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 a _UV − a _IR. Then, we consider a specific 3D example of RG flow whose UV asymptotics is normalizable and admits a 6D lifting. We promote a modulus ρ appearing in the geometry to a function of boundary coordinates. In a 6D description ρ is the scale of an SU(2) instanton. We determine the smooth deformed background up to second order in the space-time derivatives of ρ and find that the 3D on-shell action reproduces a boundary kinetic term for the massless field τ = log ρ with the correct coefficient δc = c _UV − c _IR. We further analyze the linearized fluctuations around the deformed background geometry and compute the one-point functions < T _μν > and show that they are reproduced by a Liouville-type action for the massless scalar τ, with background charge due to the coupling to the 2D curvature R ⁽²⁾. The resulting central charge matches δc. We give an interpretation of this action in terms of the (4, 0) SCFT of the D1-D5 system in type I theory.