MIT OpenCourseWarehttp://ocw.mit.edu 8.821 String Theory Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.8.821 F2008 Lecture 11: CFT continued; geometry of AdS Lecturer: McGreevy 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 Conformal Symmetry 1.1 Weyl anomaly Quantumly, conformal symmetry in a curved space (with even number of dimen sions) could be anomalous, that is ds2 → Ω(x)ds2 could be no longer a symmetry of the f ull quantum theory. This anomaly can be evaluated from the following diagram with operator Tµµ inserted at the left vertex. Figure 1: A contribution to the Weyl anomaly. The conformal anomaly signals a n onzero value for the trace of the energy-momentum tensor. In a curved spacetime, it is related to the curvature: Tµ ∼ RD /2 µ 1� � where R denotes some scalar contractions of curvature tensors and D is the number of spacetime dimensions; the power is determined by d imen s ional analysis. For the special case of D = 2, this is Tµ = − c R(2) (1) µ 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 Tµµ = aW + cGB where W and B are defined as R....R − 2R..RW = (Weyl tensor)2 = .... .. +1 R2 (2) 3 R....R − 4R..RGB = Euler density = .... .. + R2; (3) The Gauss-Bonnet tensor is the ‘Euler density’ in the sense that GB = χ(M) = (−1)pbp(M) M where χ(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 4hT (z)T (0)i. (4) z→0 The Zamolodchikov c-theorem says that this qu antity is equal to the c d efi 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 should decrease un der RG flow, b ut there are now some counterexamples 1 . 1.2 OPE Any local QFT has an Operator Product Exansion (OPE). The idea is th at any local 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| < ǫ which is their distance from nearby operators. Then we can squint the local d isturbance by a superposition of local operators � nO1(x1)O2(x2) = c12On(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 C FT, we can plug (5) into a three-point function and we find that the OPE coefficients are determined to be n c n = c12 (6) 12(x − y)|x − y|Δ1+Δ2−Δn where c12 n are proportional to the 3-point interaction coefficient. Shapere and Tachikawa, 0809.3238 2 1� �1.3 Conformal Dimensions Let’s give a few examples of operators with definite conformal dimension: a) Energy-momentum tensor: The conformal dimension is Δ = D which is guaranteed by di-mensional analysis. The energy-momentum tensor must be coupled to gµν 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, th ere is a conserved curr ent jµ which has Δ = 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 ∂µjµ = 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 Δscalar operator ≥ = free field dimension 2 D − 1 Δspin 1/2 operaor ≥ 2 = free field dimension Δspin 1 ≥ D − 1 = conserved current dimension In the last equation, spin 1 is meant to be in (12, 12) rep resentation as opposed to (1, 0). 1.4 Thermodynamics of a CFT As a last r emark on CFT we discuss th e thermodynamics of a CFT. The p artition function is defined as ZCF T = TrCF T (exp(−H/T )) . In the thermod ynamic limit, ln Z is proportional to the volume of the space. ln Z is a dimensionless quantity. Hence, we must have ln Z ∼ V Td (d is the number of s patial dimensions) in the abs en ce of any other energy scales (such as a chemical potential for some conserved charge). The fr ee energy then will be F = −T ln Z = cV Td+1 . where this c is also somehow proportional to the number of degrees of f reedom of CFT. Exploiting simple facts about CFT we can derive some interesting results. We must regard Tµµ = 0 as an operator equation in the full quantum th eory. So we might be tempted to put it inside Tr(e−H/T). The operator equation then translates into the following equation 0 = Tr(Tµµe −H/T) = hT00i − hTiii = E − dP. This last relation gives the speed of the sound ��∂P 1 cs = = (7) ∂E S d 32 Sphere, Hyperboloid and AdS The AdS space has a constant negative curvature. I t is actually the most symmetrical space with a n egative curvature. Th e 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.