Brown-Henneaux's Canonical Approach to Topologically Massive Gravity

TL;DR

Addresses the asymptotic symmetry of AdS$_3$ gravity with higher derivatives including the gravitational Chern-Simons term, using the canonical formalism to derive Virasoro algebras with left-right asymmetric central charges. Introduces a conformal factor $\Omega$ and CS coupling $\beta$ to capture all higher-derivative corrections, yielding central charges $c_L=\frac{3\ell}{2G_N}(\Omega+\frac{\beta}{\ell})$ and $c_R=\frac{3\ell}{2G_N}(\Omega-\frac{\beta}{\ell})$, and BTZ mass/angular momentum $(M,J)=(\Omega m+\frac{\beta}{\ell^2}j,\; \Omega j+\beta m)$. Demonstrates that Cardy entropy matches the macroscopic entropy for BTZ in the presence of higher-derivative corrections, and connects these results to M-theory via M5-brane CY$_3$ compactifications, reinforcing the AdS$_3$/CFT$_2$ correspondence in more general gravity theories.

Abstract

We analyze the symmetry realized asymptotically on the two dimensional boundary of AdS_3 geometry in topologically massive gravity, which consists of the gravitational Chern-Simons term as well as the usual Einstein-Hilbert and negative cosmological constant terms. Our analysis is based on the conventional canonical method and proceeds along the line completely parallel to the original Brown and Henneaux's. In spite of the presence of the gravitational Chern-Simons term, it is confirmed by the canonical method that the boundary theory actually has the conformal symmetry satisfying the left and right moving Virasoro algebras. The central charges of the Virasoro algebras are computed explicitly and are shown to be left-right asymmetric due to the gravitational Chern-Simons term. It is also argued that the Cardy's formula for the BTZ black hole entropy capturing all higher derivative corrections agrees with the extended version of the Wald's entropy formula. The M5-brane system is illustrated as an application of the present calculation.