Lecture 20: Wess-Zumino-Witten CFT and Fusion Algebrassporadic.stanford.edu/conformal/lecture20.pdf · This is discussed further in two lectures from the previous course on quantum

Review: Affine Kac-Moody Lie Algebras WZW Theories

Lecture 20: Wess-Zumino-Witten CFT andFusion Algebras

Daniel Bump

January 1, 2020


Overview

We have so far emphasized the BPZ minimal models, in which ctakes particular values < 1. These are special in that the Hilbertspace H decomposes into a finite number of Vir⊕Vir modules.

For c > 1, in order to obtain such a finite decomposition it isnecessary to enlarge Vir. If this can be done, we speak of arational CFT: see Lecture 19 for a more careful definition of thisconcept, due to Moore and Seiberg.

One natural such extension enlarges Vir to include an affineLie algebra. This leads to the Wess-Zumino-Witten (WZW)models. A further generalization of this theory, the cosetmodels of Goddard-Kent-Olive ([GKO]) contains both theminimal models and the WZW models.


Lecture 6 review: A note on notation

References: [DMS] Chapter 14Kac, Infinite-Dimensional Lie algebras, Chapters 6,7,12,13Kac and Peterson, Adv. in Math. 53 (1984).

We begin by reviewing the construction of affine Lie algebrasfrom Lecture 6, changing the notation slightly.

Let g be a finite-dimensional semisimple complex Lie algebrawith Cartan subalgebra h. We will define the untwisted affineLie algebra g by first making a central extension, then adjoininga derivation d. Finally we will make a semidirect product of gwith Vir in which d is identified with −L0.


Review: central extensions

Let g be a Lie algebra and a an abelian Lie algebra. A bilinearmap σ : g× g→ a is called a 2-cocycle if it is skew-symmetricand satisfies

σ([X,Y],Z) + σ([X,Y],Z) + σ([X,Y],Z) = 0, X,Y,Z ∈ a.

In this case we may define a Lie algebra structure on g⊕ a by

[(X, a), (Y, b)] = ([X,Y],σ(X,Y)).

Denoting this Lie algebra g ′ we have a central extension

0 −→ a −→ g ′ −→ g −→ 0.


Review: Adding a derivation

Suppose we have a derivation d of a Lie algebra g. This means

d([x, y]) = [dx, y] + [x, dy].

We may then construct a Lie algebra g ′ = g⊕ Cd in which[d, x] = d(x) for x ∈ g.

This is a special case of a more general construction, thesemidirect product.


Review: Affine Lie algebras as Central Extensions

Although the affine Lie algebra g may be constructed from itsCartan matrix, another construction described in Chapter 7 ofKac begins with the finite-dimensional simple Lie algebra g ofrank `. Tensoring with the Laurent polynomial ring gives theloop Lie algebra g⊗ C[t, t−1]. Then one may make a centralextension:

0→ C · K → g ′ → C[t, t−1]⊗ g→ 0.

After that, one usually adjoins another basis element, whichacts on g ′ as a derivation d. This gives the full affine Liealgebra g.


The dual Coxeter number

The dual Coxeter number, denoted h∨ will appear frequently informulas. We omit the definition, for which see Kac,Infinite-dimensional Lie algebras Section 6.1. But here are thevalues as a function of the Cartan type of g.

Cartan Type h∨

A` l + 1B` 2l − 1C` l + 1D` 2l − 2E6 12E7 18E8 30F4 9G2 4


The root system and Weyl group

The root system is the set of nonzero elements of h∗ that occurin the adjoint representation of h on g. Particular roots are asfollows.

There exist α∨i ∈ h and αi ∈ h∗ called simple coroots and

simple roots respectively. For i = 0, · · · , r (with r the rank of gthere is simple reflection acting on h∗ defined by

si(λ) = λ− λ(α∨i )αi

and the group generated by the si is the affine Weyl group W.


Real and imaginary roots

As usual, roots may be partitioned into positive and negative.

A root is called real if it is in the W-orbit of a simple root. A rootis imaginary if it is not real. There is a minimal positiveimaginary root δ. The roots of g are all roots of g. Let ∆ be thefinite root system of g. The real roots of g are

{α+ nδ|α ∈ ∆, n ∈ Z}.

The imaginary roots are nδ with 0 6= n ∈ Z.


Other data

The Cartan subalgebra h of g is h with the central element Kand the derivation d adjoined.

Other data that are described in Kac, Infinite-Dimensional LieAlgebras, Chapters 6 and 12. There are coroots α∨

i ∈ h fori = 0, 1, · · · , r, where r is the rank of g. There are positiveintegers a∨i dual to the ai that we will call comarks such that

K =

r∑i=0

a∨i α∨i , h∨ =

r∑i=0

a.i

The numbers a∨i (sometimes called comarks) are all 1 ifg = slr+1, and for all g at least a∨0 = 1.


Enlargement by Vir

Instead of (or more precisely, in addition to) enlarging g ′ by thederivation d we may form the semidirect product of g ′ by theentire Virasoro algebra

[Li,Lj] = (i − j)Li+j +112

(i3 − i)δi,−j · C

where Li acs as the derivation −ti+1d/dt. This semidirectproduct contains the full affine Lie algebra g = g ′ ⊕ Cd withd = −L0.

It is also possible to embed Vir in the universal envelopingalgebra of g. This is the Sugawara construction which we willmake precise later.


Integrable highest weight representations

Let Λi ∈ h∗ be the fundamental weights characterized byΛi(α

∨j ) = δij and Λi(d) = 0. The lattice P spanned by the Λi is

called the weight lattice. If λ =∑

i ciΛi with ci nonnegativeintegers, then λ is called dominant.

Given λ ∈ h it is possible to construct a Verma module M(λ)that is highest weight representation of g. It has a uniqueirreducible quotient L(λ).

Let P+ be the cone of dominant weights. If λ ∈ P+, then L(λ) isintegrable, namely exponentiates to a representation of theloop group. The character is therefore invariant under the affineWeyl group.


String functions

Kac and Peterson proved that the characters of integrablehighest weight representations are modular forms. Let λ be adominant weight and let µ be a weight. Let mλ(µ) be themultiplicity of µ in L(λ).

The function m(µ− nδ) is monotone, and zero outside the cone

(λ+ ρ|λ+ ρ) − (µ+ ρ,µ+ ρ) > 0.

Therefore we may find a smallest n such that m(µ− nδ) isnonzero. We call µ− nδ a maximal weight. Define the stringfunction

bλµ =∑

k

m(µ− kδ)e−kδ.

replacing µ by µ− nδ just multiplies the series by ekδ so there isno loss of generality in assuming that µ is maximal.


String functions (continued)

The characterχλ =

∑µ maximal

bλµ.

Moreover using the fact that δ = w(δ) for all w ∈ W we mayhave w(bλµ) = bλwµ and therefore we may rewrite bλµ by finding aWeyl group element such that wµ is dominant. There are only afinite number of dominant maximal weights. Then

χλ =∑

µ maximal, dominant

∑w∈W/Wµ

ewµbλµ.

(To avoid overcounting Wµ is the stabilizer of the dominantmaximal weight µ.)


Modular characteristics and the character

Define the modular characteristic (modular anamoly)

sλ =|λ+ ρ|2

2(k + h∨)−

|ρ|2

2k,

We are using the notation ( | ) for an invariant inner product onΛ. Here h∨ is the dual Coxeter number and k = (δ|λ). Also

sλ(µ) = sλ −|µ|2

2k.

Now define

cλµ = e−sλ(µ)δbλµ = e−sλ(µ)δ∑n>0

m(λ− nδ)e−nδ.

These modified string functions are theta functions which aremodular forms, as was proved by Kac and Peterson. Becauseof this, the character χλ, which is the sum of all string functions,is a modular form.


The level k alcove

The level of a weight λ is

k = 〈λ,K〉 =∑

a∨i λ(α∨i ).

There are a finite number of dominant weights of level k, andthese may be visualized as follows.

We may identify the fundamental weights Λ1, · · · ,Λr with theweights of the finite–dimensional Lie algebra g. Then theweights of level k are identified with weights of g by projectingon the span of Λ1, · · · ,Λr. The positive Weyl chamber for g isthe set of weights λ such that λ(α∨

i ) > 0 for 1 6 i 6 r. The levelk alcove then adds one inequality, 〈λ,K〉 6 k, where θ is thehighest root of g.


The level 3 alcove for sl3

For example, if g = sl3, here is the level 3 alcove inside thepositive Weyl chamber, showing that g has 10 positive weightsof level 3:

The SL(3) case


The affine Weyl group (continued)

k = 0

k = 1

k = 2

k = 3

The affine Weyl group fixes the origin. But parallel to the level 0plane, it induces affine linear maps on each level (except level0). The fundamental alcove on the k level set is the intersectionwith the shaded area, consisting of dominant weights.


The Hilbert space

To define a WZW CFT we pick a finite-dimensional Lie algebrag with corresponding affine Lie algebra g and a level k. In placeof the Virasoro algebra we have the semidirect productA = g⊕ Vir. This acts on L(λ) for any dominant weight λ, dueto the Sugawara construction. The Hilbert space of the model is⊕

λ of level k

L(λ)⊗ L(λ).

As usual, any field can be decomposed into a sum of fieldsφ(z)⊗ φ (z) where φ is holomorphic and φ is antiholomorphic.The holomorphic fields are called chiral.


Affine primary fields and currents

Let Φλ be the highest weight element of L(λ), identified with achiral field. The field Φλ(z)Φλ (z) in L(λ)⊗ L(λ) is called anaffine primary field. We will concentrate on the chiral fieldΦλ(z), which we will also call affine primary fields.


Currents

I addition to the primary fields and their descendents, there areother fields called currents. Let J ∈ g. The current J(z) is theformal expression

J(z) =∞∑

n=−∞ z−n−1Jn, Jn = J ⊗ tn ∈ g.

Currents had their origin in particle physics where they wereintroduced by Gell-Mann in 1964.


The Sugawara construction

Another field is the energy-momentum tensor, which is madeexplicit through the Sugawara construction. Indeed theSugawara construction which embeds Vir in the universalenveloping algebra of g can be written succinctly as

T(z) =∑n∈Z

z−n−2Ln =1

2(k + h∨)

∑a

: JaJa : (z)

where the sum is over an orthonormal basis of g with respect tothe inner product given by the Killing form divided by 2h∨.


Summary

This gives a conformal field theory with central charge

c =k dim(g)

k + h∨.

Through the OPE expansion, the primary fields acquire acomposition Φλ ?Φµ making a Fusion ring or Verlinde algebra.This is discussed further in two lectures from the previouscourse on quantum groups:

Lecture 15 The Racah-Speiser algorithm. The Kac-Waltonformula.

Lecture 17 Affine Lie algebras. The fusion ring.

Modularity considerations are closely associated with the fusionrules, as we mention at the end of Lecture 17.

http://sporadic.stanford.edu/quantum/lecture15.pdf

http://sporadic.stanford.edu/quantum/lecture17.pdf

Documents

Lecture 20: Wess-Zumino-Witten CFT and Fusion Algebrassporadic.stanford.edu/conformal/lecture20.pdf · This is discussed further in two lectures from the previous course on quantum