Holography and Conformal Anomaly Matching

Abstract

We discuss various issues related to the understanding of the conformal anomaly matching in CFT from the dual holographic viewpoint. First, we act with a PBH diffeomorphism on a generic 5D RG flow geometry and show that the corresponding on-shell bulk action reproduces the Wess-Zumino term for the dilaton of broken conformal symmetry, with the expected coefficient aUV-aIR. Then we consider a specific 3D example of RG flow whose UV asymptotics is normalizable and admits a 6D lifting. We promote a modulus \rho appearing in the geometry to a function of boundary coordinates. In a 6D description {\rho} is the scale of an SU(2) instanton. We determine the smooth deformed background up to second order in the space-time derivatives of \rho and find that the 3D on-shell action reproduces a boundary kinetic term for the massless field \tau= log(\rho) with the correct coefficient \delta c=cUV-cIR. We further analyze the linearized fluctuations around the deformed background geometry and compute the one-point functions <T\mu\nu> and show that they are reproduced by a Liouville-type action for the massless scalar \tau, with background charge due to the coupling to the 2D curvature R. The resulting central charge matches \delta c. We give an interpretation of this action in terms of the (4,0) SCFT of the D1-D5 system in type I theory.