8.821 F2008 Lecture 11: CFT continued; geometry of AdSLecturer: McGreevy Scribe: Mohamad MagrebiOctober 17, 2008In this session, we are going to talk about the following topics.1. We are making a few comments about CFT.2. We are discussing spheres and hyperboloids.3. Finally we are focusing on Lorentzian AdS and its boundary.1 Conformal Symmetry1.1 Weyl anomalyQuantumly, conformal symmetry in a curved space (with even number of dimensions) could beanomalous, that is ds2→ Ω(x)ds2could be no longer a s ymmetry of the full quantum theory. Thisanomaly can be evaluated from the following diagram with operator Tµµinserted at the left vertex.Figure 1: A contribution to the Weyl anomaly.The conformal anomaly signals a nonzero value for the trace of the energy-momentum tensor. Ina curved spacetime, it is related to the curvature:Tµµ∼ RD /21where R denotes some scalar contractions of curvature tensors and D is the number of spacetimedimensions; the power is determined by dimensional analysis.For the special case of D = 2, th is isTµµ= −c12R(2)(1)where R(2)is the Ricci scalar in two dimensions and c is the central charge of the Virasoro algebraof the 2d CFT. Also in D = 4, th e anomaly is given by Tµµ= aW + cGB where W and B aredefined asW = (Weyl tensor)2= R....R....− 2R..R..+13R2(2)GB = Euler density = R....R....− 4R..R..+ R2; (3)The Gauss-Bonnet tensor is the ‘Euler density’ in the sense thatZMGB = χ(M) =X(−1)pbp(M)where χ(M) is the Euler character of M. c and a are “central charges” which are proportional tothe number of degrees of freedom of CFT.In D = 2 the central charge can be defined even away from critical theories as follows:c = limz→0z4hT (z)T (0)i. (4)The Zamolodchikov c-theorem says that this quantity is equal to the c defi ned above when eval-uated at a RG fixed point, and is monotonically decreasing under RG flow. In four dimensions,a longstanding conjecture of Cardy suggests that a s hould decrease under RG flow, but there arenow some counterexamples1.1.2 OPEAny local QFT has an Operator Product Exansion (OPE). The idea is that any local d isturbance iscreated by local operators. Cons ider some correlation function of lo cal operators hQOi(xi)i; focuson one local operator O1(x) inserted at point x and there is another one O2(y) at y. Consideran imaginary line around them and suppose that |x − y| < ǫ which is their distance from nearbyoperators. Then we can squint the local disturbance by a superposition of local operatorsO1(x1)O2(x2) =Xn, all operators with same quantum numberscn12On(y) (5)The coefficients are independent of the other operators in the correlator. The coefficients are ingeneral hard to kn ow. In a CFT, we can plug (5) into a three-point function and we find that theOPE coefficients are determined to becn12(x − y) =cn12|x − y|∆1+∆2−∆n(6)where cn12are proportional to the 3-point interaction coefficient.1Shapere and Tachikawa, 0809.323821.3 Conformal DimensionsLet’s give a few examples of operators with definite conformal dimension:a) Energy-momentum tensor: The conformal dimension is ∆ = D which is guaranteed by di-mensional analysis. The energy-momentu m tensor must be coupled to gµνby definition, and th emetric components are dimensionless. There are other m ore algebraic arguments for this (use theConformal Ward identities to constrain the OPE of the stress tensor with itself).b) For a global s ymmetry, there is a conserved current jµwhich has ∆ = D − 1. The current can becoupled to a gauge fi eld in which case these is a gauge invariance that guarantees the conservationof the current, that is ∂µjµ= 0, and fixes the dimension.In general, the conformal dimensions are very hard to know, however there are some lower bounds onthem in unitary CFTs. The lower bound constrains the possible values of the conformal dimensionaccording to their spin s. A few examples of this is as follows (for a full analysis see hep-th/9712074)∆scalar operator≥D − 22= free field dimension∆spin 1/2 operaor≥D − 12= free field dimension∆spin 1≥ D − 1 = conserved current dimens ionIn the last equation, spin 1 is meant to be in (12,12) representation as opposed to (1, 0).1.4 Thermodynamics of a CFTAs a last r emark on CFT we discuss the thermodynamics of a CFT. The partition function isdefined asZCF T= TrCF T(exp(−H/T )) .In the therm odynamic limit, ln Z is p roportional to the volume of the space. ln Z is a dimensionlessquantity. Hence, we must have ln Z ∼ V Td(d is the number of spatial dimen sions) in the absenceof any other energy scales (such as a chemical potential for s ome conserved charge). Th e free energythen will beF = −T ln Z = cV Td+1.where this c is also somehow proportional to the number of degrees of freedom of CFT.Exploiting simple f acts about CFT we can derive some interesting results. We must regard Tµµ= 0as an operator equation in the full quantum theory. So we might be tempted to put it insideTr(e−H/T). The operator equation then translates into the following equation0 = Tr(Tµµe−H/T) = hT00i − hTiii = E − dP.This last relation gives the speed of th e soundcs=s∂P∂ES=r1d(7)32 Sphere, Hyperboloid and AdSThe AdS space has a constant negative curvature. It is actually the most symmetrical space witha negative curvature. The most symmetric (Euclidean) space with a positive curvature is obviouslya sphere. A useful and immediate description of s pheres and their metrics arises by embedding ina higher dimensional space. Below we will use the same logic to investigate the AdS space, but asa warmup we start with a sphereSd= {d+1Xi=1x2i= L2} ⊂ IRn(8)with the flat metric ds2=Pd+1i=1dx2i. Note that the defining equation of the manifold respects thesymmetries of the ambient space. That is, under the transformation xi→ Λjixjfor Λ ⊂ SO(d+1),the manifold will be mapped to itself. In other words, th e embedding is “isometric”.We can solve equation (8) to find a set of global coordinates. In two dimensions, f or example, wefind the familiar sph erical coordinatesx1= L cos θ cos φ,x2= L cos θ sin φ,x3= L sin θThe metric then would look likeds2S2= L2(dθ2+ cos2θdφ2).2.1 Euclidean AdS = hyperbolic spaceOur next goal is to describe the (Euclidean) hyperbolic space in a higher dimensional space. Inthree dimensions this is like the familiar hyperboloid defined as{x2− y2− z2= R2} ⊂ IR3, ds2= dx2+ dy2+ dz2(9)However, it will m ore useful to embed hyberbolic