Entropy of the 3-brane and the AdS/CFT duality from a two-dimensional perspective

Abstract

A two-dimensional (2D) scalar–tensor gravity theory is used to describe the near-horizon, near-extremal behaviour of black 3-brane solutions of type IIB string theory. The asymptotic symmetry group of the 2D, asymptotically anti de Sitter (AdS) metric, is generated by a Virasoro algebra. The (non-constant) configuration of the 2D scalar field, which parametrizes the volume of the 3-brane, breaks the conformal symmetry and produces a divergent central charge in the Virasoro algebra. Using a renormalization procedure, we find a finite value for the central charge and, by means of the Cardy formula, the entropy of the black 3-brane in terms of microstates of the conformal field theory living on the boundary of the 2D spacetime. We find for the entropy as a function of the temperature the power-law behaviour S ∝ T3. Unfortunately, owing to a scale symmetry of the 2D model the proportionality constant is undetermined. The black 3-brane case is a nice example of how finite temperature effects in higher dimensional AdS/CFT dualities can be described by a AdS2/CFT1 duality endowed with a scalar field that breaks the conformal symmetry and produces a non-vanishing central charge.