8 821 F2008 Lecture 11 CFT continued geometry of AdS Lecturer McGreevy Scribe Mohamad Magrebi October 17 2008 In 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 1 1 Conformal Symmetry Weyl anomaly Quantumly conformal symmetry in a curved space with even number of dimensions could be anomalous that is ds2 x ds2 could be no longer a symmetry of the full quantum theory This anomaly can be evaluated from the following diagram with operator T inserted 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 In a curved spacetime it is related to the curvature T RD 2 1 where R denotes some scalar contractions of curvature tensors and D is the number of spacetime dimensions the power is determined by dimensional analysis For the special case of D 2 this is c 2 R 1 12 where R 2 is the Ricci scalar in two dimensions and c is the central charge of the Virasoro algebra of the 2d CFT Also in D 4 the anomaly is given by T aW cGB where W and B are defined as 1 2 W Weyl tensor 2 R R 2R R R2 3 GB Euler density R R 4R R R2 3 T The Gauss Bonnet tensor is the Euler density in the sense that Z X GB M 1 p bp M M where M is the Euler character of M c and a are central charges which are proportional to the number of degrees of freedom of CFT In D 2 the central charge can be defined even away from critical theories as follows c lim z 4 hT z T 0 i z 0 4 The Zamolodchikov c theorem says that this quantity is equal to the c defined above when evaluated at a RG fixed point and is monotonically decreasing under RG flow In four dimensions a longstanding conjecture of Cardy suggests that a should decrease under RG flow but there are now some counterexamples 1 1 2 OPE Any local QFT has an Operator Product Exansion OPE The idea is that any local Q disturbance is created by local operators Consider some correlation function of local operators h Oi xi i focus on one local operator O1 x inserted at point x and there is another one O2 y at y Consider an imaginary line around them and suppose that x y which is their distance from nearby operators Then we can squint the local disturbance by a superposition of local operators X O1 x1 O2 x2 cn12 On y 5 n all operators with same quantum numbers The coefficients are independent of the other operators in the correlator The coefficients are in general hard to know In a CFT we can plug 5 into a three point function and we find that the OPE coefficients are determined to be cn12 x y cn12 x y 1 2 n where cn12 are proportional to the 3 point interaction coefficient 1 Shapere and Tachikawa 0809 3238 2 6 1 3 Conformal Dimensions Let s give a few examples of operators with definite conformal dimension a Energy momentum tensor The conformal dimension is D which is guaranteed by dimensional analysis The energy momentum tensor must be coupled to g by definition and the metric components are dimensionless There are other more algebraic arguments for this use the Conformal Ward identities to constrain the OPE of the stress tensor with itself b For a global symmetry there is a conserved current j which has D 1 The current can be coupled to a gauge field in which case these is a gauge invariance that guarantees the conservation of the current that is j 0 and fixes the dimension In general the conformal dimensions are very hard to know however there are some lower bounds on them in unitary CFTs The lower bound constrains the possible values of the conformal dimension according to their spins A few examples of this is as follows for a full analysis see hep th 9712074 D 2 free field dimension 2 D 1 free field dimension spin 1 2 operaor 2 spin 1 D 1 conserved current dimension scalar operator In the last equation spin 1 is meant to be in 21 12 representation as opposed to 1 0 1 4 Thermodynamics of a CFT As a last remark on CFT we discuss the thermodynamics of a CFT The partition function is defined as ZCF T TrCF T exp H T In the thermodynamic limit ln Z is proportional to the volume of the space ln Z is a dimensionless quantity Hence we must have ln Z V T d d is the number of spatial dimensions in the absence of any other energy scales such as a chemical potential for some conserved charge The free energy then will be F T ln Z cV T d 1 where this c is also somehow proportional to the number of degrees of freedom of CFT Exploiting simple facts about CFT we can derive some interesting results We must regard T 0 as an operator equation in the full quantum theory So we might be tempted to put it inside Tr e H T The operator equation then translates into the following equation 0 Tr T e H T hT00 i hTii i E dP This last relation gives the speed of the sound s r 1 P cs E S d 3 7 2 Sphere Hyperboloid and AdS The AdS space has a constant negative curvature It is actually the most symmetrical space with a negative curvature The most symmetric Euclidean space with a positive curvature is obviously a sphere A useful and immediate description of spheres and their metrics arises by embedding in a higher dimensional space Below we will use the same logic to investigate the AdS space but as a warmup we start with a sphere d 1 X x2i L2 IRn S d 8 i 1 P 2 with the flat metric ds2 d 1 i 1 dxi Note that the defining equation of the manifold respects the symmetries of the ambient space That is under the transformation xi i j xj for SO d 1 the manifold will be mapped to itself In other words the embedding is isometric We can solve equation 8 to find a set of global coordinates In two dimensions for example we find the familiar spherical coordinates x1 L cos cos x2 L cos sin x3 L sin The metric then would look like ds2S 2 L2 d 2 cos2 d 2 2 1 Euclidean AdS hyperbolic space Our next goal is to describe the Euclidean hyperbolic space in a higher dimensional space In three dimensions this is like the familiar hyperboloid defined as x2 y 2 z 2 R2 IR3 ds2 dx2 dy 2 dz 2 9 However it will more useful to embed …