MIT COURSE ON STRING THEORYA one-semester class about gauge/gravity duality (often called AdS/CFT) and its applications

Prof. John McGreevy

Fall 2008

LECTURE NOTES

1. What you need to know about string theory 2. D3-branes at small and large coupling 3. AdS/CFT from scratch, no strings 4. What holds up the throat? 5. How to defend yourself from a supersymmetric field theory 6. About the N = 4 SYM theory 7. Big picture of correspondence; strings and strong interactions 8. 't Hooft counting 9. Scale and conformal invariance in field theory 10. CFT in D > 2 (cont.) 11. Geometry of AdS 12. Geometry of AdS (cont.); Poincar patch; wave equation in AdS 13. Masses of fields and dimensions of operators; BF-allowed tachyons 14. How to compute two-point correlators of scalar operators 15. Preview of real-time issues; two-point functions in momentum space (cont.); more on low-mass2 fields in AdS 16. Three-point functions, anomalies; expectation values 17. Wilson loops 18. Wilson loops (cont.) 19. Pointlike probes of the bulk; Baryons and branes in AdS; 'Nonspherical horizons' 20. Brief survey of other examples of the correspondence (M2, M5, D1-D5, Dp, branes at singularities); a model of confinement 21. Confinement (cont.): how to measure the spectrum, proof of mass gap 22. Black hole mechanics, classical and quantum 23. AdS black holes and thermal gauge theory; equation of state, free energy and stress tensor 24. Hawking effect for interacting field theories and BH thermodynamics 25. AdS black holes and thermal gauge theory: Polyakov-Susskind loop, screening, quasinormal modes, Hawking-page transition

8.821 Lecture 01: What you need to know about string theoryLecturer: McGreevy Scribe: McGreevy

Since I learned that we would get to have this class, Ive been torn between a) starting over with string perturbation theory and b) continuing where we left o. Option a) is favored by some people who didnt take last years class, and by me when Im feeling like I should do more research. But because of a sneaky and perhaps surprising fact of nature and the history of science there is a way to do option b) which doesnt leave out the people who missed last years class (and is only not in the interest of the lazy version of me). The fact is this. It is actually rare that the structure of worldsheet perturbation theory is directly used in research in the subject that is called string theory. And further, one of the most important developments in this subject, which is usually called, synecdochically, the AdS/CFT correspondence, can be discussed without actually using this machinery. The most well-developed results involve only classical gravity and quantum eld theory. So heres my crazy plan: we will study the AdS/CFT correspondence and its applications and generalizations, without relying on string perturbation theory. Why should we do this? You may have heard that string theory promises to put an end once and for all to that pesky business of physical science. Maybe something like it unies particle physics and gravity and cooks your breakfast. Frankly, in this capacity, it is at best an idea machine at the moment. But this AdS/CFT correspondence, whereby the string theory under discussion lives not in the space in which our quantum elds are local, but in an auxiliary curved extra-dimensional space (like a souped-up fourier transform space), is where string theory comes the closest to physics. The reason: it oers otherwise-unavailable insight into strongly-coupled eld theories (examples of which: QCD in the infrared, high-temperature superconductors, cold atoms at unitarity), and into quantum gravity (questions about which include the black-hole information paradox and the resolution of singularities), and because through this correspondence, gauge theories provide a better description of string theory than the perturbative one. The role of string theory in our discussion will be like its role in the lives of practitioners of the subject: a source of power, a source of inspiration, a source of mystery and a source of vexation. The choice of subjects is motivated mainly by what I want to learn better. After describing how to do calculations using the correspondence, we will focus on physics at nite temperature. For what I think will happen after that, see the syllabus on the course webpage. Suggestions in the spirit described above are very welcome. 1

ADMINISTRIVIA please look at course homepage for announcements, syllabus, reading assignments please register I promise to try to go more slowly than last year. coursework: 1. psets. less work than last year. hand them in at lecture or at my oce. pset 0 posted, due tomorrow (survey). 2. scribe notes. theres no textbook. this is a brilliant idea from quantum computing. method of assigning scribe TBD. as you can see, I am writing the scribe notes for the rst lecture. 3. end of term project: a brief presentation (or short paper) summarizing a topic of interest. a list of candidate topics will be posted. goal: give some context, say what the crucial point is, say what the implications are. try to save the rest of us from having to read the paper. (benets: you will learn this subject much better, you will have a chance to practice giving a talk in a friendly environment) Next time, we will start from scratch, and motivate the shocking statement of AdS/CFT duality without reference to string theory. It will be useful, however, for you to have some big picture of the epistemological status of string theory. Todays lecture will contain an unusually high density of statements that I will not explain. I explained many of them last fall; experts please be patient. What you need to know about string theory for this class: 1) Its a quantum theory which at low energy and low curvature 1 reduces to general relativity coupled to some other elds plus calculable higher-derivative corrections. 2) It contains D-branes. These have Yang-Mills theories living on them. We will now discuss these statements in just a little more detail.

0.1

how to do string theory (textbook fantasy cartoon version)

Pick a background spacetime M, endowed with a metric which in some local coords looks like ds2 = g (x)dx dx ; there are some other elds to specify, too, but lets ignore them for now.1

compared to the string scale Ms , which well introduce below

2

Consider the set of maps X : worldsheet target spacetime (, ) X (, ) where , are local coordinates on the worldsheet. Now try to compute the following kind of path integral I [DX(, )] exp (iSws [X..]). This is meant to be a proxy for a physical quantity like a scattering amplitude for two strings to go to two strings; the data about the external states are hidden in the measure, hence the . The subscript ws stands for worldsheet; more on the action below. An analog to keep rmly in mind is the rst quantized description of quantum eld theory. The Feynman-Kac formula says that a transition amplitude takes the form x(2 )=x2 eiSwl [x( )] = x2 , 2 |x1 , 1 (1)x(1 )=x1

M

Here x : worldline target spacetime ( ) x ( ). can be written as a sum over trajectories interpolating between specied inital and nal states. Here x( ) is a map from the world line of the particle into the target space, which has some coordinates x . For example, for a massless charged scalar particle, propagating in a background spacetime with metric g and background (abelian) gauge eld A , the worldline action takes the form Swl [x] = d g (x)x x + d x A where x x. Note that the minimal coupling to the gauge eld can be written as A d x A = wl

where the second expression is meant to indicate an integral of the one-form A over the image of the worldline in the target space (The expression on the LHS is often called the pullback to the worldline. Those are words.). A few comments about this worldline integral. 1) The spacetime gauge symmetry A A + d requires the worldline not to end, except at a place where weve explicitly stuck some non-gauge-invariant source (x1 , x2 in the expression above). 2) The path integral in (1) is that of a one-dimensional QFT with scalar elds x ( ), and coupling constants determined by g , A . To understand this last statement, Taylor expand around some point in the target space, which lets call 0. Then for example A (x) = A (0) + x A (0) + ... and the coupling to the gauge eld becomes d (x A (0) + x x A (0) + ...) 3

so the Taylor coecients are literally coupling constants, and an innite number of them are encoded in the background elds g, A. After all this discussion of the worldline analog, and before we explain more about the , why might it be good to make this replacement of worldlines with worldsheets? Good Thing number 1: many particles from vibrational modes. From far away, meaning when probed at energies E Ms , they look like particles. This suggests that there might be some kind of unication, where all the particles (dierent spin, dierent quantum numbers) might be made up of one kind of string. To see other potential Good Things about this replacement of particles with strings, lets consider how interactions are included. In eld theory, we can include interactions by specifying initial and nal particle states (say, localized wavepackets), and summing over all ways of connecting them, e.g. by position-space feynman diagrams. To do this, we need to include all possible interaction vertices, and integrate over the spacetime points where the interactions occur. UV divergences arise when these points collide. To do the same thing in string theory, we see that we can now draw smooth diagrams with no special points (see gure). For closed strings, this is just the sewing together of pants. A similar procedure can be done for open strings (though in fact we must also attach handles there). So, Good Thing Number Two is the fact that one string diagram corresponds to many feynman diagrams. Next lets ask Q: where is the interaction point? A: nowhere. Its in dierent places in dierent Lorentz frames. This is a rst (correct) indication that strings will have soft UV behavior (Good Thing Number Three). They are oppy at high energies. The crucial point: near any point in the worldsheet (contributing to some amplitude), it is just free propagation. This implies a certain uniqueness (Good Thing Number Four): once the free theory is specied, the interactions are determined. The structure of string perturbation theory is very rigid; it is very dicult to mess with it (e.g. by including nonlocal interactions) without destroying it. 2

BUT: Only for some choices of (M, ) can we even dene I. To see why, lets consider the simplest example where M = IRD1,1 with g = diag(1, 1, 1, 1...1). Then the worldsheet action is 1 d2 X X . (2) Sws [X] = 4 Here = 1, 2 is a worldsheet index and = 0...D 1 indexes the spacetime coords.1 The quantity 4 has dimensions of length2 . It is the tension of the string, because it suppresses congurations with large gradients of X ; when it is large, the string is sti. This is the mass scale2 Why stop at one-dimensional objects? Why not move up to membranes? Its still true that the interaction points get smoothed out. But the divergences on the worldvolume are worse. d = 2 is a sort of compromise. In fact, some sense can be made of the case of 3-dimensional base space, using the BFSS matrix theory. They turn out to be already 2d quantized! The spectrum is a continuum, corresponding to the degree of freedom of bubbling o parts of the worldvolume. Its not entirely clear to me what to make of this.

4

Crudely rendered perturbative interactions of closed strings. Time goes upwards. Slicing this picture by horizontal planes shows that these are diagrams which contribute to the path integral for 2 to 2 scattering. Tilting the horizontal plane corresponds to boosting your frame of reference and moves the point where you think the splitting and joining happened.

alluded to twice so far:

1 ; 4 people will disagree about the factors of 2 and here.2 Ms

where . This is solved by superpositions of right-movers and left-movers: X (, ) = XL ( + ) + XR ( ).

(2) is the action for free bosons. We can compute anything we want about them. The equations of motion 0 = Sws give the massless 2d wave equation: X 2 2 0 = + X = + X

To study closed strings, we can treat the spatial direction of the worldsheet as a circle + 2, in which case we can expand in modes: + X (, ) = ein + ein . n nnZ Z nZ Z

For open strings, the boundary conditions will relate the right-moving and left-moving modes: . We can quantize this canonically, and nd [ , ] = nn+m , n m [ , ] = nn+m , [, = 0. n m ]

This says that we can think of n0 as annihilation operators. Let |0 be annihilated by all the n>0 . The zeromodes 0 , 0 are related to the center-of-mass position of the string. The worldsheet hamiltonian is and momentum p Hws = p2 + n n a,n>0

where a is a normal-ordering constant, which turns out to be 1. 5

Here comes the problem: consider the state |BAD =0 |0 for n > 0. Because of the commutator n [0 , 0 ] = n, we face the choice n n |BAD2

0, the black hole is not a point particle, but is also extended in p spatial dimensions.

The case of p = 3 The 3-brane will ll 3 + 1 dimensions (e.g., x=0,1,2,3 ). We can put the brane at y 1 = = y 6 = 0, 2 i.e, the R3,1 is at a point in the transverse R6 . Let r 2 = yi be the distance from the brane, then the metric of the brane will take the formR6

ds2 = and

1 H(r)

6 2 dx dx + H(r) dyi ,R3,1 i=1

(4)

L2 , dy 2 = dr 2 + r 2 d5 , r4 where H(r) 1 for large r and L is like an ADM mass. The RR F 5 satises F5 = N. H(r) = 1 +S 5 at constant r

(5)

(6)

The solution looks like an RN black hole (Figure 2). Far away (large r with H(r) = 1), the solution will asymptote to R9,1 . Near the horizon (r L with H(r) = L2 /r 4 ), the metric will take the form of an AdS5 S5 / r2 r2 dr 2 (7) ds2 = 2 dx dx + L2 2 + L2 2 d5 . / L r r AdS5 S5

The ultra-low energy excitations near the brane cant escape the potential well (throat) and the stu from innity cant get in. The absorption cross section of the brane goes like 3 L8 . 2 (8)

R3,1 r

Gravitational potential well Cross-section of S5

Raduis is constant L

Figure 2:

The above conclusion can be reached in another way by noting that for low , the wavelength of the excitations will be too large to t in the throat which has xed size. Comparison of low energy decoupling at large and small ( = N gs ) Large Throat states = IIB strings in AdS5 S5 + Closed strings in R9,1related by variation of , i.e, dual

Small 4D N = 4, SU(N ) YM + Closed strings in R9,1

Maldacena: Subtract Closed strings in R9,1 from both sides.

Matching of parameters The parameters of the gauge theory are g2 = gs (we will see that it is really a parameter in N = 4 YM YM) and the number of colors N . By Gausss law F5 = N (quantized by Dirac). (9) F5 =S5 S5

The supergravity equations of motion relate L (size of space) and N (number of branes) as follows:5 5... R = GN F... F ,

(10) (11)

where2 GN gs 4 , 5 5... and F... F N 2 .

3

Dimensional analysis then says R L8 . This gives L4 = gs N = (t Hooft coupling). 2 In terms of gravitational parameters, this says2 GN gs ( )4 ,

(12)

L = N 1/4 GN GN

1/2

1 N2

(in units of AdS raduis, i.e. at xed ). (13)

A picture of what has happened here which can sometimes be useful is the Picture of the Tents. We draw the distance from the branes as the vertical direction, and keep track only of the size of the longitudinal and transverse directions as a function of this radial coordinate. Flat space looks like this: (Hence, tents.) The picture of the near-horizon limit looks instead like:R r6 3,1

r

S

5

R3,1

R

S

5

R

3,1

S

5

Figure 3: Left: Flat space. Right: AdS5 S 5 . Now its the Minkowski space IR3,1 which shrinks at r = 0, and sphere stays nite size.

A Bold AssertionNow we back up and try to understand what is being suggested here, without using string theory. We will follow the interesting logic of: Reference: Horowitz-Polchinski, gr-qc/0602037 Assertion: Hidden inside any non-abelian gauge theory is a quantum theory of gravity. What can this possibly mean?? Three hints from the Elders: 1) At the least it must mean that there is some spin-two graviton particle, that is somehow a composite object made of gauge theory degrees of freedom. This statement seems to run afoul of the Weinberg-Witten no-go theorem, which says: Theorem (Weinberg-Witten): A QFT with a Poincar covariant conserved stress tensor T e forbids massless particles of spin j > 1 which carry momentum (i.e. with P = dD xT 0 = 0). GR gets around this because the total stress tensor (including the gravitational bit) vanishes by S the metric EOM g = 0. (Alternatively, the matter stress tensor which doesnt vanish is not general-coordinate invariant.) 4

Like any good no-go theorem, it is best considered a sign pointing away from wrong directions. The loophole in this case is blindingly obvious in retrospect: the graviton neednt live in the same spacetime as the QFT. 2) Hint number two comes from the Holographic Principle [t Hooft, Susskind]. This is a far-reaching consequence of black hole thermodynamics. The basic fact is that a black hole must be assigned an entropy proportional to the area of its horizon in planck units (much more later). On the other hand, dense matter will collapse into a black hole. The combination of these two observations leads to the following crazy thing: The maximum entropy in a region of space is its area in Planck units. To see this, suppose you have in a volume V (bounded by an area A) a conguration with entropy A S > SBH = 4GN (where SBH is the entropy of the biggest blackhole ttable in V ), but which has less energy. Then by throwing in more stu (as arbitrarily non-adiabatically as necessary, i.e. you can increase the entropy) you can make a black hole. This would violate the second law of thermodynamics, and you can use it to save the planet from the Humans. This probably means you cant do it, and instead we conclude that the black hole is the most entropic conguration of the theory in this volume. But its entropy goes like the area! This is much smaller than the entropy of a local quantum eld theory, even with some UV cuto, which would have a number of states Ns eV ( maximum entropy = ln Ns ) Indeed it is smaller (when the linear dimensions are large!) than any system with local degrees of freedom, such as a bunch of spins on a spacetime lattice. We conclude from this that a quantum theory of gravity must have a number of degrees of freedom which scales like that of a QFT in a smaller number of dimensions. This crazy thing is actually true, and AdS/CFT is a precise implementation of it. Actually, we already know some examples like this in low dimensions. One denition of a quantum gravity is a generally-covariant quantum theory. This means that observables (for example the eective action) are independent of the metric: 0= Sef f = T . g

We know two ways to accomplish this: 1) Integrate over all metrics. This is how GR works. 2) Dont ever introduce a metric. Such a thing is generally called a topological eld theory. The best-understood example is Chern-Simons gauge theory in three dimensions, where the variable is a gauge eld and the action is SCS = tr A dA + ...M

(where the dots is extra stu to make the nonabelian case gauge invariant); note that theres no metric anywhere here. With either option (1) or (2) there are no local observables. But if you put the theory on a space with boundary, there are local observables which live on the boundary. Chern-Simons theory on some manifold M induces a WZW model (a 2d CFT) on the boundary of M . We will see that the same thing happens for more dynamical quantum gravities. 3) A beautiful hint as to the possible identity of the extra dimensions is this. Wilson taught us that a QFT is best thought of as being sliced up by length (or energy) scale, as a family of trajectories of the renormalization group (RG). A remarkable fact about this is that the RG equations for the 5

behavior of the coupling constants as a function of RG scale z is local in scale: zz g = (g(z)) the RHS depends only on physics at scale z. It is determined by the coupling constant evaluated at the scale z, and we dont need to know its behavior in the deep UV or IR. This fact is not completely independent of locality in spacetime. This opens the possibility that we can associate the extra dimensions raised by the Holographic idea with energy scale. Next we will make simplifying assumptions in an eort to nd concrete examples.

6

8.821 F2008 Lecture 0 4Lecturer: McGreevy September 22, 2008

Today 1. Finish hindsight derivation 2. What holds up the throat? 3. Initial checks (counting of states) 4. next time: Supersymmetry self-defense Previously, we had three hints for interpreting the bold Assertion that we started with, that hidden inside any non-Abelian gauge theory is a quantum theory of gravity.

1. The Weinberg-Witten theorem suggests that the graviton lives on a dierent space than the Quantum Field Theory. 2. This comes from the holographic principle. The idea, motivated by black hole thermodynam ics, is that the theory of gravity should have a number of degrees of freedom that grows more slowly than the volume (non-extensive) suggesting that the quantum gravity should live is some extra (more) dimensions than the QFT. 3. The structure of the Renormalization Group suggests that we can identify one of these extra dimensions as the RG-scale.

In order to make these statements a little more concrete, we will make a number of simplifying assumptions. That means we will not start by writing down the more general case of the corre spondence, but we will start with a special case.

1. There are many colors in our non-Abelian gauge theory. The motivation for this is best given by the quote: You can hide a lot in a large-N matrix. 1

Shenker The idea is that at large N the QFT has many degrees of freedom, thus corresponding to the limit where the extra dimensions will be macroscopic. 2. We will work in the limit of strong coupling. The motivation for that lies in the fact that we know some things about weakly coupled Field Theories, and they dont seem like Quantum Gravity. This constrains the manner in which we take N to be large, since we would still like the system to be interacting. This means that we dont want vector models. We want the theory to have as much symmetry as possible. However there is a theorem that stands in the way of this. Theorem (Coleman-Mandula) : All bosonic symmetries of a sensible S-matrix (nontrivial, having nite matrix elements) belong to the Poincare group. We are going to use all possible loopholes to circumvent this theorem. The rst lies in the word bosonic and the second one in sensible. 3. SuperSymmetry (SUSY) (a) It constrains the form of the interactions. This means that there are fewer candidates for the dual. (b) Supersymmetric theories have more adiabatic invariants, meaning observables that are independent of the coupling. So there are more ways to check the duality. (c) It controls the strong-coupling behavior. The argument is that in non-SUSY theories, if one takes the strong-coupling limit, it tends not to exist. The examples include QED (where one hits the landau Pole for strong couplings) and the Thirring model, which is exactly solvable, yet doesnt exist above some coupling. (d) Nima says its simpler (http://arxiv.org/abs/0808.1446). The most supersymmetric theory in d=4 is the N = 4 SYM and well be focusing on this theory for a little while. It has the strange property that, unlike other gauge theories, the beta function of the gauge coupling is exactly zero. This means that the gauge coupling is really a dimensionless parameter, and so, reason number (e) to like SUSY is... (e) SUSY allows a line of xed points. This means that there is a dimensionless parameter which interpolates between weak and strong coupling. 4. Conformal Invariance: The fact the coupling constant is dimensionless (even quantum-mechanically) says that there is scale invariance. The S-matrix is not nite! Everything is soft gluons. Scale invariance applies to both space and time, so X X ,where = 0, 1, 2, 3. As we said, the extra dimension coordinate is to be thought of as an energy scale. Dimensional anal z ysis suggests that this will scale under the scale transformation, so z . The most general ve-dimensional metric (one extra dimension) with this symmetry and Poincare invariance is of the following form:z ds2 = ( L )2 dx dx + dz 2 2 L z2

. Let z =

z familiar form (by change of coordinates) = ( L )2 dx dx + dz2 L2 , which is AdS5 . It z turns out that this metric also has conformal invariance. So scale and Poincar implies e conformal invariance, at least when there is a gravity dual. This is believed to be true more

L Lz ds2

L = L We can now bring it into the more2

2

generally, but there is no proof. Without Poincar invariance, scale invariance denitely does e not imply conformal invariance. We now formulate the guess that a 4-dimensional Conformal eld Theory is related to a theory of Gravity on AdS5 Note: On a theory of gravity space-time is a dynamical variable, where you specify asymp totics. This CFT denes a theory of gravity on spaces that are asymptotically AdS5 . Why is AdS5 a solution? 1. Check on PSet 2. Use eective eld theory (which in this case just means dimensional reduction) to elaborate on my previous statement that the reason why this is a solution is that the ux holds it up. To do this lets consider the relevant part of the supergravity action, which is S = D d x[ GR(D) Fg Fg ] As an example consider the case where D=10 and g=5 (note that we are working in dimensions (D) where GN = 1 = 1/5! = 2 = 4). [Note: The reference for this is Denef et al hep-th/0701050] Lets make an Ansatz (called Freund-Rubin) The metric is some space-time (x) plus a qsphere of radius R, so that ds2 = g dx dx + R(x)2 Gmn dy m dy n and furthermore for the ux though the q-sphere we have S q Fq = N To nd the form of this part of the metric as a result of the ux, we integrate over the q-sphere and get the action in D-dimensions, which will contain a term which is GGm1 n1 ...Gmq nq (Fg )m1 ...mq (Fg )n1 ...nq Sq We should then think of R(x) as a moduli eld. Now we have to evaluate the following a integral S q GR(D) = gRq [R + R2 + ...] We will now go to something called the FRAME GAME To make the D-dimensional Einstein-Hilbert term canonical, we can do a Weyl-rescaling of E the metric, meaning to dene a new metric g = (x)g . So the new metric is some function of the space-time variables times the old metric. So |gE | = d/2 g , RE = 1 R. Well pick to absorb the factor of R, so gE RE = d/21 gR(d) = Rq gR(d) , = R2q/d2 . N We know that the ux is the integral over the q-sphere, so S q F = N N Rq , so the 2 n ux term of the potential is Vf lux (R) S q F F = V ol(S q ) gF = Rg d/2 gE ( Rq )2 At small R the rst term in the eective potential dominates, while for R large, the eective potential approaches zero from below. In between there is some minimum. To nd the actual value of this we take 0 = V (Rmin ) N 2 Rmin . So Vf lux (R) N2 R

A specic example is N=4 SYM which is related to IIB Supergravity on AdS5 S 5 .

1 . R

where > > 0

The fact that the potential goes to zero at large R is unavoidable. It is because when R is large, the curvature is small and ux is dilute. So all terms in the potential go away. 4 For special values of and , we get that N Rmin = L4 3

This endeavor is called ux compactication and it is the way to nd vacua of string theory without massless scalars. Furthermore, we can add some eective action for scalars with m2 V (Rmin ), which R 1 looks like this : S = 2 dd x g e (RE 2)R . The cosmological constant is determined as = Vef f (Rmin ) < 0 The fact the the cosmological constant exists, means that the metric is not at. The equation of motion is the following: 0 = R =2d d2 S g 1 R = g ( 2 R ) dr 2 + dx dx r2

R ( 2 ) 2d

We know the solution for AdSd , which has the metric ds2 = L2

Now I will describe the right way to compute curvatures, which is to use what is called tetrad or vielbein or Cartan-Weyl method. There are three steps

Rewrite the metric as ds2 = (er )2 + e e ,where I dene er L dr and ex L dx r r

Cartan 1 demands that they are covariantly constant and is formulated as follows: dea = ab eb , where the is called the spin connection.1 1 L For example der = 0 and e = r2 dr dx = L er e = L e = r

Why do we care about ? Heres why:ab Cartan 2 Rab = d ab + ac cb = R dx dx

From that you can get the Ricci tensor, which appears in the equation of motion, by contracting some indices R = R Lets do some examples: 1 R = r r = L2 e e Rr = d r = 1 r e e L2

1 R = L2 ( ) r 1 Rr = L2

R =

d g L2

2 2d

=

d L2

=

d(d+1) 2L2

Counting of degrees of freedom [Susskind-Witten, hep-th/9805114] The holographic principle tells us that the area of the boundary in planck units is the number of degrees of freedom (maximum entropy). 4

Area of boundary ? = 4GN

number of dofs of QFT Nd

Both sides are equal to innity, so we need to regulate our counting somehow. The picture we have of AdS is a collection of copies of Minkowski space of varying size and the 3 boundary is the locus where they get really big. The area is dened as A = R3 gd3 x = R3 d3 x L3 r This is innite for two reasons: from the integral over x and from the fact that r is going to zero. Lets regulate the Field Theory rst. We put the thing on a lattice, introducing a low distance cut-o and we put it also in a box of size R. The number of degrees of freedom is the number of 3 cells times N 2 , so Nd = R3 N 2 . To regulate the integral we write before, we stop not at r = 0 but 3 L3 3 L3 we cut it o at r = . A = d x r3 |r= = R3 Now we can translate thisL2 G10 N

N 2 GN =L3 R3 (5) GN 3

(5)

GN L5

(10)

Nd =

=

A (5) GN

So AdS/CFT is indeed an implementation of the holographic principle.

5

8.821 F2008 Lecture 5: SUSY Self-DefenseLecturer: McGreevy September 22, 2008

Todays lecture will teach you enough supersymmetry to defend yourself against a hostile supersymmetric eld theory, should you meet one down a dark alleyway. Topics will include 1. SUSY representation theory, including basic ideas regarding the algebra and an explanation of why N = 4 is maximal (a proof due to Nahm) 2. Properties of N = 4 Super Yang-Mills theory, such as the spectrum and where it comes from in string theory. Before plunging in, we might be inclined to wonder: why is supersymmetry so wonderful? In addition to the delightful properties that will be explored in the remainder of this course, there are various reasons arising from pure particle physics: for example, it stabilizes the electroweak hierarchy, changes the trajectory of RG ows such that the gauge couplings unify at very high energies, and halves the exponent in the cosmological constant problem. These are issues that we will not explore further in this lecture.

1

SUSY Representation Theory in d = 4

We begin with a quick review of old-fashioned Poincare symmetry in d = 4:

1.1

Poincare group

Recall that the isometry group of Minkowski space is the Poincare group, consisting of translations, rotations, and boosts. The various charges associated with the subgroups of the Poincare group are quite familiar and are listed below Charge P M 1 Subgroup R3,1 SO(3,1) (SU(2) SU(2))

Translations Rotations/boosts

Luckily the rotation/boost part of the algebra SO(3,1) decomposes (at least in the sense of the sloppy representation theory that we are doing today) into two copies of SU(2). We are all familiar with representations of SU(2), each of which are labeled by a half-integer si 0, 1/2, 1, ...; the product of two of these reps gives us a rep of SO(3,1):

Scalar 4-vector Symmetric tensor Weyl fermions Self-dual,anti-self-dual tensors

(s1 , s2 ) (0, 0) (1/2, 1/2) (1, 1) (1/2, 0), (0, 1/2) (1, 0), (0, 1)

Having mastered this group, we wonder whether we could possibly have a larger spacetime symme try group to deal with, leading us to the idea of enlarging the Poincare group by adding fermionic generators!

1.2

Supercharges

Lets call these fermionic generators QI ; is a two-compnent Weyl spinor index belonging to the (1/2, 0) spinor representation above; essentially this simply denes the commutation relations of Q with M , telling us that it transforms like a spinor. I labels the number of supersymmetries and thus goes from 1 to N . The total number of supercharges is thus equal to the number N multiplied by the dimension of the spinor representation and so in four dimensions is 4N . The adjoint of this operator is called QI (QI ) . We note that [P , Q] = 0 and the anticommu tators of Q with itself are given by I J I QI , QJ = 2 P k Q , Q = 2 Z IJ (1) Here Z IJ is antisymmetric in its indices and thus exists only for N > 1; playing with the SUSY algebra assures us that it commutes with everything and so is just a number called the central charge. Note that this algebra is preserved under the action of a U (N ) called (inexplicably) an R-symmetry.I QI UJ QJ I UJ U (N )

(2)

Whether or not the diagonal phase rotation of this U (N ) survives in the full quantum theory depends on dertails of the theory. Finally, the current jR of this symmetry does not commute with the supercharges [jR , Q] = 0, meaning dierent elements of the same supermultiplet can have dierent R-charges.

2

1.3

Representations of the SUSY Algebra: Massless states

Next, let us construct representations of this algebra, starting with massless states. Consider the Hilbert space H of a single massless particle, and pick a frame where the momentum is P = (E, 0, 0, E), E > 0. Now using (1) and standard formulas for ( 0 = I2 and i are the Pauli matrices) we see that 4E 0 I I Q , QJ = J (3) 0 0 Focusing on the bottom left corner of this matrix, we see that QI , (QJ ) = 0 ||QI |||2 = 0, 2 2 2 (4)

for all | H. If we assume this zero-norm state QI | is actually the zero state (which is the 2 case in a unitary Hilbert space), then we conclude that half of the Qs annihilate every state in the space, implying also that Z IJ = 0 in this subspace. We are left with the QI s; note each QI 1 1 lowers the helicity by 1/2 and each QI raises the helicity by 1/2. Finally, if we normalize the Q1 s 1 appropriately we get I Q1 Q1J I , = J (5) 2 E 2 E From now on we will suppress the 1 subscript. This is simply the algebra of a fermionic oscillator, which we know how to deal with: start with a highest-helicity state that is annihilated by QI |h = 0 and hit it with QI s to build the Hilbert space: |h, QI1 |h, QI1 QI2 |h, ... QI1 QI2 ...QIN |h (6) There are a total of 2N states because (as pointed out by T. Senthil), each operator QI may or may not be present in the string of Qs. Now we wonder: how big can N be? A theorem by Nahm answers this question. A useful lemma (related to the Weinberg-Witten theorem discussed previously) is: A quantum eld theory without gravity cannot contain massless states with helicity |h| > 1. This means that we cannot apply Q too many times, as eventually we will end up with a state with helicity more negative than -1. This is a constraint on N and allows us to summarize all possibilities in the following (nite) table, in which each entry is the number of states with that given helicity in the corresponding supermultiplet: Helicity 1 0 1 2 11 2

N = 1 gauge 1 1 1 1

N = 1 chiral 0 1 (1+1) 1 0

N = 2 gauge 1 2 (1+1) 2 1

N = 2 hyper 0 (1+1) (2+2) (1+1) 0

N = 4 gauge 1 4 6 4 1

Note that if N > 4, then we will necessarily have states with helicity greater than 1. Thus the maximal supersymmetry for a nongravitational theory in d = 4 is N = 4, corresponding to 16 supercharges. If gravity is allowed, the corresponding bound on helicity is |h| < 2 N < 8, and thus we can have N = 8 supergravity with 32 supercharges in four dimensions. 3

1.4

Representations of the SUSY Algebra II: Massive states

It is time to move on to massive states, which turn out to have larger representations. Following a procedure analogous to that above, we boost to the rest-frame of the particle where we have P = (M, 0, 0, 0). In that case (1) becomes I J I J I Q , (Q ) = 2M Q , Q = 2 Z IJ (7) J It is now necessary to introduce some unfortunate notation. We take a deep breath and block diagonalize Z IJ so that it takes the form 0 Z1 Z1 0 IJ (8) 0 Z2 Z = Z2 0 ... To keep track of this structure we split the index I into two parts I = (I, I), where I runs over each of the subblocks in the large matrix I = 1, 2, ...Floor(N /2) and I = 1, 2 and runs over the two entries in each subblock. We now form the following linear combinations of Q s: 1 (I =2,I ) I (I =1,I ) 0 Q = Q Q (9) 2 These supercharges obey the anticommutation relations I J I Q , Q = (M ZI ), J

(10)

for all I . Note that if the bound is saturated and M = |ZI |, then one of the supercharges QI or + QI kills the state, implying that this particular representation is smaller; we say that BPS states live in short multiplets.

where the are correlated on all sides of this equation. Once again, a unitary representation implies that the right-hand side of this expression should be greater than zero, leaving us with the celebrated BPS bound: M |ZI | (11)

Why is this useful? Suppose we nd a short multiplet with M = |ZI |. Then for M to stray from this special value (say under a variation of a parameter of the theory, a coupling constant etc.), the short multiplet must become long, which means that it must pair up with its conjugate short multiplet. Typically this is not possible and the mass of the state is restricted to be |ZI | always.1 With no further ado we present a table of the various possible states for massive supermultiplets: Note further that because Z is a central element of the superalgebra, which includes P , it is actually a c-number constant, rather than an operator like P . This makes the fact that Z is determined by the algebra much more powerful. Thanks to Michael Kiermaier for emphasizing this distinction.1

4

Helicity 1 01 2

N =1 gauge 1 2 1

N =1 chiral 0 1 2

N = 2 gauge 1 4 6

N = 2 BPS gauge 1 2 1

N = 2 BPS hyper 0 (1+1) (2+2)

N = 4 BPS gauge 1 4 6

Here all states will be present with their CPT conjugates of opposite helicity. Note that the massive gauge multiplet contains precisely the same states as (massless chiral + massless gauge) multiplets. Compare this to the familiar non-supersymmetric construction via the Higgs mechanism of a massive vector boson from (massless scalar + massless gauge) bosons. It should not be surprising that the supersymmetric version of this is called the Super-Higgs mechanism.

2

N = 4 Spectrum

We now turn to a closer examination of the spectrum for the maximally supersymmetric N = 4 case, which can be formulated with any gauge group (but we will restrict ourselves to U (N )). Here the eld content is (compare with the massless N = 4 gauge multiplet in the table): 1. Gauge eld A : = 0...3 is a Lorentz vector index. 2. Weyl spinors I : I = 1...(N = 4), a spinor index. 3. Scalars X i : here i = 1...6, as will be explained shortly. As the gauge eld is necessarily in the adjoint of the gauge group, all of these elds are N N matrices in the adjoint of U (N ). Note that the R-symmetry is SU (4) (the diagonal U (1) is broken, for reasons which we will presumably learn later), which acts on the fermions I in the fundamental of SU (4). To understand the action on the scalars, it is useful to know that SU (4) SO(6) (12)

The fundamental of SU (4) is the spinor representation of SO(6). As we obtained the Xs by the action of two antisymmetrized Qs on the (SU (4) singlet) A , we expect it to transform in the antisymmetric tensor representation of SU (4). This works out to be the standard vector 6 of SO(6), which acts on the i index on X i . If we stray into string theory briey, we see that this theory is also the worldvolume theory on a stack of N D3 branes. This gives us another way to understand the spectrum. Imagine placing a single D3 brane in R3,1 . This breaks 10 (total) 4 (along the brane) = 6 translational symmetries, and so we expect to have 6 Goldstone bosons in the worldvolume theory of the D3 branesthese are the X i s! Similarly, the full ten-dimensional theory had 32 supercharges (from N = 2 supergravity in ten dimensions), but we only see 16 supercharges remaining in the N = 4 theory. To understand where they went, consider the supercharges Q: {Q, Q} P 5 (13)

Those Q whose anticommutators generate translations along the 6 broken translational symmetries are broken, and this leaves us with 16 unbroken supersymmetries. By Goldstinos theorem, this breaking results in 16 massless fermions or goldstinos (goldstini ?), which can be rearranged into the 4I s. Finally, we note that another way to produce this theory is to start with maximal superymmetry in d = 10 without gravity. This is a pure N = 1 gauge multiplet, and contains 16 supercharges; we are no longer allowed to break these supersymmetries with D3 branes (since we need all of them!). What we can do is consider eld congurations that are independent of the 6 extra dimensions and simply integrate over them in the 10 dimensional action. This procedure is called dimensional reduction and will give us a 4-dimensional action containing precisely the eld content of N = 4 SYM.

6

8.821 F2008 Lecture 06: Supersymmetric Lagrangians and Basic Checks of AdS/CFTLecturer: McGreevy September 24, 2008

We are on our way to talking about really awesome things about supercool stu. Before we get there, though, we need to develop some very powerful technology. To that end, today we will talk about 1. SUSY Lagrangians and a whirlwind tour of the beauties of superspace. 2. more on N = 4 SYM. 3. Back to the Big Picture: Some basic checks of AdS/CFT

Looking past this lecture, we will be talking about strings from gauge theory next.

1

N = 4 SYM and Other Supersymmetric Lagrangians

Recall that the eld content of N = 4 SYM is a vector A , gaugini I=1...4 , and six scalars X i , all in the adjoint of the gauge group. The Lagrangian density (which is completely determined by the amount of SUSY, up to two parameters, (gY M , )), is

L = +

12 gY M 6 i .) The perturbation J will be encoded in the boundary condition on bulk elds. Incidentally, this resolves a huge confusion in GR literature dating back to 1970s, which claims that Cauchy problem3 in AdS is not well-posed. What happens is that signals can reach from the surface where initial data is supplied to the boundary in nite time4 , and hence boundary condition is needed for the problem to be well-posed. The idea (GKPW) for computing Z[J] is then: Z[J] = eR LJ CF T

= Zstrings [b.c. depends on J]

=???

gs 0 L2 /

eSSUGRA EOM, b.c. depend on J

Note that the limit gs 0, L2 / in AdS corresponds to the limit N , in QFT.2 Note that Tr (X {i1 X i } ) are chiral (QO = 0) primaries (KO = 0), and hence are short (BPS) multiplets. The correspondences of long multiplets is another story which well get to soon. 3 A Cauchy problem is one in which initial data is specied in some space-like slice, and one is asked to determine the future evolution of the system. 4 We will study the geometry of AdS in detail next week.

2

As an example of the application of this formula, one can compute the 3-point function of supergravity states. It turns out that the results agree exactly with N = 4 SYM (at leading order in 1/N ). A priori this is a surprising statement, since the eld theory calculation is done by pertur 2 bation theory at small = gY M N , while the gravity calculation is meant to work at large . This happens because all the supergravity states correspond to BPS (= chiral primary) operators, and hence their 3-point functions actually do not depend on . This specialness of the states weve been able to map so far may worry you: how do we describe the rest of the eld theory operators using the bulk theory? To approach this question, we take a small detour.

2

Strings from Gauge Fields

There are several long-suspected connection between strings and gauge elds.

2.1

Strings as model of hadrons (late 60s)

For hadron with largest spin J for a given mass, J = m2 (J) + const. Regge trajectory

which is precisely the form of the spectrum of the vibration modes of a spinning quantum relativistic string in RD .

2.2

Flux tubes in QCD

We know that the gauge eld in QCD is in the conned phase (as opposed to the Coulomb phase, see Fig. 1(a),(b) for illustration). Consequently, the eld lines bunch together and behave like a tensionful ux tube. More concretely, consider the area law for Wilson loop: Tr PeiR

A

= e Area()tension

?

Here denotes the worldsheet as shown in Fig. 1(c). This is an order parameter for connement and indicates that ux tube has a nite tension if and only if the gauge theory is in the conned phase. The above suggests that we may try to use ux tubes as microscopic variables5 , which should behave as strings in R3,1 . However, a result from Polyakov suggests that this is problematic. Polyakovs result is this: in quantizing the string worldsheet with metric , the transformation e() is a gauge symmetry and hence e = 0. There is however a conformal anomaly, 5 It will turn out that a string description does not require connement. The loophole is that one may consider funny shaped worldsheets.

3

x

loop

t

worldsheet

(a)

(b)

(c)

Figure 1: Flux tubes in QCD: (a) meson in conned phase, (b) meson in Coulomb phase, i.e. a dipole (c) the world sheet of ux tube for computing Wilson loop.

which generates kinetic term of in e . Hence should be treated as an extra dimension. For QFT in d 1, the procedure for quantizing is known (Liouville), and this is really the correct idea (this leads to what is sometimes called old matrix models, including the c = 1 matrix quantum mechanics). In d = 4 this result (so far) serves as more inspiration for our extra-dimensional picture..

3

t Hooft Counting

The most explicit evidence that gauge theory leads to string theory comes from t Hooft counting. Consider a (any) quantum eld theory with matrix elds6 b=1,...,N (for concreteness well take the a=1,...,N matrix group to be U (N )), and consider a Lagrangian of the form: 12 gYM

L Rescale = gYM , L Tr

Tr ()2 + 2 + 3 + 4 + . . .

2 ( )2 + 2 + gYM 3 + gYM 4 + . . .

Now consider propagators. It is convenient to adopt the double line notation, in which oriented index lines follow conserved color ow, so that, for propagator7 :a a d a d c b = c b c6

c d

By matrix eld we mean that their products appear in the Lagrangian only in the form of matrix multiplication, e.g. (2 )c = b c . a a b 7 a d a c b c Had we been considering SU (N ), the result would be a d c b b d /N 2 =

4

And similarly for vertices:

a c a b b a a d2 gYM

gYMc b b c c d

2 Now consider the t Hooft limit in which N and gYM 0 but which = gYM N = const. Is this limit classical/free? The answer turns out to be no. The loophole is that even though the coupling goes to zero, the number of modes diverges.

N2

N 2

3 N 2

(a)

(b)

(c)

Figure 2: Planer graphs that contribute to the vacuumvacuum amplitude.

2 gYM N = N 0

Figure 3: Non-planer graph that contributes to the vacuumvacuum amplitude. To see this more concretely, consider vacuumvacuum diagrams (see Fig. 2 and 3 for illustration). In Fig. 2 we have a set of planer graphs, whose contributions take the general form n N 2 . However, there are also contributing non-planer graphs, such as the one in Fig. 3, whose contribution does the take that general form. Every double-line graph specics a triangulation of a 2-dimensional surface . There are two ways to construct the explicit mapping: Method 1 (direct surface) Fill in index loops with little plaquettes. Method 2 (dual surface) (1) draw a vertex in every index loop and (2) draw an edge across every propagator. These constructions are illustrated in Fig. 4 and 5. Comparing the resulting surface and the corresponding contribution, it can be seen that the contri bution of a graph with h handles is proportional to N 22h n . Thus, the partition function (which is also the vacuumvacuum amplitude) must take the form: 5

S2

T2

(a)

(b)

Figure 4: Direct surfaces constructed from the vacuum diagram in (a) Fig. 2a and (b) Fig. 3.

S2

Figure 5: Dual surface constructed from the vaccum diagram in Fig. 2c. Note that points at innity are identied.

ln Z =

h=0

N 22h fh ()

which is exactly like the result from string perturbation expansion.

6

8.821 F2008 Lecture 8: Large N CountingLecturer: McGreevy October 4, 2008

1

Introduction

Today well continue our discussion of large N scaling in the tHooft limit of quantum matrix models. In particular, well review the N counting for vacuum diagrams and learn how to do N counting for correlators. Our perspective on the large N limit continues to be one of convenience. It is one of several assumptions that enable the simplest realization of holography and gauge/gravity duality. Next time well continue talking about simplifying assumptions and turn our attention to conformal invariance.

2

Vacuum tHooft counting

Lets recall our starting point. We were thinking about matrix eld theories, eld theories in which all the elds are N x N matrices in addition to whatever other labels they may have. In particular, we write our theory schematically in terms of one big eld which we think of as potentially including scalars , gauge elds A , and fermions all of which are N x N matrices. The schematic Lagrangian is 1 L = 2 Tr ( V ()) (1) g where V () 2 + 4 + ... and all products should be thought of as matrix products. This schematic representation is meant to convey two pieces of information. First, that the Lagrangian is invariant under U (N ) rotations which act on the matrix elds by conjugation: U U 1 . Second, that the elds are scaled so that the coupling g 2 sits out in front of the entire Lagrangian. In particular, we have to very careful about eld normalizations when doing large N counting for correlation functions. The tHooft limit is the limit where N and g 2 0 while = g 2 N remains xed. is the tHooft coupling. Note that we are adopting a dierent normalization condition than in the previous lecture on large N counting (there we used the eld = (1/g) which had kinetic terms with the usual eld theory normalization). With our current normalization the propagator goes like g 2 = N . The propagator in the double line notation appears in Fig. 1. Remember that for the purposes of 1

Figure 1: Propagator, /N

Figure 2: Three point vertex, N/

large N counting we are instructed to think only about counting powers of N, so we will suppress positions of elds, spin indices, etc in our analysis. In a generic matrix theory there will be three point (Fig. 2) and four point (Fig. 3) interaction vertices. Both types of vertices come with factors of 1/g 2 = N/ from the overall 1/g 2 multiplying the Lagrangian. A general diagram consists of propagators, interaction vertices, and index loops, and gives a contribution no. of prop. no. of int. vert. N N no. of index loops . (2) diagram N For example, the planar diagram in Fig. 4 has 4 three point vertices, 6 propagators, and 4 index loops giving the nal result N 2 2 . We learned last time how we to associate a triangulated surface with a Feynman diagram in two ways. The rst way is to let the interaction vertices of the diagram be the vertices of the triangu lation. In this case the index loops of the diagram bound pieces of surface. The alternative is the dual triangulation in which we place vertices in the middle of the index loops of the Feynman dia gram. Two vertices are connected if their respective index loops share a propagator in the Feynman diagram. Dont be confused by multiple uses of the word vertex. There are interaction vertices of various kinds in the Feynman diagrams and these correspond to vertices in the triangulation only in the rst formulation. If E = no. of prop., V = no. of int. vert., and F = no. of index loops then we showed last time that the diagram gives a contribution N F E+V EV . The letters refer to the rst way to triangulate

Figure 3: Four point vertex, N/

2

Figure 4: This diagram consists of 4 three point vertices, 6 propagators, and 4 index loops

the surface in which interaction vertices are triangulation vertices. Then we interpret E as the number of edges, F as the number of faces, and V as the number of vertices in the triangulation. In the dual triangulation there are dual faces F , dual edges E , and dual vertices V . The relationship , V = F , and F = V . In the dual formulation we between the original and dual variables is E = E found that a diagram contributes N F E+V EF . Its not a coincidence that the powers of N agree in both formulations. The exponent = F E + V = F E + V is the Euler character and it is a topological invariant of two dimensional surfaces. In general it is given by = 2 2h b where h is the number of handles (the genus) and b is the number of boundaries. Note that the exponent of , E V or E F is not a topological invariant and depends on the triangulation (Feynman diagram). Before we continue with the analysis its worth pointing out that there are other large N limits one might want to consider. The motivation for the tHooft limit comes in part from thinking about radiative corrections which grow with N and decrease with g. This suggests some sort of balancing act might be possible which keeps the theory non-trivial in the large N limit. We encode this scaling by xing = g 2 N as we make N large. Other interesting possibilities are provided by critical points of the matrix model. Suppose when g = gc the matrix model is at a critical point. Then there may be a sensible large N limit in which we x |g gc | N as we make N large. One example of this is matrix quantum mechanics with an inverted oscillator potential. Back to the tHooft limit. Because the N counting is topological (depending only on ) we can sensibly organize the perturbation series for the free energy in terms of surfaces! Because were computing only vacuum diagrams for the moment, the surfaces were considering have no boundaries b = 0 and are classied by their number of handles h. h = 0 is the two dimensional sphere, h = 1 is the torus, and so on. We may write the free energy as ln Z = h=0

N

22h

=0

c

,h

=

h=0

N 22h Fh ()

(3)

were the sum over surfaces is explicit. Now we can see some similarities between this expansion and perturbative string expansions. By writing N 22h = (1/N )2h2 we can identify the string coupling gs = g()/N where we can allow some function of in addition to the pure N dependence. Because of the proliferation of coupling 2 constants, lets rename the g 2 in our matrix model gY M in anticipation of the case of most interest 2 to us when the matrix model is a gauge theory. The tHooft coupling is now written = gY M N . 3

If 1/N plays the role of the string coupling gs (so that in the large N limit joining and splitting of strings is suppressed) then its reasonable to ask what plays the role of the worldsheet coupling. It turns out that we can think of as a sort of chemical potential for edges in our triangulation. Looking back at our diagram counting we can see that if becomes large then diagrams with lots of edges are important. Thus large encourages a smoother triangulation of the world sheet which we might interpret as fewer quantum uctuations on the worldsheet. More formally, we expect a relation of the form 1 which encodes our intuition about large suppressing uctuations. This story is very general in the sense that all matrix models dene something like a theory of two dimensional quantum gravity via these random triangulations. The connection makes even more sense when we remember all the extra labels weve been suppressing on our eld . For example, the position labeling where the eld sits plays the role of embedding coordinates on the worldsheet. Other indices (spin, etc.) indicate further worldsheet degrees of freedom. We may think of the matrix model partition function as giving the thermodynamics of the string theory in the sense that it provides the functions Fh entering the free energy. However, the microscopic details of the theory are not so easily discovered. In particular, its not at all clear what microscopic action we should integrate over moduli space to obtain the functions Fh . As a nal check on the non-triviality of the theory in the large N limit, lets see if the world sheet coupling will generically run in a matrix model. In d = 4 the one loop behavior is basically 3 gY M = g gY M N = gY M and this implies the beta function for is = = 2gY M N g 2 . Thus can still run in the large N limit and the theory is non-trivial. If d = then 3 the beta function for g now looks like g gY M N + 4d gY M and we nd that 2 + 4d 2 2 which is still nite in the large N limit.

3

Correlation functions

Lets now consider how to calculate correlation functions of operators, and in particular, how to do large N counting for such correlation functions. We will consider only single trace operators, operators O(x) that look like O(x) = c(k, N )Tr(1 (x)...k (x)) (4)

of O (to determine the constant c(k, N )) we will adopt the convention that OO c N 0 where the subscript c stands for connected. This convention is nice because states like O|vac will have nite norm (given by the correlation function) in the large N limit.

and which we will often abbreviate as Tr(k ). There are two annoying complications now. We must be careful about how we normalize the elds and we must be careful about how we normalize the operator O. The normalization of the elds will continue to be such that the Lagrangian takes the form L = g21 L = N L which L() containing no explicit factors of N. To x the normalization YM

To use our convention to determine c(k, N ) we need to know how to insert single trace operators into the tHooft counting. We do this by associating a new vertex with each single trace operator. This vertex has k legs where k propagators can be attached and looks like a big squid. An example of such a new vertex appears in Fig. 5 which corresponds to the insertion of the operator Tr(6 ). 4

Figure 5: New vertex for an operator insertion of Tr(k ) with k = 6

Figure 6: Disconnected diagram contributing to the correlation function Tr(4 )Tr(4 )

For the moment we dont associate any explicit factors of N with the new vertex. Lets consider the example Tr(4 )Tr(4 ) . We need to draw two four point vertices for the two single trace operators in the correlation function. How are we to connect these vertices with propagators? The dominate contribution actually comes from disconnected diagrams like the one shown in Fig. 6. The leading disconnected diagram has four propagators and six index loops and so gives a factor 4 N 2 N 2 . One the other hand, the leading connected diagram shown in Fig. 7 has four propagators and four index loops and so only gives a contribution 4 N 0 . The fact that disconnected diagrams win in the large N limit is general and goes by the name large N factorization. It says that single trace operators are basically classical objects in the large N limit OO O O . As an aside, another convenient way to draw the connected diagram in Fig. 7 is shown in Fig. 8 where we have deformed the two four point operator insertion vertices so that they are ready for contraction making it easier to draw propagators.

Figure 7: Connected diagram contributing to the correlation function Tr(4 )Tr(4 )

5

Figure 8: A redrawing of the connected diagram shown in Fig. 7

The leading connected contribution to the correlation function is independent of N and so OO c c2 N 0 . Requiring that OO c N 0 means we can just set c = 1. Note that the MAGOO review normalizes their single trace operators with an explicit 1/N which is wrong. Having xed the normalization of O we can now determine the N dependence of higher order correlation functions. For example, the leading connected diagram for O3 where O = Tr(2 ) is just a triangle and contributes a factor 3 N 1 N 1 . In fact, quite generally we have On c N 2n for the leading contribution. The states O|vac are called glueball states, and they correspond to excitations of the theory that are free at large N. These states also make sense from the string worldsheet point of view where the operator O is like an integrated vertex operator. We now want to study the generating function of correlation functions of single trace operators. Consider a compound operator like A OA where each OA Tr(kA ) is a single trace operator with kA N . This is a multi-trace operator because it is the product of single trace operators. We can write a generating function for correlation functions like A OA using the operator S = N A A OA where the overall factor of N is meant to normalize S in the same way as the Lagrangian. Remember that were suppressing all indices including spacetime locations of elds meaning there may be integrals involved in the proper denition of S. The generating function is dened as (5) Z = eF = eS and when calculating Z we must remember that each OA acts just like an extra vertex with the normalization we dened above. As a simple check of the AdS/CFT correspondence we would like to determine the large N scaling of F . The naive answer is just that F N 2 because there are N 2 degrees of freedom, however this counting may fail for example when the theory is in a conning phase. Let us therefore assume the theory is not in a conning phase and calculate the large N behavior of F more carefully. In the free theory we do indeed have F N 2 just by our naive counting of degrees of freedom. What about in tHooft limit? We may lump the interactions in L in with the operators in S since they are all single trace with an overall constant set by N. Then we can write the generating function as Z = e N Of ree

eN

O

f ree

(6)

because in the tHooft limit we have seen that single trace operators behave as if they are classical. Using our previous formula for the N dependence of correlation functions we nd that O N so that F N 2 because of the overall factor of N.

6

Figure 9: A quark vacuum bubble and a quark vacuum bubble with gluon exchange

Lets check this result using the GKPW formula relating the partition functions of eld theory and supergravity Z = eS = exp (SSU GRA ). (7) 5 We can determine the N dependence of SSU GRA G1 d x gR by scaling the action in terms 5 N of L the AdS scale. In ve dimensions the Newton constant has units of length cubed, and so the L3 correct scaled form of the action is SSU GRA G5 Sdimensionless . Using the fact that L3 /G5 N 2 N N we discover that SSU GRA scales in the same way as F from the eld theory. This fact follows from the relation L4 /( )4 = which we found by studying the RR soliton, from the ten dimensional 2 2 Newton constant in string units G10 = gs ( )4 , and from the denition = gY M N = gs N . The N assumption of a non-conning ground state on the eld theory side translates into the absence of additional scales in the bulk spacetime that would invalidate our scaling argument.

4

Simple generalizations

We can generalize the analysis performed so far without too much eort. One possibility is the addition of elds, quarks, in the fundamental of U (N ). We can add fermions L q D q or 2 . Because quarks are in the fundamental of U (N ) their propagator consists bosons L |D q| of only a single line. When using Feynman diagrams to triangulate surfaces we now have the possibility of surfaces with boundary. Two quark diagrams are shown in Fig. 9 both of which triangulate a disk. Notice in particular the presence of only a single outer line representing the quark propagator. We can conclude that adding quarks into our theory corresponds to admitting open strings into the string theory. We can also consider meson operators like qq or q k q in addition to single trace operators. Another direction for generalization is to consider dierent matrix groups such as SO(N ) or Sp(N ). The adjoint of U (N ) is just the fundamental times the anti-fundmental. However, the adjoint representations of SO(N ) and Sp(N ) are more complicated. For SO(N ) the adjoint is given by the anti-symmetric product of two fundamentals (vectors), and for Sp(N ) the adjoint is the symmetric product of two fundamentals. Also, lines in the double line formalism no longer have arrows. As a consequence the lines in the propagator for the matrix eld can join directly or cross and then join as shown in Fig. 10. In the string language the worldsheet can now be unoriented, an example being given by a matrix eld vacuum bubble where the lines cross giving rise to the worldsheet RP2 . 7

Figure 10: Propagator for SO(N ) (+) or Sp(N ) () matrix models

5

Large N vector models

Everything weve said for large N matrix models can be contrasted with the story for large N vector models. The dierence between these two classes of models is that large N matrix models have N 2 degrees of freedom while large N vector models have only N degrees of freedom. This changes the scaling of the coupling in the large N limit. The Lagrangian of a large N vector model might look something like L = N V ( /N ) where the whole second term has the form N V m2 v /N ( )2 . This form of the Lagrangian amounts to many more interactions than in the large N matrix model. The coupling gv = v /N 1/N (v is not the tHooft coupling!) goes to zero much faster than the coupling in the matrix model gY M 1/ N . Because of the dierent scaling, the only interactions in the standard large N vector model are cactus diagrams. This means that modulo some self energy corrections the theory is essentially free. There also dont appear to be any strings.

6

Preview: the need for strings

As a preview of coming attractions, lets see what matrix models can tell us about the appearance of string theory in gauge theory. Recall that we said every SUGRA perturbation in AdS5 S5 corresponds to some single trace operator in N = 4 SYM. In particular, we found a correspondence between symmetric scalar KK modes in SUGRA and chiral primary operators in the N = 4 theory. However, this map from SUGRA modes to N = 4 operators is not onto, in fact it represents a very small subset of the N = 4 operators. We would like to know what the rest of the N = 4 operators correspond to in the gravity theory. A clue comes from looking at single trace operators like O = Tr(XXXXXY XXDXX...Y X) (8)

which we think of as a string of J letters (with 1 J N ). Because of the cyclic property of the trace, this string of letters looks a lot like a sequence of operator insertions on a closed string!

8

8.821 F2008 Lecture 09: Preview of Strings in N = 4 SYM; Hierarchy of Scaling dimensions; Conformal Symmetry in QFTLecturer: McGreevy October 8, 2008

1

Emergence of Strings from Gauge Theory

Continuing from the previous lecture we substantiate the correspondence between the operators in N = 4 SYM theory with the excitations of some string theory. We saw that the primary chiral operators (i.e. primary operators in the conformal eld theory which also belong to the short multiplets of the supersymmetry) correspond to the SUGRA modes but we didnt discuss what happens when the operators in the gauge theory are in the long multiplet. Thus consider a more generic operator (residing in a long supersymmetry multiplet) of the matrix QFT in the large N limit O(x) = tr(XXXXY XXY....Y X) (1) We consider the limit N J 1 where J is the number of entities in the above product. Cyclicity of the trace implies that the structure of the above operator has the symmetry of a closed loop. In fact the above operator corresponds to creation operator for excited string states with X s and Y s the elds living on the worldsheet. Further, in the large J limit, J corresponds to the angular momentum of the string excitations and in fact this relation could be used to reconstruct the string theory. Having sketched the correspondence between the various operators of the SYM guage theory with SUGRA modes or string excitations, let us organize this knowledge to get a better perspective on when it could be useful.

The mass of the SUGRA mode is given by m2 GRA = 1/L2 . Using the AdS-CFT corre SU AdS spondence m2 GRA L2 = ( 4) where is the scaling dimension of the corresponding SU AdS chiral primary operator in the SYM theory. This implies that SU GRA N 0 0 . Moving on to the excited string states m2 /L2 string states 1/ AdS where 1/ is string tension. Thus the corresponding scaling dimension equals string states N 0 1/4 . Finally, there are D-branes in the string theory which correspond to baryonic states in the

1

gauge theory. Their mass is given by

1

2 m2 Dbrane 1/ gs

2 /gs L2 Dbrane = N 1/4 . AdS

Clearly, the scaling dimension for the dierent states (SUGRA, strings, D-branes) has distinctive dependence on N and . This hierarchy of s is what a QFT needs to have a weakly coupled gravity description without strings. For example the limit removes all states except SUGRA modes.2 There are various forms of AdS-CFT conjecture. For example, one may believe that it only holds in the limit , N (yielding classical gravity) or perhaps also at nite and N limit (when one obtains classical strings on small AdS radius). Remarkably, all evidence till date points to a much stronger statement that it holds for all N and all .

2

Conformal Symmetry in QFT

This is a worthy subject in itself and study of conformal invariance is relevant for understanding both UV and IR limit of various QFTs and also for the worldsheet theory of the strings in the conformal gauge. As you may already know (and we recapitulate it below) that conformal symmetry in two dimensions is very special due to existence of innite number of conserved currents. Therefore it pays to understand which aspects of a conformal eld theory (CFT) are particular to D = 2 and which apply to any dimension D > 2. We have already seen some of the constraints due to the requirements of Lorentz invariance and SUSY on the QFTs and expectedly, requirement of conformal invariance makes it even more constrained. This is also the last stop in our ruthless program to evade the loopholes in the Coleman-Mandula theorem. Some of the useful references are the book Conformal Field theory by Di Francesco, Mathieu and Senechal and the articles by Callan, Ginsparg and portions of MAGOO (links posted on the course webpage).

2.1

The Conformal Group

We dene CFT by a list of operators and their Green functions which satisfy certain constraints described below. Succintly, a CFT is a theory with symmetries generated by conformal group . Obviously, it behooves us to dene conformal group which we do now. Please note that the route we follow is not a good one for non-relativistic CFTs.The power of here depends on the specic geometry of the D-brane in question, i.e. which of the dimensions of the bulk the D-brane is wrapping. 2 It is worth noting here that gauge theories in general also contain non-local operators such as Wilson loops: tr P eiRC

1

A

where C is some closed curve in the space on which the eld theory lives. It can be thought of as the phase acquired by a charged particle dragged along the specied path by an arbitrarily powerful external force. The connection between these operators and strings turns out to be quite direct, as one might expect from the relationship between charges on D-branes and the ends of open strings.

2

Recall that isometry group of a spacetime with coordinates x and metric g is the set of coordinate transformations which leave the metric ds2 = g dx dx unchanged. Conformal group corresponds to a bigger set of transformations which preserve the metric up to an overall (possibly positiondependent) rescaling: ds2 (x)ds2 . Thus Poincare group (which is the group of isometries of at spacetime) is a subgroup of the conformal group with (x) = 1. Conformal transformations could also be seen as the successive application of a coordinate transformation x x , g g which preserve ds2 = g dx dx followed by a Weyl rescaling which takes x x so that ds2 is not preserved. Conformal transformations preserve the angles between vectors, hence the name: v w cos( ) v v w w

cos() =

(2)

Consider the space-time IRp,q with a constant metric g = and lets look at the piece of innitesimal coordinate transformations connected to the identity:

g

x x + (x)

(3) (4)

( + )

Note that above there are no terms with derivative(s) of metric since the metric is constant. The requirement of conformal invariance implies that

( + ) = f (x) 2 = d

(5) (6)

2 where the equality f (x) = d is obtained by taking trace of on both sides of the eqn. 5. Applying an extra derivative on 5 gives

2 + (1 ) ( ) = 0 d

(7)

Pausing for a moment, we note that d = 2 is clearly special since = 0 implies that any holomorphic (antiholomorphic) function (z) (()) of complex coordinates z = x1 + ix2 would z correspond to a conformal transformation. A little more thought leads to the result that conformal algebra is innite dimensional in two dimensions. Applying one more partial derivative on this equation and applying on eqn. 5 and using the both equations thus obtained together gives (2 d) = 3

(8)

Contracting with implies = 0 = and thus (x) is at most a quadratic function of x. Lets organize (x) by its degree in x:

Degree zero: Translation : (x) = a Degree one: Dilatation : (x) = x Degree two: Special Conformal : (x) = b . x x xx b. (12) Rotation : (x) = x with =

(9)

(10) (11)

The corresponding generators for the above transformations can now be immediately written down:

Translation : P = i Rotation : M Dilatation : D = ix

(13) (14) (15) (16)

= i(x x )

Special Conformal : C = i( . 2x . ) xx x The commutation relations for the above generators are obtained as:

[C , M ] = i( C C ) [D, M ] = 0 [M , M ] = ( M + M M M ) [D, P ] = iP [D, C ] = iC

[P , M ] = i( P P )

(17) (18) (19) (20) (21) (22) (23)

[C , P ] = 2i( D M )

The rst four equations mean that P and C transform as vectors under Lorentz transformations while D is a scalar and M is a rank-2 tensor. The next two equations mean that P and C act as raising and lowering operator for the eigenvectors of dilation operator D. 4

Interestingly, recombining conformal generators in suitable way reveals the very important and remarkable result that the conformal group in IRp,q is isomorphic to SO(p + 1, q + 1)!. First we do simple counting to check that the number of generators match correctly. The total number of generators of conformal group in d(= p + q) dimensions are d (translation) +d(d + 1)/2 (rotation) +d (special conformal) +1 (dilatation) which sum up correctly to (d + 1)(d + 2)/2, the number of generators for SO(p + 1, q + 1). To see the exact correspondence, dene the following generators:

J J d J d+1 Jd+1 d

= M 1 (P C ) = 2 1 = (P + C ) 2 = D

(24) (25) (26) (27)

where {0, 1, ...d 1}. These new generators follow the SO(p + 1, q + 1) commutation relations: [Jab , Jcd ] = i(ad Jbc + bc Jad ac Jbd bd Jac ) (28)

where a, b, c, d {0, 1, ...d + 1}. Since the conformal algebra closes, it is possible to exponentiate the innitesimal transformations to obtain nite ones:

Translation : x x = x + a , Lorentz : x x

(x) = 1

(29) (30) (31) (x) = (1 + 2 x + . )2 b. b.b x x (32)

Dilatation : x x Special conformal : x x

(x) = 1 1 = x , (x) = 2 x + b . xx , = 1 + 2 x + . b. b.b x x

= x ,

The last transformation (special conformal) may not seem very easy to visualize. One simpler way to rewrite its nite form is x x = + b .x . xx x

(33)

That is, it is nothing but a inversion followed by a translation followed again by inversion. As an aside, an alternative denition of the conformal group is that it is the smallest group which contains both Poincare and inversion operations.

5

2.2

Representation Theory

In a general QFT one is dealing with a large set of elds, their derivatives and products and it is dicult to organize various elds/operators. However in a CFT i.e. in a conformally invariant QFT there is a systematic way to organize various operators. Namely, we rst diagonalize the dilatation operator D and label the various operators by the their eigenvalues when being acted upon by D. Note that D commutes with M and hence we can still specify representations of the Lorentz algebra. Considering a spinless eld for simplicity, under a dilatation (x) (x ) = (0) which implies (34)

[D, (0)] = i(0)

(35)

of special interest is a conformal primary operator transforms under conformal transformations as: /d x (x ) (x) x

(36)

where x is the Jacobian of the conformal transformation of the coordinates which is related to x the scale factor by x d/2 x =

(37)

One must note that not every eld transforms this way even if it has denite scaling dimension. It turns out that the unitary representations of conformal algebra always have a lowest whose value is determined by the dimension of the spin. Physically this makes sense since correlators (x)(0) of the elds are not expected to diverge as |x| . Readers familiar with stateoperator correspondence (to be discussed in the following lecture(s)) may note that this could also be seen as a consequence of the fact that the spectrum of any unitary theory is bounded from below. Since C acts as a lowering operator for the scaling dimension (eqn. 22) , it annihilates the state lowest . This operator of lowest can be seen to be conformal primary according to the denition above (the Callan article does this somewhat explicitly). Now the strategy for organizing the whole spectrum of operators might be obvious: we label the various representations of the CFT 6

by lowest and build the complete multiplet by acting with P which acts as raising operator for scaling dimension (eqn. 21) . As a nal remark for this lecture: please note that in general primary operators are not eigen functions of the Hamiltonian H since the dilatation operator doesnt commute with H. As a consequence p p which is the Casimir of the Poincare group is not Casimir of the full conformal group.

7

8.821 F2008 Lecture 10: CFTs in D > 2Lecturer: McGreevy October 11, 2008

In todays lecture well elaborate on what conformal invariance really means for a QFT. Remember, well keep working with CFTs in greater than 2 spacetime dimensions. A very good reference is Conformal Field Theory by Di Francesco et al, p95-107.

1

Conformal Transformations

To begin, well give a few dierent perspectives on what conformal invariance is and what conformal transformations are. Our rst perspective was emphasized by Brian Swingle after last lecture. Recall from last lecture that the group structure of the conformal group is the set of coordinate transformations: x x g (x ) = (x)g (x)

(1)

However, we should point out that coordinate transformations dont do anything at all in the sense that the metric is invariant, ds2 = ds2 , and all we did was relabel points. To say something useful about physics, we really want to compare a theory at dierent distances. For example, we would like to compare the correlators of physical observables separated at dierent distances. Conformal symmetry relates these kinds of observables! To implement a conformal transformation, we need to change the metric via a Weyl rescaling xx g (x)g (2)

which changes physical distances ds2 (x)ds2 . Now follow this by a coordinate transformation such as the one in Eq.(1), so that x x and (x)g g (x ). This takes us to a coordinate system where the metric has the same form as the one we started with, but the points have all been moved around and pushed closer together or farther apart. Therefore, we can view the conformal group as those coordinate transformations which can undo a Weyl rescaling. 1

1.1

Scale transformations

Another useful perspective on conformal invariance, particularly for AdS/CFT, is to think about scale transformations. Any local (relativistic) QFT has a stress-energy tensor T which is conserved if the theory is translation invariant. T encodes the linear response of the action to a small change in the metric: S = T g (3)

If we put in gravity, then T = 0 by the equations of motion. But lets return to QFT without gravity. Consider making a scale transformation, which changes the metric g = 2g S = T 2 (4) where is a constant. Therefore we see that if T = 0 then the theory is scale invariant.

In fact, since nothing we said above depended on the fact that is a constant, and since our theory is a local theory, we can actually make the following transformation: g = (x)g S = T (x) (5) We conclude from the two equations above that, at least classically,

If T = 0 the theory has both scale invariance and conformal invariance. If the theory is conformally invariant, it is also scale invariant, and T = 0

However scale invariance does not quite imply conformal invariance. For more details please see Polchinskis paper, Scale and conformal invariance in quantum eld theory. The conserved currents and charges of the transformations above are: S = x T C = (2x x x2 g )T since both S and C are proportional to T .

D

S0 dd x C0 dd x

(6) (7)

C

1.2

Geometric Interpretation

Lastly wed like to give an alternative geometric interpretation of the conformal group. The con formal group in Rd,1 is isomorphic to SO(d + 1, 2), the Lorentz group of a space with two extra dimensions. Suppose this space Rd+1,2 has metric ab = diag( + +... + )ab 2 (8)

where the last two dimensions are the extra ones. A light ray in this space can be parameterized by d + 1 dimensional coordinates x in the following way: 1 1 a = (x , (1 x2 ), (1 + x2 )) 2 2 (9)

where is some arbitrary constant. The group SO(d + 1, 2) moves these light rays around. We can interpet these transformations as maps on x , and in fact these transformations are precisely the conformal transformations, as you can check on your own. Invariants in Rd+1,2 should also be conformal invariants, for example 1 a b 1 2 = ab 1 1 = 1 2 (x1 x2 )2 . 2 (10)

This statement is not quite true because a and a are identied with the same x , so is a redundant variable. Conformal invariants actually are cross ratios of invariants in Rd+1,2 , for example 1 2 3 4 . (11) 1 3 2 4 This perspective was lifted from a paper by Callan. It may be useful if you are in the business of relating CFTs to theories in higher dimensional spacetimes, but you are otherwise free to ignore it.

2

Constraints on correlators in a CFT

We will now attempt to say something concrete about constraints on a QFT from conformal in variance, namely constraints on correlators. Consider a Greens function G = 1 (x1 )...n (xn ) (12)

where i is a conformal primary. Recall from last lecture that this means satises [D, (0)] = is the dilatation operator dened above. i(0), where is the weight and D How does (x) transform under D ? Writing the eld as (x) = eiP x (0)e+iP x , and using the following result eiP x De+iP x = D + x P (13)

which follows from the algebra, then [D, (x)] = i( + x )(x). The Greens function thus transforms under the action of D by: G =n (xi i (xi ) + 0|D i (xi )|0 + 0| i (xi )D |0. + i ) xi i=1

(14)

Assuming that the vacuum is conformally invariant, the last two terms above vanish. We will assume this in the rest of the discussion. N = 4 is a good example where there is a conformally invariant vacuum. 3

Now lets ignore what we just did with innitesimal transformations and look at the nite trans formations of the greens function: n n n x i /d /2 G = (15) i i i (x ) i (x ) i i x x=xii=1 i=1 i=1

where

2 i = (1 + 2b xi + b2 x2 )2 i

scale transformation special conformal

Conformal invariance tells us exactly how the Greens function must transform. Therefore, up to the i factors, the Greens function can only depend on conformal invariants I made from x . What i are the conformal invariants? Assuming s are scalars, lets constrain the conformal invariants by using the symmetries of the theory: Translation invariance implies I depends only on dierences: xi xj . Therefore there are d(n 1) numbers I can depend on. Rotational invariance implies I depends only on magnitudes of dierences: rij = |xi xj |. There are n(n 1)/2 of these numbers. Scale invariance implies I depends only on ratios of magnitudes rij /rkl Finally special conformal transformations transform rij by (r12 )2 = 2 r2 r12 = 1/212 1/2 . (1 + 2b x1 + b2 x2 )(1 + 2b x2 + b2 x2 ) 1 2 1 2 rij rkl rik rjl .

(16)

This result implies that only cross ratios are invariant:

There are n(n 3)/2 of these cross ratios, so only n(n 3)/2 conformal invariants.1 Note that this implies there are no conformal invariants for n < 4, namely the 2-pt and 3-pt function. This is obvious from the fact that you cannot form cross ratios with only 2 or 3 spatial vectors.

2.1

Two-point function

In fact lets look at the 2-pt function for scalar primaries. Its spatial dependence must be completely determined by the logic above: 1 (x1 )2 (x2 ) = c12 f (|r12 |) = c121

1 r121 +2

(17)

Ginsparg proves why there are n(n 3)/2 if you are curious.

4

where the rst equality follows from translation plus rotation invariance, and the second equality follows from scale transformations on both sides. Finally lets apply a special conformal transfor mation to both sides: 1 +2 1 /2 /2 1 1 2 2 1 (x )2 (x ) = c12 1 2 1/4 1/4 r12 1 2 1 (1 +2 )/4 (1 +2 )/4 = c12 2 (18) 1 +2 1 (r )12 However using Eq.(17), we already know what 1 (x )2 (x ) is. From this we conclude either 1 2 1 = 2 , OR c12 = 0 The special conformal transformation was necessary for this result. As an aside, there is an alternate shorter argument by Cardy which gives the same result without using the annoying formula for the special conformal transformation, though we still have to do one. First we note that the two-point function 1 (x1 )2 (x2 ) depends only on |x1 x2 |, so we can perform a 180 degree rotation to obtain 2 (x1 )1 (x2 ), which must give the same answer. Performing a conformal transformation to the two Greens functions above gives 11 /2

2

2 /2

1 (x1 )2 (x2 ) = 2

1 /2

1

2 /2

2 (x1 )1 (x2 )

(19)

So either G12 = 0 or 1 = 2 . Some nal comments on the two point function: for a set of elds i with i = j = i (x1 )j (x2 ) = dij 2 r12 (20)

where dij must be symmetric by the argument of Cardy. If we assume the operators are hermitian then dij is also real, and then we can diagonalize dij . In a unitary theory, the diagonal elements of dij are norms of states, and positive. By rescaling i , we can arrive at a basis where dij = ij .

2.2

Higher point functions

There are no confo