.

Note that we can write

W (P ( , L), I) = ! log1e!"P ( ,L)eI/!

2.

Equation (†) implies that

(Q + !)L) exp(W (P ( , L), I) /!) = (Q + !)L)1e!"P ( ,L)eI/!

2

= e!"P ( ,L)(Q + !) )eI/!.

This shows that, if I satisfies the scale quantum master equation, thenW (P ( , L), I) satisfies the scale L quantum master equation. The converseto this statement follows from the fact that the operator W (P ( , L),!) isinvertible, with inverse W (P (L, ),!).

!

9.3. Now we can define the notion of theory in the BV formalism.

Definition 9.3.1. A pre-theory, given by a collection of e!ective inter-actions {I[L]}, satisfies the quantum master equation if each I[L] satisfiesthe scale L quantum master equation.

A pre-theory which satisfies the QME will be called a theory. The setof theories will be denoted by T ("), and the set of theories defined modulo!n+1 will be denoted by T (n).

If we choose a renormalization scheme, then we can identify the set ofpre-theories {I[L]} with the set of local action functionals I & O+

loc(E )[[!]].Under this correspondence, the statement that {I[L]} satisfies the quantummaster equation imposes an equation on I, which we call the renormalizedquantum master equation.



However, we will always regard the choice of renormalization schemeas unnatural. The e!ective interactions are always considered to be thefundamental objects.

9.4. The classical master equation. As we have seen in Section 3,the scale zero bracket

{I, J} = limL&0

{I, J}L

is well defined. The bracket only involves tree level Feynman diagrams, andso does not have any singularities.

Lemma 9.4.1. Let I(0) & Oloc(E ) be a local action functional which is atleast cubic. Let

I(0)[L] = W (P (0, L), I) modulo ! & O(E )

be the corresponding tree-level e!ective interaction. Then I(0)[L] satisfiesthe classical master equation

QI(0)[L] +12{I(0)[L], I(0)[L]}L = 0

if and only if the local action functional I(0) satisfies the classical masterequation

QI(0) +12{I(0), I(0)} = 0.

10. Homotopies between theories

10.1. We have already talked about homotopies between solutions ofthe quantum master equation in the BV formalism in finite dimensions (seeSection 2.7). Let us briefly recall the picture. Let V be a finite dimensionalgraded vector space equipped with a symplectic pairing of degree !1, andwith a di!erential Q preserving the symplectic pairing. A functional I &O(V ) satisfies the quantum master equation if

(Q + !))eI/! = 0.

A homotopy between two solutions I0, I1 of the quantum master equationis an element

I(t,dt) & O(V ), &+([0, 1])of cohomological degree zero, such that

(Q + dDR + !))eI(t,dt)/! = 0

and I(0, 0) = I0, I(1, 0) = I1.If we write

I(t,dt) = I(t) + J(t)dt

then the quantum master equation for I(t,dt) says that, for all values of t,I(t) & O(V ) satisfies the quantum master equation, and

(†) ddt

I(t) + QJ(t) + {I(t), J(t)} + !)J(t) = 0.


10. HOMOTOPIES BETWEEN THEORIES 179

LetX(t) = !{J(t),!}

be the one-parameter family of Hamiltonian vector fields associated to thefunctionals !J(t). There is a corresponding one-parameter family of sym-plectic di!eomorphisms

(t) : V $ V

such that(t + ) = (1 + X(t)) (t) modulo 2.

Equation (†) says that

e0v,Qv1/!+I(t)/!dV = (t)+1e0v,Qv1/!+I0/!dV

2

where dV is the Lebesgue measure on V .Thus, we see that

Two solutions I0, I1 of the quantum master equation are ho-motopic if and only if there is a symplectic di!eomorphism

= (1), in the connected component of the identity, suchthat

e0v,Qv1/!+I1/!dV = +1e0v,Qv1/!+I0/!dV

2.

Thus, the notion of homotopy for solutions of the quantum master equationis enough to tell us about changes of coordinates on the space of fields V .

10.2. We can also think in terms of homotopies to describe the depen-dence of the BV integral on the choice of a gauge fixing condition.

Recall that, in the finite dimensional setting, a gauge fixing conditionis given by choosing an isotropic subspace L ) V with the property thatthe pairing 'v,Qv( is non-degenerate on L. Let us choose such a subspace.Then, L4QL is a symplectic subspace of V , whose orthogonal complementwe will denote by H. H again has a symplectic form of degree !1.

Let us further assume that Q = 0 on H. This implies that H is thecohomology H+(E,Q) of E.

Then, given any interaction functional I satisfying the quantum masterequation, we find a solution of the quantum master on H by the formula

(#) IH(h) = ! log!

l#Lexp

#1! 'l, Ql(+ 1

!I(l + h)$

.

If we vary the isotropic subspace L ) E, then the solution IH to thequantum master equation on H changes by a homotopy. Similarly, if wevary the solution I to the quantum master equation on V by a homotopy,the corresponding solution IH on H varies by a homotopy.

One way to phrase this is to say that the set of gauge fixing conditionshas an enrichment to a simplicial set, where the n-simplices are just smoothfamilies of isotropic subspaces L ) E parameterized by the n-simplex, satis-fying the transversality condition given above. Let us denote this simplicialset by G F (E,Q). The set of solutions to the quantum master equation also



has an enrichment to a simplicial set, whose 1-simplices are homotopies, and2-simplices are homotopies between homotopies, etc. Let us denote this setby QME (E,Q).

Then, the functional integral (#) defines a map of simplicial sets

G F (E,Q)"QME (E,Q) $ QME (H, 0).

10.3. We would like to extend this picture to the infinite dimensionalspace of fields E . In finite dimensions, there are two ways to say thatsolutions to the quantum master equation are equivalent: either they arehomotopic, or they are related by some non-linear change of coordinates.These two definitions are equivalent.

In infinite dimensions, it is di#cult to make sense of non-linear changesof coordinates, as the change in the non-existent Lebesgue measure in Eshould be taken into account. Thus, in infinite dimensions, only the homo-topy picture can be used. In finite dimensions, two solutions of the QMEare homotopic if and only if they are related by a symplectic change of co-ordinates. Thus, two homotopic theories on E should be viewed as beingrelated by a renormalized version of a local symplectic change of coordinateson E .

In finite dimensions, we constructed a simplicial set of gauge fixing con-ditions, and a simplicial set of solutions to the quantum master equation.The functional integral was then a map to the simplicial set of solutions tothe quantum master equation on cohomology.

This picture translates directly to infinite dimensions, with some extrasubtleties. In infinite dimensions, the notion of theory – and so the notionof solution to the quantum master equation – depends on the choice of agauge fixing condition.

We will introduce a simplicial set G F of gauge fixing conditions which,in most examples, is contractible. The simplicial set of solutions to thequantum master equation, which we denote by QME , is a simplicial setfibred over G F .

This shows that the concept of theory is independent, up to homotopy,of the choice of gauge fixing conditions.

Like in finite dimensions, the functional integral is a map of simplicialsets from QME to a simplicial set of solutions to the QME on cohomology.

10.4. Let us fix a free BV theory (E , Q, '!,!() on M . We will beinterested in families of theories over the algebra of forms on the n-simplex.Our gauge fixing conditions will also come in families.

Definition 10.4.1. A family of gauge fixing conditions for E , over&+()n), is an &+()n) linear di!erential operator

QGF : E , &+()n)$ E , &+()n)

of cohomological degree !1, with the following properties.



(1) QGF is self-adjoint for the &+()n) linear pairing E ,E ,&+()n)$&+()n).

(2) QGF 2 = 0.(3) The operator

D = [Q + dDR, QGF ]is a generalized Laplacian. This means that the symbol of D is asmooth family of Riemannian metrics on the bundle T +M , param-eterized by )n.

Example. Let us consider Chern-Simons theory on a three-manifoldM , with coe#cients in a Lie algebra g with an invariant pairing. Then, thespace of fields is E = &+(M), g[1].

The choice of a smooth family of metrics on M parameterized by )n

leads to an operatord+ : E $ E , C"()n).

We define QGF to be the &+()n) linear extension of this to

E , &+()n)$ E , &+()n).

The choice of such a family of gauge fixing conditions over &+()n) leadsto a heat kernel and a propagator, as usual. The heat kernel is the element

Kl & E , E , &+()n)

which is the kernel for the &+()n) linear operator e$l D. The propagator isthe element

P ( , L) =! L

(QGF , 1)Kldl & E , E , &+()n).

As before, the propagator P ( , L) is a chain homotopy between the heatkernels K and KL:

(Q + ddR)P ( , L) = K !KL

where Q + ddR refers to the total di!erential on the cochain complex E ,E , &+()n).

If)m $ )n

is a face or degeneracy map, one can pull a family of gauge fixing operatorsover &+()n) to one over &+()m). In this way, gauge fixing operators for(E , Q) form a simplicial set, which we will denote G F (E , Q).

10.5. Let us fix a family of gauge fixing conditions over &+()n). Thisallows us to consider families of pre-theories over &+()n). Such a familyis given by a collection of e!ective interactions {I[L]}, where each I[L] is afunctional on E with values in &+()n):

I[L] & O+(E ,&+()n))[[!]].



Each I[L] must be of cohomological degree zero, where the cohomologi-cal degree incorporates the cohomological grading on E as well as that on&+()n). As usual, the collection {I[L]} of e!ective interactions must satisfythe renormalization group equation

I[L] = W (P ( , L), I[ ])

as well as the locality axiom, which states that I[L] has a small L asymptoticexpansion in terms of local action functionals.

As with gauge fixing operators, one can pull back pre-theories alongface and degeneracy maps of simplices. In this way, we find a simplicial setET (")(E , Q), whose n-simplices are families of gauge fixing conditions over&+()n), together with a family of pre-theories for this family of gauge fixingconditions.

Thus, there is a natural map of simplicial setsET (")(E , Q) $ G F .

The renormalization results from Chapter 2 apply in this situation, andallow us to describe the simplicial set ET (")(E , Q) explicitly. Let O+,0

loc (E )[[!]]denote the simplicial set whose n-simplices are cohomological degree zero el-ements of O+

loc(E ,&+()n))[[!]].Then, if we choose a renormalization scheme, we find an isomorphism of

simplicial setsET (")(E , Q) %= G F " O+

loc(E )[[!]].We can reformulate this statement in a renormalization-scheme indepen-

dent way, as follows. Let ET (n)(E , Q) denote the simplicial set of pre-theoriesdefined modulo !n+1. Then, the map

ET (n+1)(E , Q) $ ET (n)(E , Q)

makes ET (n+1)(E , Q) into a principal bundle for the simplicial abelian groupwhose n-simplices are closed degree zero elements of Oloc(E ,&+()n)).

10.6. There is an analog of the quantum master equation in this situa-tion.

Definition-Lemma 10.6.1. A functional

I & O+(E ,&+()k))[[!]]

satisfies the scale L quantum master equation if

(Q + dDR + !)L)eI/! = 0.

Here dDR is the de Rham di!erential on &+()k), and )L is the operatorof contraction with the heat kernel KL & E , E , &+()n). Everything, asalways, is linear over &+()n).

As before, a functional I satisfies the scale quantum master equationif and only if W (P ( , L), I) satisfies the scale L quantum master equation.



A theory is a pre-theory {I[L]} where each I[L] satisfies the scale Lquantum master equation.

The simplicial set of theories will be denoted by T (")(E , Q), as before.

Thus, a homotopy between two theories is a 1-simplex of T (") con-necting the two 0-simplices. Two theories which are homotopic should beviewed as being equivalent: the philosophy is that they di!er by a changeof coordinates on the space E .

Later we will see that the map T (")(E , Q) $ G F (E , Q) of simplicialsets is a fibration. In all examples, G F (E , Q) is a contractible simplicial set.This means that the concept of theory is independent of the choice of gaugefixing condition: we can take the fibre of T (")(E , Q) over any 0-simplex ofG F (E , Q), and this fibre is independent of the choice of point of G F (E , Q).

10.7. In finite dimensions, we have seen that the functional integralyields a map of simplicial sets from the simplicial set QME of solutionsto the quantum master equation to the simplicial set of solutions to thequantum master equation on cohomology.

A similar result holds in infinite dimensions, as long as we impose someextra constraints on our gauge fixing conditions.

Definition 10.7.1. A family of gauge fixing conditions parameterizedby )n, given by a C"()n) linear map

QGF : E , C"()n)$ E , C"()n),

is non-negative if the operator

D = [Q, QGF ]

is symmetric for some Hermitian metric on the vector bundle E, with non-negative eigenvalues, and if we have a direct sum decomposition

E = KerD4 Im Q4 Im QGF .

Here, KerD, Im QGF , and ImQ are both regarded as sub C"()n)-modulesof E , C"()n).

We will letH = KerD ) E , C"()n)

denote the sub C"()n)-module of harmonic elements of E .

In this situation, KerD is canonically isomorphic to the Q-cohomologyof E , C"()n).

Proposition 10.7.2. Let G F (E , Q)+ denote the simplicial set of pos-itive gauge fixing conditions, and let T (")(E , Q)+ denote the fibre productof T (")(E , Q) with G F (E , Q)+, over G F (E , Q).

Let QME (H+(E , Q)) denote the simplicial set of solutions to the quan-tum master equation on the graded symplectic vector space H+(E , Q).

There is a canonical map of simplicial sets

T (")(E , Q)+ $ QME (H+(E , Q))



which, on 0-simplices, takes an element {I[L]} of T (")(E , Q)+ to the re-striction of I[#] to the space of harmonic elements of E .

Proof. Suppose that

QGF : E , C"()n)$ E , C"()n)

is a family of non-negative gauge fixing conditions.Suppose that

{I[L] & O+(E ,&+()n))[[!]]}is a family of theories for this family of gauge fixing conditions.

We need to show how to construct a function in

O+(H+(E , Q),&+()n))[[!]]

which satisfies the quantum master equation. Further, this construction isrequired to be compatible with face and degeneracy maps of simplices.

The first step in the construction is to embed H+(E , Q) , &+()n) as asub &+()n)-module of E , &+()n).

The positivity conditions we imposed on the operator

[Q,QGF ] : E , C"()n) $ E , C"()n)

imply that there is a canonical isomorphism

Ker[Q,QGF ] %= H+(E , Q), C"()n).

This gives a map

$ : H+(E , Q)$ E , C"()n) ) E , &+()n)

sending & H+(E , Q) to the family $( )(t1, . . . , n) of harmonic representa-tives of for the family of Laplacians [Q,QGF ]. Here, t1 < · · · < tn refer tothe coordinates on the n-simplex.

However, the resulting map

H+(E , Q)$ E , &+()n)

is not a cochain map. Although Q$( ) = 0, ddR$( ) is not necessarily zero.We need to modify it so that it becomes a cochain map.

LetA = [Q,QGF ]$1QGF : E , &+()n)$ E , &+()n).

The positivity assumptions we impose on [Q,QGF ] implies that this operatorexists.

The modification is defined by

/$ $( )!A · ddR$( ) + (A · ddR)2$( )! · · ·

='

l'0

(!A · ddR)l$( ).

Note that (A · ddR)n+1 = 0, so that this series converges.



Since

[Q,A · ddR] = ddR

[Q,ddR] = 0Q$( ) = 0

we find that

(Q + ddR)'

l'0

(!A · ddR)n$( )

='

!(!A · ddR)k[Q,A · ddR](!A · ddR)m$( )

+ ddR

'(!A · ddR)l$( )

='

!(!A · ddR)kddR(!A · ddR)m$( )

+ ddR

'(!A · ddR)l$( )

='

!ddR(!A · ddR)l$( )

+ ddR

'(!A · ddR)l$( )

=0.

Thus, we have constructed a cochain map

H+(E , Q)$ E , &+()n).

The image of the resulting map

H+(E , Q), &+()n)$ E , &+()n)

is the kernel of [Q + ddR, QGF ].The positivity assumptions we have imposed on the operator [Q,QGF ]

imply that, if Kt is the heat kernel for the operator [Q + ddR, QGF ], thenK" exists. Further, the propagator

P ( ,#) =! "

QGF Ktdt

exists and is smooth for all > 0.It follows that if

{I[L] & O+(E ,&+()n))[[!]]}satisfies the renormalization group equation, and the quantum master equa-tion, then I[#] exists.

The kernel K" is the kernel for the operator of projection onto

Ker[Q + ddR, QGF ] = H+(E , Q), &+()n).

It follows that

K" & H+(E , Q),H+(E , Q) ) E , E , &+()n)

is the inverse to the natural degree !1 symplectic pairing on H+(E , Q).



Thus, if I[L] & O+(E ,&+()n))[!]] solves the renormalization groupequation and the quantum master equation, then

I[#] |H((E ,Q)($((%n)

solves the quantum master equation on H+(E , Q),&+()n), as desired. !

11. Obstruction theory

We have seen that the simplicial set ET (n+1)(E , Q) of pre-theories definedmodulo !n+2 is a torsor over the simplicial set ET (n)(E , Q) for the simplicialabelian group of local action functionals. We would like to have a similarresult for the simplicial set T (n+1)(E , Q) of theories.

What we find is that there is a cohomological obstruction to extending apoint of T (n)(E , Q) to a point in T (n+1)(E , Q). This obstruction is a localaction functional. The set of possible lifts is the set of ways of making theobstruction cochain exact; thus, the set of possible lifts is a torsor for theabelian group of closed elements of the space of local action functionals.

11.1. Let us fix a free BV theory (E , Q) on a compact manifold M , andlet {I[L]} & T (n)(E , Q)[k] be a k-simplex in the space of theories definedmodulo !n+1.

Let us lift, arbitrarily, {I[L]} to an element {:I[L]} of ET (n+1)(E , Q)[k],that is, a pre-theory defined modulo !n+2.

Let us define the scale L obstruction by

On+1[L] = !$n$11Q:I[L] + 1

2{:I[L], :I[L]}L + !)L:I[L]

2.

This expression is independent of !, because of the !$n$1 and the fact that:I[L] satisfies the quantum master equation modulo !n+1.

Lemma 11.1.1. Let be a parameter of square zero and cohomologicaldegree !1. Let I0[L] be I[L] modulo !. Then,

I0[L] + On+1[L]

satisfies both the scale L classical master equation and the classical renor-malization group equation. Thus, I0[L]+ On+1[L] defines a classical theoryin the BV formalism.

The set of lifts of {I[L]} to a k-simplex of T (n+1)(E , Q) is the set ofdegree 0 elements J [L] & Oloc(E ,&+()k)) such that I0[L] + J [L] satisfiesthe classical renormalization group equation and locality axiom modulo 2,and such that

QJ [L] + {I0[L], J [L]} = On+1[L].

Proof. We will prove this using a simple lemma about di!erentialgraded Lie algebras. Let g be a di!erential graded Lie algebra, and letX & g be an odd element. Let

O(X) = dgX + 12 [X,X].


11. OBSTRUCTION THEORY 187

Then,dgO(X) + [X,O(X)] = 0.

Let us makeO(E ,&+()k))[[!]]/!n+2[1]

into a di!erential graded Lie algebra, with bracket {!,!}L, and with di!er-ential Q+!)L. (Here, Q incorporates the de Rham di!erential on &+()k)).Then,

!n+1On+1[L] = (Q + !)L)I[L] + 12{I[L], I[L]}L

from which it follows that

QOn+1[L] + {I0[L], On+1[L]}L = 0

as desired.Next, we need to show that that I0[L] + On+1[L] satisfies the classical

renormalization group equation. In fact, the classical renormalization groupequation can be expressed as a classical master equation. Let us considerthe algebra

O(E ,&+()k), &+((0,#)L)).Let us give this algebra a BV structure, where the di!erential is the totaldi!erential coming from Q : E $ E and the de Rham di!erential on &+()k)and &+((0,#)L). The BV operator defined by the formula

)L + dL"(QGF(1)KL.

An elementJ [L] & O(E ,&+()k), C"((0,#)L))[[!]]

satisfies the renormalization group equation and the quantum master equa-tion if and only if, when we view it as an element of

O(E ,&+()k), &+((0,#)L)),

it satisfies the quantum master equation.This formulation is just a rephrasing of the infinitesimal form of the

renormalization group equation.Note that :I[L] satisfies the renormalization group equation. It follows

that, if we view :I[L] as an element:I[L] & O(E ,&+()k), &+((0,#)L)),

that On+1[L] is the obstruction to :I[L] satisfying the renormalization groupequation. Thus, the previous argument applies to show that I0[L]+ On+1[L]satisfies the classical RGE as well as the classical master equation.

For the final clause, observe that any other lift of I[L] to a k-simplexof ET (n+1)(E , Q) is of the form :I[L] + !n+1J [L] where I0[L] + J [L] satisfiesthe classical RGE modulo 2. The obstruction vanishes for such a lift if andonly if

QJ [L] + {I0[L], J [L]} = On+1[L].!



Since I0[L] + On+1[L] satisfies the classical master equation, it fol-lows that the L $ 0 limit of On+1[L] exists, and is an element On+1 &Oloc(E ,&+()k)). Further, if I0 & Oloc(E ,&+()k)) refers to the classicalaction at scale 0, then

QOn+1 + {I0, On+1} = 0

where the bracket {!,!} is now taken at scale zero.Thus, we have shown the following.

Corollary 11.1.2. Let {I[L]} & T (n)(E , Q)[k]. Then, there is an ob-struction

On+1 & Oloc(E ,&+()k))which is a closed, degree 1 element, where the cochain complex has the dif-ferential Q + {I0,!}.

The set of lifts of {I[L]} to a k-simplex of T (n+1)(E , Q) is the set ofdegree 0 elements J & Oloc(E ,&+()k)) making On+1 exact.

11.2. Let us consider Oloc(E ) as a cochain complex, with di!erentialQ + {I0,!}, where I0 = limL&0 I0[L] is the classical interaction. There isa simplicial abelian group associated to this cochain complex, by the Dold-Kan correspondence; the n-simplices are elements Oloc(E ,&+()k)) which areclosed and of degree 0. We will refer to this simplicial abelian group by thesame notation, Oloc(E ).

We are also interested in the simplicial abelian group Oloc(E )[1], whosen-simplices are closed, degree 1 elements of Oloc(E ,&+()k)).

Now let us choose, arbitrarily, a section ET (n)(E , Q) $ ET (n+1)(E , Q).This could come, for example, from the choice of a renormalization scheme.

The obstruction On+1 described above gives a map of simplicial sets

On+1 : T (n)(E , Q)$ Oloc(E )[1].

We can rephrase the obstruction theory results above in more homotopy-theoretic terms as follows.

Corollary 11.2.1. There is a homotopy fibre diagram of simplicial sets

T (n+1)(E , Q) !!

""

0

""

T (n)(E , Q)On+1 !! Oloc(E , Q)[1]

This corollary just says that a point of T (n+1)(E , Q) is the same as apoint of T (n)(E , Q) together with a homotopy between the obstruction andzero.

Proof. Let us define a simplicial set POloc(E , Q)[1] whose k-simplicesare pairs

, & Oloc(E ,&+()k))


12. BV THEORIES ON Rn 189

which are of degrees 0 and 1 respectively, such that is the coboundary of,

(Q + ddr) + {I0, } = .

There is a map

POloc(E , Q)[1]$ Oloc(E , Q)[1]

sending ( , ) $ . This map is a fibration, and POloc(E , Q)[1] is con-tractible.

The obstruction theory results above show that

T (n+1)(E , Q) = T (n)(E , Q)"Oloc(E ,Q)[1] POloc(E , Q)[1].

It follows from the fact that the map POloc(E , Q)[1] $ Oloc(E , Q)[1] is afibration that this fibre product is in fact a homotopy fibre product. SincePOloc(E , Q)[1] is contractible, we find that T (n+1)(E , Q) is the homotopyfibre product

T (n+1)(E , Q) . T (n)(E , Q)"LOloc(E ,Q)[1] {0}

as desired. !Corollary 11.2.2. The map

T (n+1)(E , Q) $ T (n)(E , Q)

is a fibration of simplicial sets, as is the map

T (")(E , Q)$ G F (E , Q).

Proof. This follows from the previous corollary, and the fact thatT (0)(E , Q) is a product of G F (E , Q) with the simplicial set whose k-simplices are elements of Oloc(E ,&+()k)) satisfying the scale 0 classicalmaster equation. !

12. BV theories on Rn

Now we will see how to define theories in the BV formalism on Rn. Themain di!erence from the discussion on a compact manifold is that on Rn

we can talk about renormalizability. This requires us to assume that ourbundle of fields has an extra grading, by dimension; this will allow us toapply the renormalization group flow.

Definition 12.0.3. A free theory on Rn in the BV formalism is describedby the following data:

(1) A bigraded vector space E. The first grading is called the cohomo-logical grading, and the second grading is the dimension grading.

We will think of E as a trivial vector bundle on Rn, and let

E = E ,S (Rn)

be the space of Schwartz sections.



The grading by dimension on E induces an R>0 action on E =S (Rn), E, by

Rl (f(x)e) = f( x) ie

where f(x) & S (Rn) and e is of dimension i.(2) E is equipped with a non-degenerate degree !1 anti-symmetric pair-

ing'!,!( : E , E $ det(Rn)

which respects dimension, so that Ei pairs with En$i.The pairing on E induces an integration pairing

'!,!( : E , E $ C

'f1e1, f2e2( =!

Rnf1f2 'e1, e2(

where ei & E and fi & S (Rn). This pairing is of dimension zero.(3) A di!erential operator Q : E $ E which is translation invariant,

of cohomological degree one, preserves dimension, is of square zero,and is skew self adjoint for the pairing.

Throughout, we will abuse notation and refer to the totality of data ofa free BV theory by E .

12.1. As on a compact manifold, we need to choose a gauge fixing oper-ator QGF : E $ E in order to discuss the renormalization group flow. Theset of possible gauge fixing conditions will form a simplicial set G F ; thefollowing definition describes the set of m-simplices.

Definition 12.1.1. Let (E , Q) be a free BV theory. A family of gaugefixing operators on E , parameterized by &+()m), is an &+()m) linear dif-ferential operator

QGF : E , &+()m)$ E , &+()m)

such that:(1) QGF is of cohomological degree !1, translation-invariant, of square

zero, and self-adjoint for the pairing '!,!(.(2) QGF is of dimension !2.(3) The commutator

D = [Q + ddR, QGF ]

is a sum of two terms

D = D! +D!!,

where:(a) D! is the tensor product of the Laplacian on Rn with the iden-

tity on E.(b) D!! is a nilpotent operator commuting with D!.



12.2. We will use the notation of Chapter 3, Section 3. Thus,

O(E ,&+()m) =/

k>0

Hom(Dg(Rnk,&+()m)), E(k)Sk

where Dg(Rnk,&+()m)) refers to the space &+()m)-valued distributions onRnk, which are invariant under the action of Rn by translation, and of rapiddecay away from the diagonal, as detailed in Chapter 3, Section 3.

We will letOloc(E ,&+()m)) ) O(E ,&+()m))

denote the subspace of local action functionals, which are translation invari-ant, as always.

12.3. There is a heat kernel

Kl & C"(Rn " Rn), E ,E , &+()m)

for the operator e$l D. If we pick a basis ei for E, then

Kl ='

$i,j(x, y, l)ei , ej

where, for all basis elements ek and functions g(x) & S (Rn),

(!1)|ek|'

i,j

!

y#Rn$i,j(x, y, l)g(y)ei 'ej , ek( = e$lDg(x)ek

Here |ek| denotes the parity of the basis element ek.The functions $i,j(x, y, l) appearing in the heat kernel are of the form

$i,j(x, y, l) = ((l1/2, l$1/2, x! y)e$.x$y.2/l

where( & &+()m)[l1/2, l$1/2, x! y]

is a polynomial in the variables l±1/2 and x!y, with coe#cients in &+()m).The propagator of our theory is given by

P ( , L) =! L

l=(QGF , 1)Kldl.

This propagator satisfies the conditions of Chapter 3, Section 3, so thatthe bijection between collections of e!ective interactions and local actionfunctionals holds in this situation.

Further, the propagator P ( , L) scales as

RlP ( , L) = P (l$2 , l$2L).

Thus, the conditions of Chapter 4, Section 6 are satisfied.

Definition 12.3.1. As on a compact manifold, a pre-theory is a col-lection of e!ective interactions {I[L]} satisfying renormalization group flowand the locality axiom as defined in Chapter 3, Section 3.

More formally, we define a simplicial set ET (")(E , Q) of pre-theories.An m-simplex of ET (")(E , Q) consists of



(1) An m-simplex QGF of the simplicial set of gauge fixing operators.(2) A collection

I[L] & O+,0(E ,&+()m))[[!]]

of e!ective interactions, which are of cohomological degree 0 and atleast cubic modulo !.

The following axioms must be satisfied:(1) The renormalization group equation

I[L] = W (P ( , L), I[ ])

is satisfied.(2) The locality axiom of Chapter 3, Section 3 holds. If we consider

Ii,k[L] as an element

Ii,k[L] & Dg(Rnk,&+()m)), (E ))(k,

then, if e & S (Rnk) , E(k has compact support away from thesmall diagonal,

Ii,k[L](e) $ 0 as L $ 0.

We let ET (")(E , Q) denote the simplicial set of theories, and ET (n)(E , Q)denote the simplicial set of theories defined modulo !n+1.

We will reserve the word theory for collections of e!ective interactions{I[L]} which satisfy the quantum master equation, as defined below.

12.4. As in Chapter 4, there is a natural action of R>0 on the set ofpre-theories. This action is called the local renormalization group flow.

The local renormalization group flow also acts on families of theoriesover the n-simplex, and so gives an action of R>0 on the simplicial setET (")(E , Q).

As before, if {I[L]} & ET (")E , Q[m] is an m-simplex in the space ofpre-theories, then

RGl(I[L]) & O+(E ,&+()m))[[!]], C[l, l$1, log l].

We let:R(")(E , Q) ) ET (")(E , Q)EM (")(E , Q) ) ET (")(E , Q)

be the sub simplicial sets of relevant and marginal pre-theories, respectively;where, as before, an m-simplex {I[L]} in the space of pre-theories is relevantif

RGl(I[L]) & O+(E ,&+()m))[[!]], C[l, log l].and marginal if

RGl(I[L]) & O+(E ,&+()m))[[!]], C[log l].

(Note that all theories we are considering are translation invariant)



12.5. Let us define a simplicial abelian group whose m-simplices aredegree 0 elements of Oloc(E ,&+()m)). We will denote this simplicial abeliangroup by Oloc(E ). We will let

O iloc(E ) ) Oloc(E )

denote the sub simplicial abelian group whose m simplices are degree 0 anddimension i elements of Oloc(E ,&+()m)). These elements are not requiredto be closed, as we are not currently dealing with the quantum masterequation. Similarly, we will denote by O'0

loc (E ) the simplicial abelian groupof local action functionals of dimension 5 0.

The results of Chapter 4 show the following:

Theorem 12.5.1. (1) The simplicial set ET (n+1)(E , Q) is a torsorover ET (n)(E , Q) for the simplicial abelian group Oloc(E ).

(2) The simplicial set :R(n+1)(E , Q) is a torsor over :R(n)(E , Q) for thesimplicial abelian group O'0

loc .(3) The simplicial set EM (n+1)(E , Q) is a torsor over EM (n)(E , Q) for

the simplicial abelian group O'0loc .

12.6. Next we will define a simplicial set of theories, which are pre-theories satisfying the quantum master equation. As on a compact manifold,we can define a scale L BV operator )L and a scale L bracket {!,!}L onthe space

O+(E ,&+()m))[[!]].We say that an element I & O+(E ,&+()m))[[!]] satisfies the scale L quan-tum master equation if

(Q + ddR)I[L] + !)LI[L] + {I[L], I[L]}L

where ddR is the de Rham di!erential on &+()m).As on a compact manifold, we have the following.

Lemma 12.6.1. A functional

I & O+(E ,&+()m))[[!]]

satisfies the scale quantum master equation if and only if

W (P ( , L), I)

satisfies the scale L quantum master equation.

We can now define the notion of theory in the BV formalism.

Definition 12.6.2. Define a sub-simplicial set

T (")(E , Q) ) ET (")(E , Q)

of theories, inside the simplicial set of pre-theories, as follows.An m-simplex in T (")(E , Q) is an m-simplex in ET (")(E , Q), described

by a collection of e!ective interactions

{I[L] & O+(E ,&+()m))[[!]]}



such that each I[L] satisfies the scale L quantum master equation.Similarly, let us define simplicial sets R(")(E , Q) and M (")(E , Q) of

relevant and marginal theories, respectively, by

R(")(E , Q) = :R(")(E , Q) 3T (")(E , Q)

M (")(E , Q) = EM (")(E , Q) 3T (")(E , Q).

12.7. Suppose we have an m-simplex {I[L]} in the space T (n)(E , Q) oftheories defined modulo !n+1. Then, as in Section 11, there is an obstruction

On+1({I[L]}) & Oloc(E ,&+()m))

to lifting {I[L]} to an m-simplex of T (n+1)(E , Q). This obstruction is closed,

QOn+1({I[L]}) + ddROn+1({I[L]}) + {I0, On+1({I[L]})} = 0,

and of cohomological degree 1. Here I0 is the scale zero classical action.Lifts of {I[L]} to T (n+1)(E , Q) are the same as degree zero cochains ofOloc(E ,&+()m)) which bound On+1({I[L]}).

12.8. Recall that the obstruction On+1({I[L]}) depends on a lift of{I[L]} to a pre-theory defined modulo !n+2. (The cohomology class of theobstruction, however, is independent of such a lift).

If {I[L]} & R(n)(E , Q) is a relevant theory defined modulo !n+1, thenwe can lift it to a relevant pre-theory :I[L], defined modulo !n+2.

Lemma 12.8.1. In this situation, the obstruction On+1({I[L]} (definedusing a relevant lift :I[L]) is of dimension 5 0.

Proof. This is a simple calculation. Recall that there is a scale Lobstruction defined by

!n+1On+1[L] = (Q + ddR):I[L] + !)L:I[L] + {:I[L], :I[L]}L.

The scale zero obstruction we have been using is the L$ 0 limit of On+1[L].Let R+

l denote the operation induced from the rescaling map Rl : E $ E ,on all spaces of functionals on E .

We need to show that

R+l On+1[l2L] & O(E ,&+()m)), C[l].

In other words, we need to check that only positive powers of l appear.Because On+1[L] satisfies a version of the classical renormalization group

equation, it is automatic that no powers of log l can appear in this expression.The fact that :I[L] is a relevant theory means that

R+l:I[l2L] & O(E ,&+()m)), C[l, log l].

In addition, the bracket {!,!}L and the BV operator )L have a compati-bility with the operator R+

l , namely

R+l {I, J}l2L = {R+

l I,R+l J}L

R+l ()l2LI) = )LR+

l I.



Putting these identities together yields the proof. !

In a similar way, if {I[L]} is an n-simplex in the space of marginal theo-ries, we can arrange it so that the obstruction On+1({I[L]}) is a dimensionzero local action functional.

From this lemma, we see that to lift an element {I[L]} & R(n)(E , Q) toan element of R(n+1)(E , Q) is the same as giving a local action functional

J & Oloc(E ,&+()m)),

of cohomological degree 0 and dimension 5 0, which kills the obstructionOn+1:

(Q + ddR)J + {I0, J} = On+1.

A similar statement holds for marginal theories.

12.9. Let us denote by Oloc(E , Q) the simplicial abelian group whosem-simplices are closed, degree 0 elements of Oloc(E ,&+()m)). Similarly,let O'0

loc (E , Q) denote the simplicial abelian groups whose n-simplices areclosed degree 0 elements of O'0

loc (E ,&+()m)), and let O0loc(E , Q) be the

simplicial abelian group whose n-simplices are closed degree 0 elements ofO'0

loc (E ,&+()m)). Recall that the superscripts 5 0 and 0 indicate that weare looking at elements of dimension 5 0 or 0 respectively.

The obstructions discussed above are maps of simplicial sets

On+1 : T (n)(E , Q) $ Oloc(E , Q)[1]

On+1 : R(n)(E , Q) $ O'0loc (E , Q)[1]

On+1 : M (n)(E , Q) $ O0loc(E , Q)[1].

Just like on a compact manifold, we have the following theorem.

Theorem 12.9.1. There are homotopy Cartesian diagrams

T (n+1)(E , Q) !!

""

0

""

T (n)(E , Q)On+1 !! Oloc(E , Q)[1]

for all theories,

R(n+1)(E , Q) !!

""

0

""

R(n)(E , Q)On+1 !! O'0

loc (E , Q)[1]



for relevant theories, and

M (n+1)(E , Q) !!

""

0

""

M (n)(E , Q)On+1 !! O0

loc(E , Q)[1]

for marginal theories.

13. The sheaf of BV theories on a manifold

In this section, we will describe the BV formalism in the most generalset-up we will need. We will show that there is a simplicial sheaf of theories,on any manifold, compact or not. On a compact manifold, global sectionswill be theories in the sense we described earlier.

13.1. We will set things up in somewhat greater generality than wehave used before. Everything in this chapter will depend on some auxiliarydi!erential graded ring.

Definition 13.1.1. A di!erential graded manifold is a manifold X, pos-sibly with corners, equipped with a sheaf of commutative graded algebras Aover the sheaf C"

X of smooth functions on X, together with a di!erentialoperator

dA : A $ A

where A = '(X,A), with the following properties.(1) As a C"

X -module, A is locally trivial of finite rank, so that A arisesas the sheaf of sections of some graded vector bundle.

(2) dA is a derivation of degree 1 and square zero.(3) There is a sheaf of ideals m ) A which is nilpotent (that is, mk = 0

for some k ; 0), such that A/m = C"X .

A di!erential graded vector bundle on (X,A) is a sheaf E of locallyfree, finite rank A-modules, whose space of global sections will be denotedE ; together with a di!erential operator dE : E $ E which makes E into adi!erential graded A -module.

If (X,A), (Y,B) are dg manifolds, a map (X,A) $ (Y,B) is a smoothmap f : X $ Y , together with a map of sheaves of graded C"

X -algebrasf+B $ A, compatible with di!erentials.

If (X,A) and (Y,B) are dg manifolds, we can take their product; this isthe dg manifold (X"Y,A"B). Any ordinary manifold M is a dg manifold,where A = C"

M .

Definition 13.1.2. Let M be a manifold. A family of free BV theorieson M , parameterized by a dg manifold (X,A), is the following data.

(1) A dg bundle E on (M " X,A), whose di!erential will be denotedQ : E $ E .


13. THE SHEAF OF BV THEORIES ON A MANIFOLD 197

(2) A mapE ,A E $ DensM ,A

which is of degree !1, anti-symmetric, and leads to an isomorphism

HomA(E,A),DensM $ E

of sheaves of A-modules on M "X.This pairing leads to a degree !1 anti-symmetric pairing

'!,!( : E ,A E $ A .

(3) If e, e! & E ,

dA+e, e!

,=+Qe, e!

,+ (!1)|e|

+e,Qe!

,.

Definition 13.1.3. Let (E,Q, '!,!() be a family of free BV theorieson M parameterized by A . A gauge fixing condition on E is an A -lineardi!erential operator

QGF : A $ A

such thatD = [Q,QGF ] : E $ E

is a generalized Laplacian in the sense of (BGV92). This means that thesymbol (D) of D, is a smooth family of metrics on T +M , parameterized byX, times the identity on E.

Throughout this section, we will fix a family of free theories on M ,parameterized by A . We will take A to be our base ring throughout; thus,tensor products, etc. will be taken over A .

If U ) M is an open subset, then a family of free theories (E , Q) onM , parameterized by A , restricts to one (E |U , Q) on U . Similarly, a gaugefixing condition for (E , Q) restricts to a gauge fixing condition for (E |U , Q).Thus, we have a sheaf G F sh(E , Q) of gauge fixing conditions for (E , Q) onM .

All of our constructions will be functorial with respect to change ofthe base ring A . Note that the forms on the n-simplex &+()n) providea cosimplicial dg manifold. Thus, the sheaf of gauge fixing conditions on(E , Q) over A has a natural enrichment to a simplicial presheaf, whose n-simplices are gauge fixing conditions on E ,&+()n) over A ,&+()n). Thissimplicial presheaf will be denote G F sh,%(E , Q).

A similar remark holds for theories over A , which we will define shortly.

13.2. If we were working on a compact manifold, one could constructthe propagator for the theory using the heat kernel for D. Heat kernels onnon-compact manifolds are more di#cult to deal with. In Chapter 2, Section14, we dealt with this problem by using a “fake heat kernel” to define thepropagator.

A fake heat kernel is defined in Chapter 2, Section 14 to be an element

Kl & E ,A E , C"((0,#)l)



satisfying the same small l asymptotics one expects for the actual heat ker-nel.

Let us choose such a fake heat kernel. This allows us to define the fakepropagator

P ( , L) =! L

(QGF , 1)Kldl & E ,A E .

We will assume that our fake heat kernel Kl is such that the fake propagatorP ( , L) is symmetric.

13.3. Let Ec denote the space of sections of the bundle E on M " Xwhich are supported on a subset of M which is proper over X. Recall thatin Chapter 2, Section 14 we defined a subspace

Homp(E (A nc , R) ) Hom(E (A n

c , R)

of properly supported E)-valued distributions on Mn "X. An element ofHom(E (A n

c , R) is in Homp(E (A nc , R) if its support C ) Mn " X has the

property that all of the projection maps C $ M "X are proper.We will let

Op(Ec) =/

n>0

Homp(E (A nc , R)Sn .

Definition 13.3.1. A pre-theory for (E , Q) on M consists of a gaugefixing condition QGF : E $ E , and a collection of e!ective interactions

I[L] & O+p (Ec)[[!]]

satisfying the axioms of Definition 14.6.1. That is, the renormalizationgroup equation

I[L] = W (P ( , L), I[ ]) .

must hold, and I[L] must admit a small L asymptotic expansion in terms oflocal action functionals.

We will let ET (n)(E , Q) denote the set of pre-theories defined modulo !n.

In Chapter 2, Section 14 we showed the following.(1) The notion of pre-theory is independent of the choice of fake heat

kernel.(2) If I[L] defines a pre-theory on M , then we can restrict I[L] in a

canonical way to get a pre-theory on any open subset U ) M . Inthis way, the set of pre-theories forms a sheaf on the manifold M ,which we will denote by ET (n)

sh (E , Q).(3) The map ET (n)

sh (E , Q)$ ET (n$1)sh (E , Q) makes the sheaf ET (n)

sh (E , Q)into a torsor for the sheaf Oloc(E ) of cohomological degree zero localaction functionals on M .

(4) The choice of renormalization scheme gives rise to an isomorphismof sheaves

ET (")sh (E ) %= O+

loc(E )[[!]]



between the sheaf of pre-theories and the sheaf of local action func-tionals which are at least cubic modulo !.

13.4. Next, we will define a subsheaf of the sheaf of pre-theories con-sisting of those pre-theories which satisfy the quantum master equation, i.e.,which are theories.

Let us fix, as above, a fake heat kernel Kl. Unfortunately, as the fakeheat kernel does not satisfy the heat equation (except in an asymptotic senseas l $ 0), we do not have the identity

(Q, 1 + 1,Q)P ( , L) = K !KL.

This makes it a little more tricky to define the quantum master equationthan on a compact manifold. It turns out that the correct BV operator touse is not that defined by K , but something slightly di!erent.

Lemma 13.4.1. The limit

lim&0

(K ! (Q, 1 + 1,Q)P ( , L))

exists, and is an element of E ,A E (that is, it is a smooth section of E"AEon M2 "X).

Let us denote this limit by K !L. Note that

(Q, 1 + 1,Q)P ( , L) = K ! !K !L.

Let)L : O(Ep)$ O(Ep)

be the operator of contraction with !K !L. As in Section 7, one can also

define a bracket {!,!}L associated to the kernel KL.

Definition 13.4.2. A functional I & Op(E )+[[!]] satisfies the scale Lquantum master equation if

QI[L] + 12{I[L], I[L]} + !)LI[L] = 0.

A pre-theory {I[L]} is a theory if each I[L] satisfies the scale L quantummaster equation. As before, the renormalization group equation implies thatif I[L] satisfies the scale L QME for one value of L, then it does for allothers.

We will let T (n)(E , Q) denote the space of theories on M .

13.5. Everything, so far, has depended on the choice of a fake heatkernel. However, we have seen in Chapter 2, Section 14 that the notion ofpre-theory is independent of the choice of fake heat kernel.

Indeed, suppose Kl, :Kl are two fake heat kernels, with associated fakepropagators P ( , L), :P ( , L). Suppose that {I[L]} defines a theory for thefake propagator P ( , L). Then,

:I[L] = W1:P (0, L)! P (0, L), I[L]

2



defines a theory for the fake propagator :P (0, L). This expression is well-defined because :P (0, L)! P (0, L) is smooth, that is, an element of E , E .

Lemma 13.5.1. Suppose that the theory {I[L]} satisfies the quantummaster equation defined using the fake heat kernel Kl. Then {:I[L]} sat-isfies the quantum master equation defined using the fake heat kernel :Kl.

Proof. Recall that we defined

K !l = lim

&0(K ! (Q, 1 + 1,Q)P ( , L)) .

Let us define :K !l in the same way, using :Kl and :P ( , L) in place of Kl and

P ( , L).Then,

(Q, 1 + 1,Q)(P (0, L)! :P (0, L)) = :K !L !K !

L.

Let )L, :)L denote the two BV operators. It follows from this identity that

[Q, "P (0,L)$ eP (0,L)] = :)L !)L

where, as usual, "P (0,L)$ eP (0,L) refers to the operator on Op(Ec) defined by

contracting with P (0, L)! :P (0, L).This identity says that "P (0,L)$ eP (0,L) is a chain homotopy between the

two BV operators. As in Lemma 9.2.2, it follows that if I[L] satisfies the)L quantum master equation, then

W1:P (0, L)! P (0, L), I[L]

2

satisfies the :)L quantum master equation.!

13.6. So far, we have seen that the set of pre-theories ET (n)(E , Q) arisesas the set of global sections of a sheaf ET (n)

sh (E , Q) on M . We have not yetshown that the set of theories forms a subsheaf. We need to show thatthe quantum master equation is local in nature, and thus compatible withrestriction.

This we will do using the obstruction techniques of 11.Let {I[L]} be a theory on M , defined modulo !n+1. In Section 11, we

showed how to construct an obstruction

On+1({I[L]}) & Oloc(E )

to the theory {I[L]} satisfying the quantum master equation modulo !n+2.This obstruction is of degree 1, and satisfies the equation

QOn+1({I[L]}) + {I0, On+1({I[L]})} = 0.

The formula for the obstruction given there applies in this more generalsetting, also, and we find, as in Section 11, that lifts of {I[L]} to a theory



defined modulo !n+2 are the same as local action functionals J & Oloc(E )such that

QJ + {I0, J} = On+1({I[L]}).The cochain representative of the obstruction depends on a lift of I[L] to

a pre-theory :I[L] defined modulo !n+2, even though the cohomology classof the obstruction is independent of such a lift. However, the choice ofa renormalization scheme gives us a section of the map ET (n+1) $ ET (n),thus giving a lift of any theory defined modulo !n+1 to a pre-theory definedmodulo !n+2. When we speak about the obstruction, we will mean theobstruction associated to the choice of a renormalization scheme in thisway.

The following lemma shows that the obstruction does not depend on thechoice of a fake heat kernel, and is local in nature, so that the set of theoriesforms a subsheaf of the sheaf of pre-theories.

Lemma 13.6.1. (1) Let {I[L]} be a theory on M , defined modulo!n+1. Then, the restriction {IU [L]} of {I[L]} to any open subsetof U is a theory, that is, satisfies the quantum master equation.

(2) The restriction of the obstruction On+1({I[L]}) to U is the obstruc-tion to the restriction {IU [L]}, that is,

On+1 ({IU [L]}) = On+1 ({I[L]}) |U & Oloc(E |U ).

(3) The obstruction On+1({I[L]}) is independent of the choice of a fakeheat kernel.

Proof. We will assume, by induction, that the first statement holds:that is, that the restriction of a theory {I[L]} defined modulo !n+1 to U isa theory on U .

Let :I[L] denote the lift of I[L] to a pre-theory defined modulo !n+2,which arises from the choice of a renormalization scheme.

Recall that we have a scale L obstruction defined by

On+1[L] = !$n$11Q:I[L] + {:I[L], :I[L]}L + !)L

:I[L]2

.

This scale L obstruction is such that I0[L] + On+1[L] satisfies the classicalrenormalization group equation and the classical master equation, whereis a parameter of degree !1. The L $ 0 limit of this scale L obstructionexists, and is the local obstruction On+1({I[L]}) & Oloc(E ).

Let us suppose that the pre-theory :I[L] corresponds to a local actionfunctional I & Oloc(E )[!]/!n+2, under the bijection between pre-theoriesand local action functionals arising from the choice of a renormalizationscheme. It follows that :I[L] restricted to U corresponds to I restricted toU .

It is immediate from the graphical expression of On+1[L] that the smallL asymptotics of On+1[L], when restricted to U , only depends on the smallL asymptotics of the heat kernel KL and on the Ii,k restricted to U .



This proves the second and third statements of the lemma, and we cancontinue by induction. !

Thus, the set T (n)(E , Q) of theories arises as the global sections of asheaf of theories on M , which we will denote by T (n)

sh (E , Q). The set ofsections of this sheaf over an open subset U ) M is the set T (n)(E |U , Q) oftheories for the restriction of the free theory (E , Q) to U .

13.7. So far we have been fixing a base dg manifold (X,A). If f :(Y,B) $ (X,A) is a map of dg manifolds, then we can pull back E to a dgvector bundle on M " (X,A). Global sections of this dg vector bundle are

f+E = E ,A B.

The sheaves of theories and pre-theories are natural with respect to thesepull-backs; there are maps

f+ : T (n)sh (E , Q)$ T (n)

sh (f+(E , Q)).

This allows us to enrich our sheaf T (n)sh (E , Q) into a simplicial presheaf,

which we will denote by T (n)sh,%(E , Q). The set T (n)

sh,%(E , Q)(U)[k] of k-simplices of this simplicial set of sections over U will be defined by

T (n)sh,%(E , Q)(U)[k] = T (n)(E |U , &+()k), Q + ddR).

Here, E |U , &+()k) is viewed as a module for A , &+()k), and we workrelative to the base ring A , &+()k).

13.8. Let us view Oloc(E , Q) as a simplicial presheaf on M , whosek-simplices, on an open subset U ) M , are closed degree 0 elements ofOloc(E |U ,&+()k)).

Then, the obstruction map

On+1 : T (n)sh,%(E , Q)$ Oloc(E , Q)[1]

is a map of simplicial presheaves.When we talked about the simplicial set of theories on a compact man-

ifold, we saw that the obstruction map allows us to represent T (n+1)(E , Q)as a homotopy fibre product of simplicial sets. We would like to have asimilar statement in this context.

Instead of delving into the homotopy theory of simplicial presheaves– which can be somewhat technical – I will simply explain how to writeT (n+1)

sh,% (E , Q) as an actual fibre product.Let us define a simplicial POloc(E , Q)[1] by saying that the sections of

the presheaf of k-simplices, on an open subset U , are pairs of elements

, & Oloc(E |U ,&+()k))

such that is of degree 0, is of degree 1, and

(Q + ddR) + {I0, } = .


14. QUANTIZING CHERN-SIMONS THEORY 203

There is a map

POloc(E , Q)[1]$ Oloc(E , Q)[1]( , ) /$ .

Further, POloc(E , Q)[1] is contractible.

Theorem 13.8.1. There is a fibre diagram of simplicial presheaves

T (n+1)sh,% (E , Q) !!

""

POloc(E , Q)[1]

""

T (n)(E , Q) !! Oloc(E , Q)[1]

Proof. This is an immediate corollary of the obstruction theoretic re-sults above. !

Note that since POloc(E , Q)[1] is contractible, to show that we have ahomotopy fibre diagram

T (n+1)sh,% (E , Q) !!

""

0

""

T (n)(E , Q) !! Oloc(E , Q)[1]

it su#ces to show that the map

POloc(E , Q)[1]$ Oloc(E , Q)[1]

is a fibration in the appropriate model structure.

14. Quantizing Chern-Simons theory

In this section, we will give a very simple obstruction theoretic proof ofthe existence of a quantization of Chern-Simons theory on any 3-manifold,modulo constants. For a more direct approach, including stronger results,see (Cos07; CM08a; Iac08). This section is written as an example in theobstruction-theoretic techniques developed in this book; I do not prove anynew results about Chern-Simons theory.

14.1. Let M be a 3-manifold and let g be a Lie algebra with an invariantpairing. Let us assume that H i(g) = 0 if i is 1, 2 or 4. (This is satisfied, forexample, if g is semi-simple).

The space of fields for Chern-Simons theory with values in g is

ECS(M) = &+(M), g[1].

The operator Q : ECS(M)$ ECS(M) arising from the quadratic part of theChern-Simons operator is simply the de Rham di!erential, and the cubic



part of the action is, as usual,

ICS = 16

!

M' , [ , ](g

where '!,!(g refers to the chosen invariant pairing on g.

14.2. Recall that if we choose a Riemannian metric on M , we get anassociated gauge fixing condition for Chern-Simons theory on M , where thegauge fixing operator is QGF = d+.

In Section 10 we showed how to construct a simplicial set of gauge fixingconditions, and a simplicial set of theories living over the simplicial set ofgauge fixing conditions. We will apply this construction to Chern-Simonstheory.

Let Met(M) denote the simplicial set of Riemannian metrics on M .Thus, the n-simplices are smooth families of Riemannian metrics parame-terized by 6n. Let T (")(ECS(M)) $ Met(M) denote the simplicial set,constructed in Section 10, of quantizations of our classical Chern-Simonstheory. Thus, a 0-simplex of T (")(ECS(M)) is a metric on M , and a quan-tization of Chern-Simons theory with the corresponding gauge fixing condi-tion. An n-simplex of T (")(ECS(M)) is a smooth family of metrics on Mand a family of quantizations for this family of metrics.

In Section 11 we showed that the map

T (")(ECS(M)) $ Met(M)

is a fibration of simplicial sets.Let

CS"(M) = '(Met(M),T (")(ECS(M)))

denote the simplicial set of sections of T (")(ECS(M)).A point in CS"(M) is a “universal” quantization of Chern-Simons theory

on M ; that is, a quantization of Chern-Simons theory for every metric onM , which varies in a homotopically trivial fashion when the metric varies.

Let CSn(M) be defined in the same way as CS"(M), except that quan-tizations are only considered modulo !n+1.

In this section, we will prove the following result.

Theorem 14.2.1. For any connected 3 manifold M , there is an isomor-phism of simplicial sets

CS"(M) %= !H3(g)[[!]]

where the right hand side is regarded as a constant simplicial set.

Note that H3(g) is the natural home of the level of Chern-Simons theory.Indeed, if g is a semi-simple Lie algebra with no Abelian factors, then H3(g)is canonically isomorphic to the tangent space to the space of non-degenerateinvariant bilinear forms on g. Thus, this result shows that specifying a


14. QUANTIZING CHERN-SIMONS THEORY 205

quantization of Chern-Simons theory is equivalent to specifying an R[[!]]-valued invariant form on g, which modulo ! is a given form; or equivalently,to specify a !-dependent level.

This is certainly not the strongest result in this direction; see (Cos07;CM08a; Iac08) for stronger results. However, the result proved here is prob-ably the strongest result one can prove which does not require a detailedanalysis of the Feynman graphs appearing in Chern-Simons theory. Theproof given below is a simple application of the obstruction theoretic tech-niques we developed earlier.

14.3. Let us now turn to the proof of Theorem 14.2.1. First, we willstate some simple results regarding the group of obstructions to the quanti-zation of Chern-Simons theory.

The deformation-obstruction complex for quantizing Chern-Simons the-ory is, as we have seen in 11, the complex Oloc(ECS(M))/R with di!erentialQ + {ICS ,!}. (Here we are working modulo constants).

In Section 6, we showed how to compute the deformation-obstructioncomplex for quantizing any theory using an auxiliary DM Lie algebra, ormore generally, a DM L"-algebra. For Chern-Simons theory, the DM L"-algebra is J(&+(M) , g)), the DM L"-algebra of jets of forms with coe#-cients in the Lie algebra g. This is quasi-isomorphic to the DM L"-algebrag, C"

M , which arises from the locally constant sheaf of Lie algebras g.Thus, the results of Section 11 show the following.

Lemma 14.3.1. There is a quasi-isomorphism

(Oloc(ECS(M)), Q + {ICS ,!} . M ,DM (C+red(g), C"

M ) .

It follows from this and from our assumptions about the cohomology ofg that the space of obstructions to quantizing & CSn(M) to order n + 1 isH1(M,H3(g)), and that the space of deformations of any such quantizationis H0(M,H3(g)).

14.4. Now we are ready to prove 14.2.1. We will prove the result byinduction. Thus, let us assume, by induction, that for all 3-manifolds M ,every

k$1 & CSk$1(M)extends to an element

k & CSk(M).

To prove the theorem, we need to show that for every k & CSk(M) theobstruction

O( k) & H1(M,H3(g))to extending k to the next order vanishes.

Recall that, if U )M is an open subset, there is a restriction map

CSk(M)$ CSk(U).



Further, this restriction map is compatible with the formation of the ob-struction class. That is, the following diagram commutes:

CSk(M) !!

""

CSk(U)

""

H1(M,H3(g)) !! H1(U,H3(g))

where the vertical arrows are the obstruction maps.We will use this fact to deduce that the obstruction O( k) is zero.Let us choose a basis ai of H1(M, R) where each ai is the fundamental

class of an embedded circle in M . Let Ui be an embedded torus in M whichsupports the homology class ai. Let k |Ui denote the restriction of k toUi. We need to show that

O( k |Ui) = 0 & H1(Ui,H3(g)).

Let us choose an embedding of Ui into R3. There is a commutativesquare

CSk(R3) !!

""

CSk(Ui)

""

CSk$1(R3) !! CSk$1(Ui).

The vertical maps are torsors for the same abelian group, H0(Ui,H3(g)) =H0(R3,H3(g)). It follows that this is a fibre square. This allows us todeduce, by induction, that the map

CSk(R3) $ CSk(Ui)

is a bijection.From this, we see that the obstruction O( k |Ui) & H1(Ui,H3(g)) is the

restriction of a class in H1(R3, H3(g)), and is thus zero, as desired.This completes the proof of Theorem 14.2.1.


CHAPTER 6

Renormalizability of Yang-Mills theory

1. Introduction

In this chapter we prove the following theorem.

Theorem 1.0.1. Pure Yang-Mills theory on R4, or on any four-manifoldwith a flat metric, with coe"cients in any semi-simple Lie algebra g, isperturbatively renormalizable.

The proof is more conceptual than existing proofs in the physics litera-ture (for instance, we don’t rely on any graph combinatorics). The proof isan application of the results developed in Chapter 5, in particular of Theo-rem 12.9.1. To apply this result to Yang-Mills theory, we need to verify thatcertain cohomology groups of the complex of classical observables vanish.This turns out to be a computation in Gel’fand-Fuchs cohomology. Thisobstruction theoretic argument is in some ways unsatisfactory; it relies onapparently fortuitous vanishing of certain Lie algebra cohomology groups(the fact that H5(su(3))Out(su(3)) = 0 plays a crucial role). However, I don’tknow of any more direct argument.

If our Lie algebra g is a direct sum of k simple Lie algebras, we find thatthere are k coupling constants. This means that the space of renormalizablequantizations of pure Yang-Mills theory we construct depends on k parame-ters, term-by-term in !; the space of such quantizations is the space of series!R[[!]]/k.

In addition, we show that the renormalizable quantizations of pure Yang-Mills theory constructed here are universal in the Wilsonian sense describedin Chapter 4: any deformation is equivalent, in the low energy limit, to arenormalizable deformation. However, the usefulness of Wilsonian univer-sality in this situation is counteracted by the phenomenon of asymptoticfreedom, which implies that at low energy the perturbation expansion forthe theory is not well-behaved.

We only need to prove the result on R4, as any theory on R4 which isinvariant under Euclidean symmetries gives a theory on any four-manifoldwith a flat metric.

2. First-order Yang-Mills theory

We have seen in Chapter 5 how to put the usual formulation of Yang-Mills theory into the Batalin-Vilkovisky formalism. Further, we have seen

207


208 6. RENORMALIZABILITY OF YANG-MILLS THEORY

that the lack of a suitable gauge fixing condition prevents us from applyingour renormalization techniques to this theory. In this section we will in-troduce the first-order formulation of Yang-Mills theory, which resolves thisproblem.

I should emphasize that the first-order formulation of Yang-Mills theoryis completely equivalent to the usual second-order formulation. In the courseof this chapter we will give a careful proof of the equivalence of classicalYang-Mills theory in the first and second order formulations. The first-order formulation is a familiar device in the physics literature, where theequivalence of first and second order formulations is taken as established(see, for example, (CCRF+98)).

Let g be a Lie algebra equipped with an invariant pairing. Throughoutthis chapter, &i(Rn) will denote the space of Schwartz i-forms on Rn.

The first-order formulation of Yang-Mills theory has two fields: a con-nection A & &1(R4) , g, as before, as well as a self-dual two form B &&2

+(R4),g. The Lie algebra of infinitesimal gauge symmetries is &0(R4),g.If X & &0(R4), g, the fields A and B transform as

A /$ [X,A] + dX

B /$ [X,B].

In other words, A transforms as a connection, and B transforms as a form.The action, with coupling constant c, is

S(A,B) = 'F (A), B(+ c 'B,B( = 'F (A)+, B(+ c 'B,B(

where '!,!( denotes the inner product on the space &+(R4), g given by

' 1 ,X1, 2 ,X2( =!

R41 9 2 'X1, X2(g .

The functional integral for this first-order treatment of Yang-Mills isthus

!

(A,B)#($1(R4)(g/$2+(R4)(g)/G

eS(A,B)/!

where we integrate over the quotient of the space &1(R4), g4&2+(R4), g

of fields by the action of the gauge group G = Maps(R4, G).The quadratic part of the action S(A, B) is

'dA,B(+ c 'B,B( .

2.1. Let us now apply the Batalin-Vilkovisky machine to first-orderYang-Mills theory. The extended space of fields we end up with is described


2. FIRST-ORDER YANG-MILLS THEORY 209

in the following diagram:

&0(R4), g ghosts, degree ! 1

&1(R4), g4 &2+(R4), g fields, degree 0

&2+(R4), g4 &3(R4), g anti-fields, degree 1

&4(R4), g anti-ghosts, degree 2

Let us denote this extended space of fields by E .The graded vector space E is naturally an odd symplectic vector space;

the ghosts pair with the anti-ghosts and the fields with the anti-fields.To describe the action, we need names for variables living in the various

direct summands of E . Let us denote by X a ghost variable, that is, anelement of &0(R4), g; the field variables will be denoted by A & &1(R4), gand B & &2

+(R4) , g; the anti-field variables are A) & &2+(R4) , g and

B) & &2+(R4), g; and finally the anti-ghost variable X) & &2

+(R4), g.Then the action, with coupling constant c, can be written

SFO(c) = 12

+[X,X], X),+

+[X,B], B),+

+dX,A),

++[X,A], A),+ 'F (A), B(+ c 'B,B( .

The subscript FO indicates “first-order”. The first term in this expressiongives the Lie bracket on the space of ghosts; the next three terms accountfor the action of this Lie algebra on the space of fields; and the final twoterms give the first-order Yang-Mills action on the space of fields.

One convenient way to describe this BV action is to introduce an aux-iliary di!erential graded algebra, Y . We define Y and its di!erential Q bythe following diagram:

Y 0

.

Y 1

.

Y 2

.

Y 3

.

&0(R4) d !! &1(R4) d !!

L

&2+(R4)

L

&2+(R4) d !!

2c Id$$!!!!!!!!!

&3(R4) d !! &4(R4)

In this diagram, and throughout this chapter, we abuse notation and write

d : &1(R4) $ &2+(R4)

for the de Rham di!erential composed with the projetion onto the summand&2

+(R4) of &2(R4).The algebra structure on Y is as follows. The middle row of the diagram

forms an algebra in a natural way: if , & &1(R4), their product is

( 9 ) & &2+(R4)



where is the projection from &2(R4) onto &2+(R4). Thus, the middle row

of the diagram is the quotient of the ordinary de Rham complex of R4 bythe di!erential ideal consisting of &2

$(R4), &3(R4) and &4(R4).The bottom row of this diagram is a module over the middle row, in an

evident way. This defines the commutative algebra structure.The algebra Y has a trace

Tr : Y $ Rof degree !3, defined by

Tr(a) =!

R4a

if a & Y 3 = &4(R4), and Tr a = 0 otherwise.We can identify the space Y , g[1] with the Batalin-Vilkovisky space

of fields for the first-order formulation of Yang-Mills theory. Note that Yhas an odd symmetric pairing defined by Tr(ab). Thus, Y , g has an oddsymmetric pairing defined by

+a, E, a! , E!, = Tr(aa!)

+E,E!,

g.

Thus, Y ,g[1] has an odd symplectic pairing; this is the same as the Batalin-Vilkovisky odd symplectic pairing.

The general BV procedure produces an action on Y , g[1].

Lemma 2.1.1. The Batalin-Vilkovisky action for first-order Yang-Millstheory is the Chern-Simons type action

SFO(a, E) = 12 'a, E,Qa, E(+ 1

6 'a, E, [a, E, a, E]( .

This is a simple direct computation.

3. Equivalence of first-order and second-order formulations

3.1. The idea behind the equivalence between the first- and second-orderformulations of Yang-Mills is very simple. In the first-order formulation, thefields in cohomological degree zero are a connection A and a self-dual twoform B. If we perform the gauge invariant change of variables

B /$ B ! 12cF (A)+

then the first-order action

'F (A)+, B(+ c 'B,B(changes to

! 14c 'F (A)+, F (A)+(+ c 'B,B( .

The first term is the usual Yang-Mills action. Since this change of coordi-nates is upper triangular, and therefore formally preserves the measure, wecan formally conclude that the theory given by the first-order action is equiv-alent to the theory given by the new action ! 1

4c 'F (A)+, F (A)+(+ c 'B,B(.The field B is non-interacting in this new action; thus, we can integrate itout to leave the field A with the usual Yang-Mills action.


3. EQUIVALENCE OF FIRST-ORDER AND SECOND-ORDER FORMULATIONS 211

3.2. In this section we will show how to make this argument more precisein the context of the BV formalism. Thus, let us return to considering thespace Y , g[1] of BV fields for the first-order Yang-Mills theory. As before,let us denote a ghost by X & &0(R4),g, an anti-ghost by X) & &4(R4),g,an anti-field for B by B) & &2

+(R4) , g, and an anti-field for A by A) &&3(R4), g.

Recall that we can write the second-order Yang-Mills action, in the BVformalism, with coupling constant g, as

SY M (g) = 12

+[X,X], X),+

+[X,A], X),+

+dX,A),! 1

g'F (A)+, F (A)+( .

LetH = ! 1

2c

+F (A)+, B),

In the classical BV formalism, the first-order Yang-Mills action SFO(c) isequivalent to the action

SFO(c) + {H,SFO(c)}.Let us consider the one-parameter family of equivalent actions

S(t) & Oloc(Y , g[1])

where

S0 = SFO(c)ddt

S(t) = {H,S(t)}.

Lemma 3.2.1. The action S1 is

S1(X,A,B,A), B), X)) = SY M (4c) + c 'B,B(++[X,B], B), .

where SY M (4c) is the standard second-order Yang-Mills action in the BVformalism with coupling constant 4c, as above.

Proof. Recall that the action S can be written as

SFO(c) = SGauge + 'F (A)+, B(+ c 'B,B(where SGauge encodes the Lie bracket on the space of ghosts, as well as theaction of this Lie algebra on the space of fields A and B.

Because the functional

H = ! 12c

+F (A)+, B),

is gauge invariant,{SGauge,H} = 0.

Note also that

{H, 'F (A)+, B(} = ! 12c 'F (A)+, F (A)+(

{H, 'B,B(} = !1c 'F (A)+, B(

Thus,

St = SGauge + a(t) 'F (A)+, F (A)+(+ b(t) 'F (A)+, B(+ c 'B,B(



whereddt

a(t) = ! 12cb(t)

ddt

b(t) = !1

b(0) = 1a(0) = 0.

Thus,

b(t) = 1! t

a(t) = ! 12c t + 1

4c t2

so that

S1 = SGauge ! 14c 'F (A)+, F (A)+(+ c 'B,B(

= SY M (4c) ++[X,B], B),+ c 'B,B(

as desired. !

3.3. Thus, we see that at the classical level, the action SFO(c) for thefirst-order formulation of Yang-Mills is equivalent, in the BV formalism, tothe action SY M (4c) + '[X,B], B)(+ c 'B,B(.

People often argue, heuristically, that this equation is true at the quan-tum level as well, by saying that the change of coordinates we have performedis upper triangular and thus does not a!ect the (non-existent) “Lebesguemeasure”.

For us, these considerations are of no importance. The main theorem inthis chapter is a classification of all possible renormalizable quantizations ofclassical Yang-Mills theory. The proof of this classification is of a purely co-homological nature, and works for any version of classical Yang-Mills theory.The only di#culty with the most familiar version is the lack of a suitablegauge fixing condition.

3.4. The BV formalism allows us to “integrate out” some of the fields ina theory. If we split our odd symplectic vector space as a symplectic directsum of two such spaces, and then integrate over a Lagrangian subspace ofone of them, we are left with an action functional on the other symplecticvector space.

Let us apply this procedure to integrate out the field B (and its anti-fieldB)). We will see that we are left with the usual second-order formulationof Yang-Mills theory, in the BV formalism. This argument is somewhatheuristic, as we are dealing with infinite dimensional integrals. However, itis rather straightforward to see that this integrating out procedure works atthe classical level, and yields an equivalence between classical Yang-Mills inits usual formulation and the classical first-order Yang-Mills we are using.


4. GAUGE FIXING 213

Let us write the space of fields Y , g[1] as a direct sum

Y , g[1] = E1 4 E2

where

E1 =-&0(R4), g[1]

.4-&1(R4), g

.4-&3(R4), g[!1]

.4-&4(R4), g[!2]

.

andE2 =

-&2

+(R4), g.4-&2

+(R4), g[!1]..

This is a symplectic direct sum, and E1 is the space of BV fields forsecond-order Yang-Mills theory.

LetL ) E2

be the obvious Lagrangian subspace,

L = &2+(R4), g.

Recall that the action on the space Y , g[1] we are using is

SY M (4c) ++[X,B], B),+ c 'B,B( .

Integrating out the fields in E2 leaves the functional on E1 defined by

! log#!

B#LeSY M (4c)/!+c0B,B1/!

$= SY M (4c) + C(3.4.1)

where C is a constant.Thus, we see that integrating out the field B leaves the second-order

version of Yang-Mills theory in the BV formalism.

4. Gauge fixing

As we have seen, the main di#cult with working with second-order Yang-Mills theory is that there seems to be no gauge fixing condition suitable forour renormalization techniques. In this section we will construct such agauge fixing condition for first-order Yang-Mills theory.

LetQGF : Y $ Y

be the di!erential operator defined by the diagram

&0(R4) &1(R4)d(##

L

&2+(R4)2d(

##

L

&2+(R4) &3(R4)

2d(## &4(R4)

d(##

.

In the map &3(R4) $ &2+(R4), we are composing 2d+ with the projection

onto &2+(R4) but, as usual, we omit this from hte notation.

Then, the Laplacian-type operator

D = [Q, QGF ]



can be decomposed asD = D! + 4cD!!

where both operators D! and D!! are independent of the coupling constantc. An elementary calculation shows that D! is the usual Laplacian on thespaces of forms, and that D!! is given by the diagram

degree ! 1 degree 0 degree 1 degree 2

&0(R4) &1(R4) &2+(R4)

&2+(R4)

d(

%%

&3(R4)

d(

%%

&4(R4)

Note that

[D!, D!!] = 0-D!!.2 = 0.

This implies that the gauge fixing condition satisfies the technical require-ments of Chapter 5, and so that there is a propagator given by the integralof a heat kernel satisfying the various axioms we need.

5. Renormalizability

We have already defined pure Yang-Mills theory in the first-order formu-lation at the classical level. In this section we will state the main theorem ofthis chapter, which allows us to construct this theory at the quantum level.

Let us split up the first-order action SFO as a sum

SFO(a, E) = 'a,E,Qa, E(+ I(0)(a, E)

where, as before, Q is described by the diagram

&0(R4) d !! &1(R4) d !!

L

&2+(R4)

L

&2+(R4) d !!

2c Id$$!!!!!!!!!

&3(R4) d !! &4(R4).

The functionalI(0)(a,E) & Oloc(Y , g[1])

is the interacting part of the action. I(0) is cubic, of cohomological degreezero, and satisfies

QI(0) + 12{I

(0), I(0)} = 0.

We would like to turn this into a quantum theory. At the classical level,Yang-Mills theory is conformally invariant. Thus, I(0) & M (0). We wouldlike to classify all lifts of I(0) to elements of R("), that is, to a relevantquantum theory.


5. RENORMALIZABILITY 215

LetH i,j(Oloc(Y , g[1])R4

)

denote the cohomology of the complex Oloc(Y ,g[1])R4 of translation invari-ant local action functionals, with di!erential Q+{I(0),!}, in cohomologicaldegree i and scaling dimension j.

Suppose we have an element X(n) & R(n) which is a lift of the classi-cal theory to a theory defined modulo !n+1. Theorem 12.9.1 implies thatthe obstruction to lifting X(n) to R(n+1) is an element of H1,'0(Oloc(Y ,g[1])R4). If the obstruction vanishes, the moduli space of lifts up to equiva-lence is a quotient of H0,'0(Oloc(Y , g[1])R4) by some action of the spaceH$1,'0(Oloc(Y , g[1])R4). If H$1,'0(Oloc(Y , g[1])R4) also vanishes, thenthe space of lifts up to equivalence is isomorphic to H0,'0(Oloc(Y ,g[1])R4).

Thus, the key question in understanding renormalizability of Yang-Millsis the computation of the cohomology groups H i,j(Oloc(Y , g[1])R4) whenj 5 0 and i = !1, 0, 1. Since, at the classical level, the theory is invari-ant under SO(4), we can restrict ourselves to considering quantizationswhich are also SO(4)-invariant. Thus, we need only compute the groupsH i,j

1Oloc(Y , g[1])R4"SO(4)

2when j 5 0 and i = !1, 0, 1.

Theorem 5.0.1. Let g be a semi-simple Lie algebra. For any non-zerovalue of the coupling constant c, there are natural isomorphisms

H i,01Oloc(Y , g[1])R4"SO(4)

2=

JKL

KM

0 if i < 0H0(g, Sym2 g) if i = 0H5(g) if i = 1

Further,

H i,j1Oloc(Y , g[1])R4"SO(4)

2= 0 if j > 0 and i * 2.

The proof of this theorem will occupy most of this chapter.I should remark that the same result holds if we replace Y , g[1] by

the space of BV fields for second-order Yang-Mills theory. This is becausethe theorem is a calculation of the cohomology of the deformation com-plex for the classical theory, and first-order and second-order Yang-Mills areequivalent at the classical level and so have quasi-isomorphic deformationcomplexes.

5.1. Since the cohomology groups H0,j1Oloc(Y , g[1])R4"SO(4)

2vanish

if j > 0, we see that any relevant lift X(n) & R(n) of classical Yang-Millsis equivalent to a marginal lift Y (n) & M (n). Hence we will consider onlymarginal lifts.



However, there are potential obstructions to constructing a marginal lift,lying in the Lie algebra cohomology group

H5(g) = H1,01Oloc(Y , g[1])R4"SO(4)

2.

Unfortunately this group is non-zero if the semi-simple Lie algebra g containsa factor of su(n) where n 5 3.

We can ensure that such obstructions must vanish by asking that ourquantizations respect an additional symmetry. Let

H ) Out g

be the group of outer automorphisms of g which preserve the decomposition

g = g1 4 g2 4 · · ·4 gk

of g into simple factors. The classical Yang-Mills action I(0) is invariantunder the action of H. Thus, we can ask for quantizations which are in-variant under H; if the quantization X(n) & M (n) is H-invariant, then theobstruction

On+1(X(n)) & H i,01Oloc(Y , g[1])R4"SO(4)

2= H5(g)

will also be H-invariant.

Lemma 5.1.1. For any semi-simple Lie algebra g,

H5(g)H = 0.

Thus, if X(n) is H-invariant, the obstruction On+1(X(n)) vanishes.

Proof of lemma. If g = g1 4 g2 4 · · · 4 gk is the decomposition of ginto simple factors, then

H5(g) = 4iH5(gi).

Thus, it su#ces to prove that for any simple Lie algebra g,

H5(g)Out(g) = 0.

Recall that if G is the compact Lie group associated to g,

H+(g) = H+(G, R).

If g 8= su(n), then standard results on the cohomology of compact Lie groupsimply that H5(g) = 0. The only problems arise when g = su(n) when n 5 3.

If n > 3, the map SU(3) $ SU(n) induces an isomorphism on H5.Further,

Out(SU(n)) = Out(su(n)) = Z/2,

and this group acts by taking a unitary matrix A & SU(n) to its complexconjugate A. Therefore the map SU(3) $ SU(n) is equivariant for theaction of Z/2.

Thus, what we need to show is that

H5(SU(3))Z/2 = 0.


6. UNIVERSALITY 217

The fundamental class in H8(SU(3)) is the cup product of a generator ofH3(SU(3)) with a generator of H5(SU(3)). The Z/2 action on H3(SU(3))is trivial; thus, to show that H5(SU(3))Z/2 = 0, it su#ces to show thatH8(SU(3))Z/2 = 0; or, in other words, that the non-trivial element of Z/2acts on SU(3) in an orientation reversing way.

Thus, it su#ces to show that the map

su(3) $ su(3)

A /$ A

is orientation reversing. This is a simple computation.!

As a corollary of this lemma and of Theorem 5.0.1, we find the following.Let

M (")Y M )M (")

R(")Y M ) R(")

denote the sub-simplicial set of marginal (respectively, relevant) theorieswhich coincide at the classical level with Yang-Mills theory.

Corollary 5.1.2. The inclusion

M (")Y M $ R(")

Y M

is an isomorphism on 0.There is a (non-canonical) bijection

0

1M (")

Y M

2%= H0(g, Sym2 g), !R[[!]].

Thus, the set of renormalizable quantizations of pure Yang-Mills theory isthe set of deformations of the chosen pairing on g to a symmetric invariantpairing

g, g $ R[[!]]

which, modulo !, is the original pairing.

6. Universality

The quantizations of Yang-Mills constructed above are universal: anyother quantization of Yang-Mills is equivalent, in the low-energy limit, toone of the quantizations constructed in Theorem 5.1.2.

The precise statement is the following.

Theorem 6.0.3. Let

I &1T (")

Y M

2R4"SO(4)



be any quantization of first-order Yang-Mills theory which is invariant underthe Euclidean symmetries of R4. Then there is a theory I ! which is equiv-alent to I – that is, in the same connected component of the simplicial set1T (")

Y M

2R4"SO(4)) – and which satisfies the following.

(1) I ! has only marginal and irrelevant terms, that is, for all L,

RGl(I ![L]) & O(Y , g[1])[[!]], R[l$1, log l]

as a function of l.(2) There is a marginal theory J &M (")

Y M which is in the same univer-sality class as I !, that is,

RGl(I ![L])! RGl(I[L]) & O(Y , g[1])[[!]], l$1R[l$1, log l],

so that RGl(I ![L])! RGl(I[L]) tends to zero as l $#.Further, the marginal theory J is uniquely determined (up to contractiblechoice) by these properties.

Proof. The first statement follows immediately from the fact that thecohomology of Oloc(Y ,g[1])R4"SO(4) vanishes in positive scaling dimension.

The second statement is also straightforward. We have

RGl(I ![L]) & O(Y , g[1])[[!]], R[l$1, log l].

Define Jl[L] by discarding those terms of RGl(I ![L]) which have negativepowers of l. Then,

Jl[L] & O(Y , g[1])[[!]], R[l$1, log l].

It is straightforward to check that Jl[L] satisfies the renormalization groupequations and the quantum master equation:

W (P ( , L), Jl[ ]) = Jl[L]RGm(Jl[L]) = Jlm[L]

(Q + !)L)eJl[L]/! = 0.

These three equations are simple consequences of the corresponding equa-tions for RGl(I ![L]).

Thus, we letJ [L] = J1[L].

This set of e!ective interactions defines a marginal theory in the same uni-versality class as I !, as desired. !

7. Cohomology calculations

In this section we will prove Theorem 5.0.1.Let

Y = (Y )R4

so thatY = Y ,S (R4).


7. COHOMOLOGY CALCULATIONS 219

Explicitly,

Y =

JKKKL

KKKM

&0 degree 0&1 4 &2

+ degree 1&2

+ 4 &3 degree 2&4 degree 3

where &i refers to the space of translation invariant forms on R4.The space

Oloc(Y , g[1])R4

of translation invariant functionals on Y , g[1] is an odd Lie algebra underthe Batalin-Vilkovisky bracket. Let

Q : Oloc(Y , g[1])R4 $ Oloc(Y , g[1])R4

be the di!erential coming from the di!erential on Y , and let

I(0) & Oloc(Y , g[1])R4

be the interaction. This satisfies the classical master equation

QI(0) +12{I(0), I(0)} = 0.

Thus, Q + {I(0),!} defines a di!erential on Oloc(Y , g[1])R4 .We are interested in the complex Oloc(Y , g[1])R4 with respect to the

di!erential Q + {I(0),!}. This complex Oloc(Y , g[1])R4 is bigraded; thefirst grading is the usual cohomological grading, the second is by scalingdimension. Let

H i,j(Oloc(Y , g[1])R4, Q + {I(0),!})

denote the cohomology in cohomological degree i and scaling dimension j.Let NY denote the formal completion of the dga Y at 0. Thus,

NY = Y [[x1, x2, x3, x4]]

with the natural di!erential Q.Observe that NY is acted on by the algebra R["1, . . . , "4]. The generators

"i act by derivations. In a similar way, any tensor power of NY is acted onby R["1, . . . , "4].

The following lemma is a special case of Lemma 6.7.1 in Chapter 5,Section 6.

Lemma 7.0.4. There is an isomorphism of complexes

Oloc(Y , g[1])R4 %= C+red(NY , g),L

R["1,...,"4] R.

Here C+red denotes the reduced Gel’fand-Fuchs cohomology. The action of

R["1, . . . , "4] on NY , g is the obvious one; the action on R is the one whereeach "i acts trivially.

Further, the subcomplex of Oloc(Y , g[1])R4 of scaling dimension k cor-responds to the subcomplex of C+

red(NY , g)R4 of scaling dimension k ! 4.



We will apply this to prove Theorem 5.0.1, which is restated here.

Theorem. If the Lie algebra g is semi-simple, then

H i,0(Oloc(Y , g[1])R4, Q + {I(0),!})SO(4) =

JKL

KM

0 if i < 0H0(g,Sym2 g) if i = 0H5(g) if i = 1

Further,

H i,j(Oloc(Y , g[1])R4, Q + {I(0),!})SO(4) = 0 if j > 0 and i * 2.

All these isomorphisms are compatible with the action of the group of outerautomorphisms of g which preserve the chosen invariant pairing on g.

This is the key result that allows us to prove renormalizability of Yang-Mills. Again, this theorem holds only for fixed non-zero values of the cou-pling constant c.

Proof. There is a spectral sequence converging to the cohomology ofR,L

R[!1,...,!4]C+

red(NY , g) whose first term is R,LR[!1,...,!4]

H+red(NY , g). Thus,

the next step is to compute the reduced Lie algebra cohomology H+red(NY ,g).

We are interested, ultimately, only in the cohomology of Oloc(Y ,g[1])R4 innon-negative scaling dimension. This implies that we only need to computethe cohomology groups H+

red(NY , g) in scaling dimension 5 !4.Thus, let H i,j

red(NY , g) denote the ith reduced Lie algebra cohomologygroup in scaling dimension j.

Lemma 7.0.5.

H i,0red(NY , g) = H i

red(g)

H i,$1red (NY , g) = 0

H i,$2red (NY , g) = 0

H i,$3red (NY , g) = 0

H i,$4red (NY , g) = H i

-g, Sym2

-g) , 92R4

..

All isomorphisms are Aut(g)" SO(4) equivariant.

Proof. LetNY (k) ) NY

be the finite-dimensional subspace consisting of elements of scaling dimen-sion k. Then, Y is a direct product

NY =/

k'0

NY (k).

Also,NY (0) = R.



A simple computation shows that

H+(NY (1)) = 0

H+(NY (2)) = 92R4[!1].

(This second equation only holds when the coupling constant c is non-zero).We will first compute the ordinary (non-reduced) Lie algebra cohomol-

ogy H+(NY , g). The reduced Lie algebra cohomology is obtained by modi-fying this in a simple way.

LetC+(NY , g)(k) ) C+(NY , g)

be the subcomplex of scaling dimension k. There is a direct product decom-position

C+(NY , g) =/

k%0

C+(NY , g)(k)

of complexes. Each subcomplex C+(NY , g)(k) is finite dimensional.Recall that NY (0), g = g, and NY (k) = 0 if k < 0. Let

Y+ ) NYbe the subspace consisting of elements of scaling dimension > 0.

Observe that

C+(NY , g)(k) = Sym+(g+[!1]), C+(NY+ , g)(k)

= Sym+(g+[!1]), Sym+(NY+ , g[1]))(k).

Filter the complex C+(NY , g)(k) by saying

F rC+(NY , g)(k) = Sym+(g+[!1]), Sym'r(NY+ , g[1]))(k).

The filtration is finite:

C+(NY , g)(k) = F 0C+(NY , g)(k) < · · · < F k+1C+(NY , g)(k) = 0.

Thus, we find a spectral sequence converging to the cohomology of the com-plex C+(NY , g)(k) whose first term is

H+(g, Sym+(H+(NY+), g[1]))(k)).

The space Sym+1<

k>0 H+(NY (k)), g[1]2)

is viewed simply as a module forthe Lie algebra g.

Putting these spectral sequences together for varying k, we find a spec-tral sequence converging to H+(NY , g) whose first term is

H+

"g, Sym+

"/

k>0

H+(NY (k)), g[1]

%)%.

We can do a little better, because

H+(NY (1)) = 0.



Also, because g is semi-simple,

H+(g, g) = H+(g, g)) = 0

so that

H+

"g,

"/

k>0

H+(NY (k)), g[1]

%)%= H+(g, g)),

/

k>0

H+(NY (k))[!1] = 0.

Thus, the first term of the spectral sequence can be rewritten as

H+

5

6g, R4 Sym'2

5

6/

k'2

H+(NY (k)), g[1]

7

8)7

8 .

Now

Sym'2

5

6/

k'2

H+(NY (k)), g[1]

7

8)

consists entirely of terms of scaling dimension * !4, and the scaling dimen-sion !4 part is Sym2(H+(NY (2)), g[1])).

As before, let H i,j(NY , g) denote the ith Lie algebra cohomology inscaling dimension j. We see that

H i,j(NY , g) =

JKL

KM

H i(g) if j = 00 if j = !1,!2,!3H i(g, Sym2(H+(NY (2)), g[1]))) if j = !4.

If we take reduced Lie algebra cohomology, we find the same expressionexcept that H+(g) is replaced by H+

red(g).The final thing we need is that

H+(NY (2)) = 92R4[!1].

This equation, which only holds when the coupling constant c appearing inthe di!erential on Y is non-zero, is straightforward to prove.

!

Next, we should look again at the complex R ,LR[!1,...,!4]

C+red(NY , g).

This complex can be identified explicitly as

4 9i R4 , C+red(NY , g)[i]

with a di!erential which is a sum of the di!erential on C+red(NY , g) and

maps9iR4 , C+

red(NY , g)$ 9i$1R4 , C+red(NY , g)

arising from the action of the Lie algebra R4 on C+red(NY ,g), via the operators

"i.



We will use the spectral sequence for the cohomology of this doublecomplex, whose first term is

4 9i R4 ,H+red(NY , g)[i].

We are only interested in the part of scaling dimension 5 !4. Since 9iR4

comes with scaling dimension !i, we are interested in the subcomplex givenby

4j$i'$4 9i R4 ,H+,jred(NY , g)[i].

Lemma 7.0.5 implies that this can be rewritten as-90R4 ,H+ -g, Sym2

-g) , 92R4

...4D

i'0

-9iR4 ,H+

red(g)[i]..

The first summand is in scaling dimension !4, the remaining summands arein scaling dimension between 0 and !4.

We would like to compute the di!erentials of the spectral sequence. Alldi!erentials preserve scaling dimension; and, if we assume that all previousdi!erentials are zero, the di!erential on the kth page maps

9iR4 ,H+red(NY , g)[i]$ 9i$kR4 ,H+

red(NY , g)[i! k].

From this, we see that the only possibly non-zero di!erential, in scalingdimension 5 !4, is on the fourth page of the scaling dimension !4 part.This di!erential is a map

d4 : 94R4 ,H+red(g)[4]$ H+ -g, Sym2

-g) , 92R4

...

of cohomological degree one. After this, all spectral sequence di!erentialsmust vanish.

We are ultimately interested in the SO(4)-invariant part of the coho-mology. Thus, we will calculate the cohomology of d4 after taking SO(4)-invariants.

The following lemma completes the proof of the theorem.

Lemma 7.0.6. The di!erential d4 on the fourth page of the spectral se-quence

d4 : 94R4 ,H3red(g)$ H0

-g, Sym2

-g) , 92R4

..SO(4)

is injective. Further, the cokernel of this map is naturally isomorphic to

H0(g, Sym2 g)).

Proof. Let us write

g = g1 4 · · ·4 gk

as a direct sum of simple Lie algebras. Note that

H3(g) = 4H3(gi)

andH0(g, g, g) = 4H0(gi, g

)i , g)i ).



Since all constructions we are doing are functorial in the Lie algebra g, thedi!erential in the spectral sequence maps

94R4 ,H3(ga) $ H0-ga, Sym2

-ga , 92R4

...

for each simple direct summand ga of g.Thus, it su#ces to prove the statement with g replaced by one of its

constituent simple Lie algebras ga. First, we show that the map is injective,or equivalently non-zero (as H3(ga) = R).

This can be seen by an explicit computation. We will write the generatorof H3(ga) as '[x, y], z]( where x, y, z & ga. Let e1, . . . , e4 be a basis for R4.Then, the generator of 94R4 ,H3(ga) can be written as

'ijkleiejekel , '[x, y], z(

where ijkl is the alternating symbol on the indices shown.To compute how this element is mapped by the di!erentials of the spec-

tral sequence, first we map it to 93R4,C3red(NY ,ga) under the natural map.

The resulting element will be exact; we pick a bounding cochain, which isthen mapped to 92R4 , C2

red(NY , ga). Again, this element is exact, so wepick a bounding cochain, map it to 91R4 , C1

red(NY , ga), and so forth.The element of 93R4 , C3

red(NY , ga) is given by the formula'

ijkleiejek , '[x, y], z"l(

where, as before, "i is short for ""xi

, which we think of as an element of thedual of R[[x1, . . . , x4]].

This is exact, and a bounding cochain is given by the expression'

ijkleiejek , 'x, y"l(

(up to a non-zero constant). This expression is mapped to'

ijkleiej , 'x"k, y"l( & 92R4 , C2red(NY , ga).

Again, this is exact; a bounding cochain is given (up to sign) by'

ijkleiej ,+x"k, ydx)

l

,

where dx)l is an element of degree !1 in NY ), dual to the element dxl &

&1 ) NY 1.This expression is now mapped to

'ijklei ,

+x"k, ydx)

l "j,& 91R4 , C1

red(NY , ga).

A bounding cochain for this is'

ijklei ,+xdx)

k , ydx)l "j

,.

Finally this element is mapped to'

ijkl ,+xdx)

k "i, ydx)l "j

,& 90R4 , C0

red(NY , ga).



This is non-zero in the cohomology

H0,$4red (NY , ga) = H0

-ga, Sym2

-g)a , 92R4

..

We have seen that the map

H3(ga)$ H0-ga, Sym2

-g)a , 92R4

..SO(4)

is injective. It remains to show that the cokernel of this map is naturallyisomorphic to H0

-ga, Sym2 (g)a )

..

It is straightforward to calculate that the map-Sym2

-92R4

..SO(4) $-92R4 , 92R4

.SO(4)

is an isomorphism. Thus,

H0-ga, Sym2

-g)a , 92R4

..SO(4)

= H0-ga,Sym2

-g)a..,-Sym2

-92R4

..SO(4).

The map

H3(ga)$ H0-ga, Sym2

-g)a..,-Sym2

-92R4

..SO(4)

arises from the tensor product of a natural isomorphism

H3(ga) %= H0(ga, Sym2 g)a )

with a canonical invariant element of Sym2-92R4

.. To complete the proof,

it thus su#ces to check that the vector space-Sym2

-92R4

..SO(4)

is two-dimensional; this is straightforward.!!



Appendix 1: Asymptotics of graph integrals

In this Appendix, we will show that certain integrals attached to graphsadmit asymptotic expansions of a certain form. This is a key result in ourconstruction of counterterms, and so underlies all the results of this book.

Everything in this chapter will depend on some auxiliary manifold withcorners X, equipped with a sheaf A of commutative superalgebras over thesheaf of algebras C"

X , with the following properties:(1) A is locally free of finite rank as a C"

X -module. In other words, Ais the sheaf of sections of some super vector bundle on X.

(2) A is equipped with an ideal I such that A/I = C"X , and Ik = 0 for

some k > 0. The ideal I, its powers I l, and the quotient sheavesA/I l, are all required to be locally free sheaves of C"

X -modules.We will fix the data (X,A) throughout. The algebra '(X,A) of C" globalsections of A will be denoted by A .

In addition, we will fix a compact manifold M with a super vector bundleE on M , whose space of global sections will be denoted by E .

1. Generalized Laplacians

Definition 1.0.7. A di!erential operator

D : '(M,E) ,Dens(M)),A $ '(M,E),A

is called a generalized Laplacian if it has the following properties:(1) D is A -linear , and is of order 2 as a di!erential operator.(2) There exists a smooth family g of Riemannian metrics on M , pa-

rameterized by X, and an isomorphism

: E) ,Dens(M)$ E

of vector bundles on M , such that the symbol (D) of D is

(D) = g , & '(Sym2 T +M, Hom(E) ,Dens(M), E)),A .

A generalized Laplacian on the vector bundle E on M (whose globalsections is E ) is specified by a metric on M , a connection on E and apotential F & '(M, End(E)). A smooth family of generalized Laplaciansis a family where this data varies smoothly; this is equivalent to sayingthat the operator varies smoothly. We are dealing with a smooth familydepending on the manifold X with the bundle of algebras A. The metric onM has coe#cients only in C"(X) – and thus has no dependence on A – but

227


228 APPENDIX 1: ASYMPTOTICS OF GRAPH INTEGRALS

the parameters for the connection and the potential will have coe#cients inA = '(X,A).

The results of (BGV92) show that there is a heat kernel

Kt & E , E ,A

for the generalized Laplacian D. Further, this heat kernel has a well-behavedsmall t asymptotic expansion, as we will explain.

Let y & M . Let U ) M " X denote the open subset of points (z, x)where d (z, y) < , where is less than the injectivity radius of M for anyof the metrics arising from points in X. Normal coordinates on U gives anisomorphism

U %= Bn "X

where Bn is the ball of radius in Rn, and n = dimM .Thus, we get an isomorphism

C"(U) = C"(Bn), C"(X).

of C"(X)-algebras.The vector bundle E becomes a vector bundle (still called E) on Bn,

and we find'(U,E) = '(Bn, E), C"(X).

The following is a variant of a result proved in (BGV92), following(MP49; BGM71; MS67).

Theorem 1.0.8. There exists a small t asymptotic expansion of Kt

which, in these coordinates, is of the form

Kt . t$dim M/2e$.x$y.2/t'

i'0

ti i

where x, y denote coordinates on the two copies of Bn, and

i & '(Bn, E), '(Bn, E),A .

If we denote

KNt = t$ dim M/2e$.x$y.2/t

N'

i=0

ti i

then for all l & Z'0,99Kt !KN

t

99l= O(tN$(dim M)/2$l).

Here +·+l denotes the C l norm on the space C"(Bn), C"(Bn),A .

This precise statement is not proved in (BGV92), as they do not use theauxiliary nilpotent ring A. But, as Ezra Getzler explained to me, the proofin (BGV92) goes through mutatis mutandis.


3. PERIODS 229

2. Polydi!erential operators

Definition 2.0.9. Let Di!(E,E!) denote the infinite rank vector bundleon M of di!erential operators between two vector bundles E and E! on M .Let

PolyDi!('(E)(n,'(E!)) = '(M, Di!(E, C)(n,E!) ) Hom('(E)(n,'(E!))

where C denotes the trivial vector bundle of rank 1. All tensor products inthis expression are fibrewise tensor products of vector bundles on M .

In a similar way, let

PolyDi!%k('(E)(n,'(E!)) = '(M, Di!%k(E, C)(n , E!)

denote the space of polydi!erential operators of order * k.

It is clear that PolyDi!('(E)(n,'(E!)) is the space of sections of aninfinite rank vector bundle on M . Further, this infinite rank vector bundleis a direct limit of the finite rank vector bundles of polydi!erential operatorsof order * k. Thus, PolyDi!('(E)(n,'(E!)) is a nuclear Frechet space.

If F is a super vector bundle on another manifold N , let

PolyDi!('(E)(n,'(E!)), '(F )

denote the completed projective tensor product, as usual, so that

PolyDi!('(E)(n,'(E!)), '(F ) = '(M "N, (Di!(E, C)(n ,E!) " F ).

If X is a manifold, we can think of PolyDi!('(E)(n,'(E!)),C"(X) as thespace of smooth families of polydi!erential operators parameterized by X.

One can give an equivalent definition of polydi!erential operators interms of local trivialisations {ei}, {e!j} of E and E!, and local coordinatesy1, . . . , yl on M . A map '(E)(n $ '(E!) is a polydi!erential operator if,locally, it is a finite sum of operators of the form

f1ei1 , · · ·, fnein /$'

j

e!j$ji1...in

(y1, . . . , yl)(DIi1f1) · · · (DIin

fn)

where $ji1...in

(y1, . . . , yl) are smooth functions of the yi, the Ik are multi-indices, and the operators DIk are the corresponding partial derivatives withrespect to the yi.

3. Periods

Let us recall the definition of the subalgebra

P((0, 1)) ) C"((0, 1))

of periods.Suppose we have the following data.(1) an algebraic variety X over Q;(2) a normal crossings divisor D ) X;(3) a Zariski open subset U ) A1

Q, defined over Q, such that U(R)containts (0, 1).



(4) a smooth map X $ U , of relative dimension d, also defined overQ, whose restriction to D is flat.

(5) a relative d-form & &d(X/U), defined over Q, and vanishingalong D.

(6) a homology class & Hd((X1/2(C), D1/2(C)), Q), where X1/2 andD1/2 are the fibres of X and D over 1/2 & U(R). We assumethat is invariant under the complex conjugation map on the pair(X1/2(C), D1/2(C)).

Let us assume that the maps

X(C)$ U(C)D(C)$ U(C)

are locally trivial fibrations. For t & (0, 1) ) U(R), we will let Xt(C) andDt(C) denote the fibre over s(t) & U(R).

We can transfer the homology class & H+(X1/2(C), D1/2(C)) to anyfibre (Xt(C), Dt(C)) for t & (0, 1). This allows us to define a function f on(0, 1) by

f(t) =!

t

t.

The function f is real analytic, and real valued.

Definition 3.0.10. Let PQ((0, 1)) ) C"((0, 1)) be the subalgebra offunctions of this form. Elements of this subalgebra will be called rationalperiods.

LetP((0, 1)) = PQ((0, 1)), R ) C"((0, 1))

be the real vector space spanned by the space of rational periods. Elementsof P((0, 1)) will be called periods.

4. Integrals attached to graphs

Definition 4.0.11. A labelled graph is a connected, oriented graph ,with some number of tails (or external edges).

For each vertex v of , let H(v) denote the set of half edges (or germsof edges) emanating from v; a tail attached at v counts as a half edge.

Also, has an ordering on the sets of vertices, edges, tails and on theset of half edges attached to each vertex.

Each vertex v of is labelled by a polydi!erential operator

Iv & PolyDi!(E (H(v), Dens(M)),A .

Let O(v) denote the order of Iv.

Let T ( ) denote the set of tails of . Fix & E (T ( ), and fix te & (0,#)for each e & E( ). Define a function f (te, ) as follows.


4. INTEGRALS ATTACHED TO GRAPHS 231

Let H( ) denote the set of half edges of , so H( ) = =v#V ( )H(v). Byputting Kte at each edge of , and at the tails, we get an element

,e#E( ) Kte & E (H( ).

On the other hand, the linear maps Iv at the vertices define a map!

MV ( ),Iv : E (H( ) $ Dens(M)(V ( ) ,A

RMV ( )!!!!$ A .

Let

f (te, ) =!

MV ( ),Iv

-,e#E( ) Kte

.& A .

Theorem 4.0.12. The integral

F ( , T, ) =!

te#( ,T )E( )f (te, )

/dte

has an asymptotic expansion as $ 0 of the form

F ( , T, ) .'

fr( )(r(T, )

where fr( ) & P((0, 1) ), and the (r are continuous linear maps

E (T ( ) $ C"((0,#)T ),A

where T is the coordinate on (0,#).Further, each (r(T, ) has a small T asymptotic expansion

(r(T, ) .'

gr(T )!

M$r,k( )

where$r,k & PolyDi!(E (T ( ), Densities(M)),A .

and gr are smooth functions of T & (0,#).

I should make a few comments clarifying the statement of this theorem.(1) The functions fr( ) appearing in this small asymptotic are sums

of integrals of the form!

t1,...,tk#U3( ,1)k

F1(t1, . . . , tk)1/2

F2(t1, . . . , tk)1/2

where Fi, Gi & Z'0[t1, . . . , tk] are polynomials with positive integercoe#cients, and U ) ( , 1)k is an open subset cut out by inequalitiesf1(ti) > 0, . . . , fn(ti) > 0 where each fm is of the form tri ! tj , forsome 1 * i, j * k and some r & Z.

Functions of this form are always periods, in the sense we use.Indeed, we can define an algebraic variety

X = { , t1, . . . , tk, y, z | F2(ti) 8= 0, y2 = F1(ti), z2 = F2(ti)},



on which the function

F 1/21

F 1/22

=y

z

is defined.We define a divisor D ) X, as the zero locus of the function

"n/

i=1

fi

%5

6k/

j=1

(tj ! )(tj ! 1)

7

8 .

The cycle& Hk(X ,D )

is the locus where fi 5 0, * tj1. The form & &k(X , D ) is

=y

zdt1 . . . dtk.

(2) Asymptotic expansion means the following.For every compact subset K ) X, let +·+K,l denote the C l norm

on the space A = '(X,A), defined by taking the supremum overK of the sum of all order * l derivatives.

Let us consider F ( , T, ) as a linear map

E (T ( ) $ A , C"({0 < < T})/$ F ( , T, ).

The precise statement is that for all R, l & Z'0, all compact subsetsK ) X and L ) (0,#), there exists m & Z'0 such that

supT#L

99999F ( , T, )!R'

r=0

fr( )(r(T, )

99999K,l

< R+1 + +m

for all T su#ciently small. Here + +m denotes the Cm norm on thespace E (T ( ).

(3) The small T asymptotic expansion in part (2) has a similar defini-tion.

4.1. There is a variant of this result which holds when M is Rn. Inthat case, E is a trivial super vector bundle on Rn; all of the polydi!erentialoperators associated to vertices of are translation invariant; and the kernelKt is an element of C"(Rn " Rn),E(2 of the form

Kt ='

ei , ejf(+x! y+2)Pi,j(x! y, t1/2, t$1/2)e$.x$y.2/t

where ei is a basis of the vector space E, Pi,j are polynomials in the variablesxk ! yk and t±1/2, and f is a smooth function on [0,#) of compact supportwhich takes value 1 in a neighbourhood of zero.

This case is in fact much easier than the general case. The generalcase, of a compact manifold, is reduced to the case of Rn using the small t


5. PROOF OF THEOREM 4.0.12 233

asymptotic expansion of the heat kernel. Thus, the proof given on a compactmanifold applies for Rn also.

4.2. A second variant of this result, which we will occasionally need,incorporates a propagator of the form P ( , T ) + K , where is an oddparameter of square zero. In this case, the graphs we use may have onespecial edge, on which we put K instead of Kte . Thus, instead of integratingover the parameter te for this edge, we specialise to .

5. Proof of Theorem 4.0.12

The function F ( , T, ), whose small asymptotic behaviour we wantto understand, is an integral

F ( , T, ) =!

te#( ,T )E( )f (te, )

/dte.

Theorem 4.0.12 will be an immediate corollary of the existence of a smallasymptotic expansion for the integrand, f (te, ). What we will show isthat we can write the region of integration (0, T )E( ) as a union of a finitenumber of closed sets, and that on each such closed set the function f hasan asymptotic expansion in terms of rational functions.

More precisely, we will show the following.

Proposition 5.0.1. Let us choose a total ordering of the set of edgesE( ) of . This allows us to enumerate this set of edges as e1, . . . , ek, andlet ti denote tei.

Then there exist(1) a finite number C1, . . . , Cr of closed sets which cover (0, T )E( ),(2) Fi, Gi & Z'0[t1, . . . , tk] \ {0} for i & Z'0,(3) polydi!erential operators

(i & PolyDi!(E (T ( ), Dens(M)), C"(X)

for i & Z'0,(4) dr,mr & Z'0 where dr $#,

such that;;;;;f (t1, . . . , tk)!r'

i=1

Fi(t1, . . . , tk)1/2

Gi(t1, . . . , tk)1/2

!

M(i( )

;;;;; * + +mr(max(ti))dr

for all{t1, . . . , tk} & Cj ) (0, T )k

with max(ti) su"ciently small. Here, max(ti) denotes the largest of thet1, . . . , tk;

The proof of this proposition will occupy the remainder of Appendix 1.Theorem 4.0.12 is an immediate corollary of this proposition, and requiresno further proof.



The strategy of the proof of proposition 5.0.1 is to first show that theintegrand f (te, ) has a good asymptotic expansion when all of the te areroughly of the same size. “Roughly of the same size” means that there issome R such that for all e, e!, tRe * te. This leaves us to understand theregions where some of the te are much smaller than others. These regionsoccur when the lengths of the edges of some subgraph ! ) collapses.These regions are dealt with by an inductive argument.

5.1. Without loss of generality, we may assume that X is a compactmanifold with boundary. Further, the sheaf of algebras A on X plays only acombinatorial role, and has no e!ect on the analysis. Thus, we may assume,without loss of generality, that A is C.

Further, we will often avoid mention of the parameter space X. Thus,if f is some expression which has depends on X, I will often abuse notationand write |f | for the C l norm of f as a function on X.

For a function I : E( ) = {1, . . . , k} $ Z'0, let |I| =*

I(i). LettI =

<tI(i)i . Similarly, if n & Z, let tn =

<tni .

LetO( ) =

'

v#V ( )

O(v)

where O(v) is the order of the polydi!erential operator Iv we place at thevertex v.

For R > 1 letAR,T ) (0, T )E( )

be the region where tRi * tj for all i, j. This means that the ti are all of asimilar size.

The first lemma is that the function f has a good asymptotic expansionon the region AR,T .

Proposition 5.1.1. Fix any R > 1.There exists(1) Fi, Gi & Z'0[t1, . . . , tk] \ {0} for i & Z'0,(2) polydi!erential operators


for i & Z'0,(3) dr & Z'0 tending to infinity,(4) A constant C( ), depending only on the combinatorial structure of

the graph and the dimension of M (but independent of the localaction functionals placed at the vertices of the graph),

such that;;;;;f (t1, . . . , tk)!r'

i=1

Fi(t1, . . . , tk)1/2

Gi(t1, . . . , tk)1/2

!

M(i( )

;;;;; * + +dr+RC( )+5RO( ) tdr1

for all {t1, . . . , tk} with t1 5 t2 5 · · · 5 tk, tR1 5 tk, and t1 su"ciently small.



Proof. As above, let

KNt (x, y) = t$ dim M/2((x, y)e$.x$y.2/t

N'

i=0

ti i(x, y)

be the approximation to the heat kernel to order tN . This expression iswritten in normal coordinates near the diagonal in M2. The i(x, y) aresections of E " E defined near the diagonal on M2. ((x, y) is a cut-o!function, which is 1 when +x! y+ < and 0 when +x! y+ > 2 .

We have the bound99Kt(x, y)!KN

t (x, y)99

l= O(tN$dim M/2$l).

The first step is to replace each Kti by KNti on each edge of the graph.

Thus, let fN (ti, ) be the function constructed like f (ti, ) except usingKN

ti in place of Kti .Each time we replace Kti by the approximation KN

ti , we get a contri-bution of tN$O( )$(dim M)/2

i from the edge ei, times the O( ) norm of thecontribution of the remaining edges, times + +O( ).

The contribution of the remaining edges to the +!+O( )-norm can be

calculated to be<

j 2=i t$O( )$(dim M)/2j . We are thus left with the bound

;;f (ti, )! fN (ti, );; < Ct$O( )$(dim M)/2tN1 + +O( )

where C is a constant. (Recall our notation : tn denotes<

tni ).In particular, if the ti are in AR,T , so that ti > tR1 , and if N is chosen to

be su#ciently large, we find that;;f (ti, )! fN (ti, )

;; < + +O( ) tN$|E( )|(O( )+(dim M)/2)R+11

for t1 su#ciently small.Next, we construct a small ti asymptotic expansion of fN (ti, ). Recall

that fN (ti, ) is defined as an integral over a small neighbourhood of thesmall diagonal in MV ( ). Let

n = dimM.

By using a partition of unity we can consider fN (ti, ) as an integral over asmall neighbourhood of zero in RnV ( ). This allows us to express fN (ti, )as a finite sum of integrals over RnV ( ), of the following form.

For each vertex v of , we have a coordinate map xv : RnV ( ) $ Rn. Fixany > 0, and let : [0,#) $ [0, 1] be a smooth function with (x) = 1 ifx < , and (x) = 0 if x > 2 . Let us define a cut-o! function on RnV ( )

by the formula

= (999'

xv

9992)

5

6'

v##V ( )

999999xv# ! |V ( )|$1

'

v#V ( )

xv

999999

27

8 .



Thus, the function is zero unless all points xv are near their centre ofmass |V ( )|$1 *xv, and is zero when this centre of mass is too far fromthe origin.

For each 1 * i * k, let Qi be the quadratic for on Rn|V ( )| defined by

Qi(x) =0 if the edge ei is a loop, i.e. is attached to only one vertex

Qi(x) = +xv1 ! xv2+2 if v1, v2 are the vertices attached to the edge ei

We can write fN as a finite sum of integrals of the form!

RnV ( )e$

PQi/ti

'

I,K

tI$(dim M)/2$O( )$I,K"K,x .

In this expression,• The sum is over I : E( ) $ Z'0, with all I(e) * N +O( )+1, and

multi-indices K : V ( )"{1, . . . , n}$ Z'0, with*

K(v, i) * O( ).The notation "K,x denotes

"K,x =/

v#V ( ),1%i%n

"

"xK(v,i)v,i

.

In this notation, we are pretending (by trivialising the vectorbundle E on some small open sets in M) that is a function onRnV ( ).

• The $I,K are smooth functions on RnV ( ).Next, we will use Wick’s lemma to compute the asymptotics of this

integral. Let c = (1/ |V ( )|)*

xv be the centre of mass function RnV ( ) $Rn. We can perform the integral in two steps, first integrating over thecoordinates yv = xv!c, and secondly by integrating over the variable c. (Ofcourse, there are |V ( )|!1 independent yv coordinates). The quadratic form*

Qi/ti on RnV ( ) is non-degenerate on the subspace of RnV ( ) of vectorswith a fixed centre of mass, for all ti & (0,#). Thus, the integral over thevariables yv can be approximated with the help of Wick’s lemma.

Let us order the set V ( ) of vertices as v1, v2, . . . , vm. We will use thecoordinates y1, . . . , ym$1, and c on RnV ( ). Then fN is a finite sum ofintegrals of the form!

w#Rn(|w|2)

!

y1,...,ym$1#Rn

@('

+yv+2)

e$P

Qi(y)/ti'

tI$O( )$(dim M)/2$I,K"K,w,yi

A.

Here we are using the same notation as before, in these new coordinates.To get an approximation to the inner integral, we take a Taylor ex-

pansion of the functions and $I,K around the point {yi = 0, w}, onlyexpanding in the variables yi. We take the expansion to order N !. We find,



as an approximation to the inner integral, an expression of the form!

yi#Rne$

PQi(y)/ti

'tI$(dim M)/2$O( )yKcK,I,L

#/ "Li

"Liyi

"Lw

"Lww

$

yi=0

where the sum is over a finite number of multi-indices I,K,L, and cK,I,L

are constants.We can calculate each such integral by Wick’s lemma. The application

of Wick’s lemma involves inverting the quadratic form*

Qi(y)/ti. LetA = A(ti) denote the matrix of the quadratic form

*Qi(y)/ti; this is a

square matrix of size (dim M) |V ( )|, whose entries are sums of t$1i . Note

that1<k

i=1 ti2

A has polynomial entries.Let

P (ti) = det

""k/

i=1

ti

%A

%.

This is the graph polynomial associated to (see (BEK06)). One importantproperty of P is that it is a sum of monomials, each with a non-negativeinteger coe#cient.

We can writeA$1 = P$1B

where the entries of B are polynomial in the ti. Note also that

det A = P t$(dim M)(|V ( )|$1)/2.

(The expansion from Wick’s lemma gives an overall factor of (detA)$1/2).Thus, we find, using Wick’s lemma, an approximation of the form

fN (ti) . P$1/2'

l'0,I:E( )&12Z%0

P$ltI$(dim M)/2$O( )!

M(l,I( )

where the (l,I are polydi!erential operators

(l,I : E (T ( ) $ Densities(M)

and the sum is finite (i.e. all but finitely many of the (l,I are zero).This expansion is of the desired form; it remains to bound the error

term.

Lemma 5.1.2. The error term in this expansion is bounded by

+ +N #+1+O( ) t(N#+1)/2$RC( )$2RO( )

1

for N ! su"ciently large and t1 su"ciently small. Here C( ) is a constantdepending only on the dimension of M and the combinatorial structure ofthe graph.

Proof. The error in this expansion arises from the error in the Taylorexpansion of the functions ,$I,K around 0. Recall that everything is Taylor



expanded to order N !. Thus, if N ! is su#ciently large, the magnitude of theerror in the expansion can be bounded by an expression of the form

t$(dim M)/2$O( ) + +N #+1+O( )

!

w#Rn

!

y1,...,ym$1#Rn

k/

i=1

e$P

Qi(y)/ti'

K

|P (y)|

where P (y) is a polynomial of order N ! + 1!O( ).The Gaussian factor

e$P

Qi(y)/ti

increases if we increase each ti. Since ti * t1, we find we can bound theintegral by the corresponding integral where we use the Gaussian using thequadratic form

*Qi(y)/t1.

Also, since each ti > tR1 , we have the bound

t$(dim M)/2$O( ) =k/

i=1

t$(dim M)/2$O( )i * t$R|E( )|((dim M)/2+O( ))

1 .

Putting these two inequalities together, we find that our error is boundedby

t$R|E( )|((dim M)/2+O( ))1 + +N #+1+O( )

!

w#Rn

!

yi#Rn

k/

i=1

e$P

Qi(y)/t1 |P (y)| .

Since P (y) is a polynomial of order N ! + 1!O( ), we see that!

y1,...,ym$1#Rn

k/

i=1

e$P

Qi(y)/t1 |P (y)| = Ct12 (dim M |V ( )|+N #+1$O( ))

1

for some constant C.Thus, we find our error is bounded by

+ +N #+1+O( ) t(N#+1)/2$RC( )$2RO( )

1

for some C( ) which only depends on the combinatorial structure of thegraph and on dimM . !

By choosing N and N ! appropriately, we can use this bound to completethe proof of the proposition. !

5.2. Next, we will discuss the asymptotic expansion of the functionf (te) when some of the te are much smaller than others.

A subgraph ! of a graph is given by the set of edges E( !) ) E( ).The vertices of the subgraph ! are the ones that adjoin edges in E( !). Thetails of ! are the half-edges of which adjoin vertices of !, but which arenot part of any edge of !.

Let us fix a proper subgraph ! of . We will consider the functionf (t1, . . . , tk) on the region where the ti corresponding to edges in ! aremuch smaller than the remaining ti.



Let us enumerate the edges of as e1, . . . , ek, where e1, . . . , el & E( ) \E( !) and el+1, . . . , ek & E( !).. Let ti = tei .

Let A#

R,T ) (0, T )E( ) be the subset where

tRe * te# if e & E( ) \ E( !) and e! & E( !)

tRe * te# if e, e! & E( !).

This means that the lengths of the edges of the subgraph ! are all aroundthe same size, and are all much smaller than the lengths of the other edges.

Lemma 5.2.1. Let , ! be as above. Assume that ! contains at leastone edge. Let R > l.

Then, there exists a finite number of closed subsets C1, . . . , Cr of A#

R,T ,which cover A

#

R,T , such that for each Cj, there exists(1) Fi, Gi & Z'0[t1, . . . , tk] \ {0} for i & Z'0,(2) polydi!erential operators


for i & Z'0,(3) dr & Z'0 tending to infinity,

such that;;;;;f (t1, . . . , tk)!

r'

i=1

Fi(t1, . . . , tk)1/2

Gi(t1, . . . , tk)1/2

!

M(i( )

;;;;;

* + +dr+RC( )+5RO( ) max(ti)dr

for all {t1, . . . , tk} & Cj with max(ti) su"ciently small. Here, max(ti) refersto the largest of the ti.

Proof. Without loss of generality let us assume that t1 5 t2 5 · · · 5 tk.We can write

f (t1, . . . , tk, ) = f #(tl+1, . . . , tk, ,Kt1 , · · ·,Ktl).

The right hand side of this equation denotes the graph integral for ! withinputs being tensor products of the heat kernels Kti , for 1 * i * l, and .

The starting point of the proof is Proposition 5.1.1 applied to each con-nected component of the graph !. This proposition implies that we canapproximate the contribution of each such connected component by a localaction functional applied to the tails of that connected component. If weapproximate the contribution of each connected component in this way, weare left with a graph integral ( / #(t1, . . . , tl, ) for the quotient graph / !,with certain local action functionals at the vertices of / !.

More formally, Proposition 5.1.1, applied to the connected componentsof the subgraph !, implies that there exist

(1) local action functionals Iv,i at each vertex v of / !, for each i &Z'0



(2) there exists Fi, Gi & Z'0[tl+1, . . . , tk] \ {0}, for each i & Z'0

with the following properties. Let (i/ #(t1, . . . , tl, ) denote the graph in-

tegral obtained by putting the local action functional Iv,i at the vertex v of/ !, the heat kernel Kti at the edges of / !, and at the tails of / !

(which are the same as the tails of ).Then, there exists a sequence dr & Z'0, tending to infinity, such that

(†)

;;;;;f (t1, . . . , tk, )!r'

i=1

Fi(tl+1, . . . , tk)1/2

Gi(tl+1, . . . , tk)1/2(i

/ #(t1, . . . , tl, )

;;;;;

< tdrl+1 + ,Kt1 , · · ·,Ktl+dr+RC( #)+5RO( #)

for all t1, . . . , tk with tl+1 5 · · · 5 tk, tRl+1 5 tk, and tl+1 su#ciently small.Here C( !) is some combinatorial factor only depending on the combinatoricsof the graph !, and O( !) is the sum of the orders of the local actionfunctionals on the vertices of !.

We are only interested in the region where tRi 5 tl+1 if 1 * i * l. Notethat

+Kt+n = O(t$(dim M)/2$n).

Thus,

+ ,Kt1 , · · ·,Ktl+dr+RC( #)+5RO( #)

* + +dr+RC( #)+5RO( #) t$ldr$RC( , #)$5lRO( #)l

if t1 is su#ciently small. Here C( , !) is some constant depending onlyon the combinatorics of the graph , its subgraph !, and on dimM . Theright hand side of this expression only has tl, because we are assuming thatt1 5 · · · 5 tl.

Putting this into equation (†), we find that the right hand side of (†)can be bounded by

tdrl+1t

$ldr$RC( , #)$5lRO( #)l + +dr+RC( #)+5RO( #) .

Recall that tRl 5 tl+1. Thus, tdRl+1 * tRdR

l , so we find that the right hand sideof equation (†) can be bounded by

tRdr$ldr$RC1( , #)$RC2( , #)O( )l + +dr+RC( #)+5RO( #) .

where Ci( , !), as before, only depend on the combinatorics of the graphs.If we take R to be greater than l, the exponent of tl in this expression

tends to infinity as r $#. In addition, since tl < t1, for r su#ciently large,we have

tRdr$ldr$RC1( , #)$RC2( , #)O( )l * tRdr$ldr$RC1( , #)$RC2( , #)O( )

1 .



Thus, we have shown that for all R > l, we can find some Fi, Gi and (i

as above, together with a sequence dr,mr tending to #, such that(‡);;;;;f (t1, . . . , tk, )!

r'

i=1

Fi(tl+1, . . . , tk)1/2

Gi(tl+1, . . . , tk)1/2(i

/ #(t1, . . . , tl, )

;;;;; < tdr1 + +mr

.

We can assume, by induction on the number of vertices of a graph, thatProposition 5.0.1 holds for the graph / !. This implies that we can coverthe set {t1 5 t2 · · · 5 tk} by a finite number of closed sets, C1, . . . , Cr; andthat for each such closed set Cm, we can find functions Hij , Lij & Z[t1, . . . , tl],local action functionals $ij of , such that we can approximate (i

/ # on theset Cm by

'

j'0

Hij(t1, . . . , tl)1/2

Lij(t1, . . . , tl)1/2$ij( ).

More precisely, there exists some sequences ns, es & Z'0 where es tends to# such that;;;;;;

(i/ #(t1, . . . , tl)!

s'

j=0

Hij(t1, . . . , tl)1/2

Lij(t1, . . . , tl)1/2$ij( )

;;;;;;* tes

1 + +ns

for t1 su#ciently small, and (t1, . . . , tl) & Cm.Plugging this estimate for (i

/ # into equation (‡) yields the desiredresult. !

We have shown that Proposition 5.0.1 holds on the following subsets of(0, T )E( ):

(1) The regions AR,T , where t1 5 · · · 5 tk and tR1 5 tk, for any R > 1.(2) The regions A

#

R,T , where t1 5 · · · 5 tk, tRl 5 tl+1, and tRl+1 5 tk,for any subgraph ! ) which contains at least one edge, andany R > k. Here tl+1, . . . , tk correspond to the edges of in thesubgraph !.

It remains to show that we can cover (0, T )E( ) with a finite number ofregions of this form.

Lemma 5.2.2. Fix any R > k = #E( ), and any 0 < T < #. Thenthe regions A

#

R2m ,T, where 0 * m * |E( )| and ! ) is non-empty, cover

(0, T )E( ). (The regions AR2k,T appear as AR2k,T

, where is considered asa subgraph of itself).

Proof. Let {te} & (0, T )E( ). As before, label the elements of E( ) by{1, 2, . . . , k}, in such a way that t1 5 t2 5 · · · 5 tk.

Either tRj 5 tk for all j < k, or, there is a smallest i1 < k such thattRi1 * tk. In the first case, we are done, as then {te} & A

#

R,T where ! is thesubgraph with the single edge corresponding to tk.



Suppose the second possibility holds. Then either for all j < i1, tRj 5 ti1 ,or there exists a smallest i2 < i1 with tRi2 * ti1 . In the first case, we’re done,as we are in A

#

R,T where ! is the subgraph whose edges correspond toti1 , ti1+1, . . . , tk.

Again, let’s suppose the second possibility holds. Then tR2

i2 * tRi1 * tk.Either, for all j < i2, tR

2

j 5 ti2 , and then we are in A#

R2,T where ! is thesubgraph whose edges correspond to ti2 , ti2+1, . . . , tk.

Otherwise, there is some smallest i3 < i2 with tR2

i3 * ti2 . Then tR4

i3 * tk.And so forth.

We eventually end up either finding ourselves in one of the regionsA

#

R2j ,T, for some non-empty proper subgraph ! ) , or we find that some

ik+1 = 1, so we are in AR2k ,T

= AR2k ,T

. !

This completes the proof of Proposition 5.0.1. Theorem 4.0.12 is animmediate corollary.


Appendix 2 : Nuclear spaces

Throughout this book, we freely use linear algebra in the symmetricmonoidal category of nuclear spaces. This appendix will recall some basicfacts about the theory of nuclear spaces, and in particular will explain howthe topological vector spaces we use throughout (such as spaces of smoothfunctions and distributions) form nuclear spaces.

1. Basic definitions

Recall that a topological vector space (sometimes abbreviated to TVS)over R or C is a vector space equipped with a topology that makes scalarmultiplication and addition continuous. All of the topological vector spaceswe consider are Hausdor!. The most important class of TVS (containingboth Banach and nuclear spaces) are the locally convex spaces. Such a spacehas a basis (for the topology) given by convex sets. Finally, a Frechet spaceis a complete, metrizable locally convex spaces.

The subtleties in functional analysis often arise from the freedom tochoose di!erent topologies on dual spaces or tensor products of spaces. ForV a TVS, let V + denote the space of all linear maps from V to the base field,and let V ) ) V + denote the subspace of continuous linear functionals. Thereare several natural topologies on the V ), of which two are the following. Theweak topology on V ) is the topology of pointwise convergence: a sequenceof functionals { n} converges to zero if the sequence { n(x) converges tozero for any x & V . The strong topology on V ) is the topology of boundedconvergence: a filter of the origin converges in the strong topology if itconverges uniformly on every bounded subset in V . We will use the strongtopology here.

There are also several natural topologies one can put on the spaceHom(E,F ) of continuous linear maps from E $ F . The topology of in-terest here is the topology of bounded convergence, described as follows.Let U ) F be an open subset of F , and B ) E a bounded subset of E. Abasis of neighbourhoods of 0 in Hom(E,F ) consists of those subsets

U(B,U) = {f : E $ F | f(B) ) U} ) Hom(E,F ).

With this topology, as long as both E and F are locally convex and Haus-dor!, so is Hom(E,F ).

We define the projective topological tensor product of two locally convexspaces V and W as follows. Let V ,algW denote the algebraic tensor product

243


244 APPENDIX 2 : NUCLEAR SPACES

of V and W . The projective topology on V ,algW is the finest locally convextopology so that the canonical map

V "W $ V ,alg W

is continuous.We will let V , W (or just V ,W when there is no ambiguity) denote

the completion of V ,alg W with respect to this topology.There is another natural topology on V ,alg W , called the injective

topology, which is coarser than the projective topology; every reasonabletopology on V ,alg W lies between these two. (We will not recall the precisedefinition of the injective topology, as it is a little technical: see (Tre67) or(Gro52) for details).

We will let V ,i W denote the completion of V ,alg W with respect tothe injective topology.

We now give the most elegant definition of a nuclear space:

Definition 1.0.3. A locally convex Hausdor! space V is nuclear if thecanonical map V ,pW $ V ,iW is an isomorphism for any locally convexHausdor! space W .

There are several equivalent definitions, although these equivalences arehighly nontrivial. For more details, see chapter 50 of (Tre67) or (Gro52).

Definition 1.0.4. Let Nuc denote the category, enriched in ordinaryvector spaces, whose objects are complete nuclear spaces and whose mor-phisms are linear continuous maps.

Remark. Although the space Hom(E,F ) of continuous linear maps be-tween nuclear spaces does have a natural topology, as described above, Nucdoes not form a symmetric monoidal category enriched in topological vectorspaces.

If E,F are nuclear spaces, then so is their completed tensor productE , F . Thus, we have

Proposition 1.0.5. Nuc is a symmetric monoidal category with respectto the completed tensor product.

2. Examples

Many function spaces in geometry are nuclear:• smooth functions on an open set in Rn,• smooth functions with compact support on an open set, or on a

compact set, in Rn,• distributions (the continuous dual space to compactly-supported

smooth functions) on an open set in Rn,• distributions with compact support on an open set in Rn,• Schwartz functions on Rn,• Schwartz distributions on Rn,


3. SUBCATEGORIES 245

• holomorphic functions on an open set in Cn,• formal power series in n variables (with the inverse limit topology),• polynomials in n variables.

In consequence, we see that, for example, smooth sections of a vector bundleon a compact manifold form a nuclear Frechet space.

Nuclear spaces enjoy the following useful properties:• A closed linear subspace of a nuclear space is nuclear.• The quotient of a nuclear space by a closed linear subspace is nu-

clear.• A countable direct sum of nuclear spaces (equipped with finest

locally convex topology) is nuclear.• Likewise, a countable colimit of nuclear spaces is nuclear, as long

as the colimit is taken in the category of Hausdor! locally convextopological vector spaces.

• A direct product of nuclear spaces (equipped with the producttopology) is nuclear.

• Likewise, a limit of nuclear spaces is nuclear.• The completed tensor product is nuclear.• The tensor product of nuclear spaces commutes with all limits.

That is, if E = lim Ei, F = lim Fj are nuclear spaces written aslimits of nuclear spaces Ei, Fj , then

E , F = limi,j

(Ei , Fj).

3. Subcategories

Let us list three important classes of nuclear spaces:(1) (NF) spaces, that is, nuclear Frechet spaces.(2) (NLF) spaces, which are countable colimits of NF-spaces.(3) (NDF) spaces, that is, nuclear (DF) spaces. A space is (NDF) if

and only if its strong dual is of type (NF).Nuclear spaces which are (NF) or (NDF) are particularly well behaved.

• If E is a nuclear Frechet space (that is, of type (NF)), then thestrong dual E) is a space of type (NDF).

• If E is of type (NDF), then the strong dual E) is of type (NF).• Countable products of spaces of type (NF) are again of type (NF).• Closed subspaces and Hausdor! quotients of spaces of type (NF)

are again of type (NF) (thus, spaces of type (NF) are closed undercountable limits and finite colimits).

• Similarly, spaces of type (NDF) are closed under finite limits andcountable colimits.

• If E,F are both of type (NF), or both of type (NDF), then

(E , F )) = E) , F).



• If E,F are both of type (NF), or both of type (NDF), then so isE , F .

• If E = colim Ei is a countable colimit of nuclear spaces, and F isof type (NDF), then

E , F = colim(Ei , F ).

These properties can be rewritten in categorical terms, as follows.Proposition 3.0.6. Let NF (respectively, NDF) denote the full subcate-

gories of Nuc consisting of spaces of type (NF) (or type (NDF), respectively).Then the categories NF and NDF are symmetric monoidal subcategories ofNuc. NF is closed under countable limits and finite colimits; NDF is closedunder countable colimits and finite limits. Sending a nuclear space to itsstrong dual induces equivalences of symmetric monoidal categories

) : NFop %= NDF) : NDFop %= NF.

3.1. Let us now list some further useful properties of nuclear spaces.• Any nuclear space is semi-reflexive: this means that the natural

map E $ (E))) is a linear isomorphism (although not necessarilya homeomorphism).

• Any nuclear space E of type (NF) is reflexive: this means that thenatural map E $ (E))) is a linear homeomorphism, where dualsare given the strong topology. This follows from the facts that (NF)spaces are barrelled ((Tre67), page 347); barrelled nuclear spacesare Montel ((Tre67), page 520); and Montel spaces are reflexive((Tre67), page 376).

• If E is a nuclear space of type (NF), then the strong and weaktopologies on bounded subsets of E) coincide ((Tre67), page 357).

• Let E,F be nuclear Frechet spaces. In the following equalities,Hom-spaces between nuclear spaces will be equipped with the topol-ogy of uniform convergence on bounded subsets, as above. Then,as we see from (Tre67), page 525:

E , F = Hom(E), F )

E) , F = Hom(E,F )

E) , F) = (E , F ))

E) , F) = Hom(E,F))

• If E is of type (NLF) and F is any nuclear space, then

E) , F = Hom(E,F ).

• Finally, for any nuclear spaces E and F , we have the followingisomorphism ((Tre67), page 522, proposition 50.4) only of vectorspaces without topology

E) , F = Hom(E,F ).


5. ALGEBRAS OF FORMAL POWER SERIES ON NUCLEAR FRECHET SPACES 247

This last isomorphism is, in fact, a topological isomorphism if we equipHom(E,F ) with a certain topology which is in general di!erent from thetopology of uniform convergence on bounded subsets.

Caution: the evaluation map

E) "E $ Ris not, in general, continuous for the product topology on E)"E; it is onlyseparately continuous. One can see this easily when E = C"(M), whereM is a compact manifold. Then, E) = D(M) is the space of supporteddistributions on M . If the evaluation map

C"(M)"D(M)$ Rwere continuous, it would extend to a continuous linear map

C"(M),D(M) = Hom(C"(M), C"(M))$ R.

On the subspaceC"(M), &n(M),

the putative evaluation map must be given by restriction to the diagonalM ) M " M and integrating. However, C"(M) , D(M) contains the-distribution on the diagonal; and we cannot restrict this to the diagonal.

This means that, although the spaces of linear maps between nuclearspaces are topological vector spaces in a natural way, Nuc does not form asymmetric monoidal category enriched in topological vector spaces with theprojective tensor product.

4. Tensor products of nuclear spaces from geometry

• If M is a manifold, and F is any locally convex Hausdor! topolog-ical vector space, then C"(M) , F is the space of smooth mapsM $ F .

• If M and N are manifolds, then

C"(M), C"(N) = C"(M "N).

• Let D(M) be the dual of C"(M), that is, the space of compactlysupported distributions on M . Then,

D(M),D(N) = D(N "M).

Similar identities hold when we consider sections of vector bundles on Mand N instead of just smooth functions.

5. Algebras of formal power series on nuclear Frechet spaces

Since the category Nuc is a symmetric monoidal category, we can per-form most of the familiar constructions of linear algebra in this category. Inparticular, if E is a nuclear space, we will let

Symn E =-E(n

.Sn

be the coinvariants for the natural symmetric group action on E(n.



Suppose that E is a nuclear Frechet space. Then, E) is a space of type(NDF), and in particular a nuclear space. We will let

O(E) =/

n'0

Symn E).

We will view O(E) as the algebra of formal power series on E. Note that,since the category of nuclear spaces is closed under limits, O(E) is a nu-clear space, and indeed a commutative algebra in the symmetric monoidalcategory of nuclear spaces.

If A & Nuc is a commutative algebra in the category of nuclear spaces,then we will view

O(E),A

as the algebra of A-valued formal power series on E. Thus, we will oftenwrite O(E,A) instead of O(E),A. Note that, because the tensor productcommutes with limits,

O(E),A =/

n

-Symn E), A

.

=/

n

Hom(E(n, A)Sn

where the space Hom(E(n, A) is equipped with the strong topology (that ofuniform convergence on bounded subsets). The last equality holds because,for all nuclear Frechet spaces E and all nuclear spaces F , Hom(E,F ) =E) , F where Hom(E,F ) is equipped with the strong topology.

As an example, if E = C"(M), then

O(C"(M)) =/

n'0

D(Mn)Sn

where D(Mn) refers to the space of distributions on Mn.


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Renormalization and Effective Field Theory · 2018-12-04 · Renormalization and Effective . Field Theory . Kevin Costello . This is a preliminary version of the book Renormalization