Path integral derivation of the Brown-Henneaux central charge

TL;DR

This paper derives the Brown-Henneaux central charge using path integral methods, clarifying its origin from boundary conditions or transformation laws, and discusses implications for black hole entropy.

Contribution

It provides a novel path integral derivation of the Brown-Henneaux central charge, distinguishing it from quantum anomalies and analyzing its relation to boundary conditions.

Findings

Central charge arises from boundary condition non-invariance or transformation law.

Distinction made between classical boundary effects and quantum anomalies.

Implications discussed for black hole entropy calculations.

Abstract

We rederive the Brown-Henneaux commutation relation and central charge in the framework of the path integral. To obtain the Ward-Takahashi identity, we can use either the asymptotic symmetry or its leading part. If we use the asymptotic symmetry, the central charge arises from the transformation law of the charge itself. Thus, this central charge is clearly different from the quantum anomaly which can be understood as the Jacobian factor of the path integral measure. Alternatively, if we use the leading transformation, the central charge arises from the fact that the boundary condition of the path integral is not invariant under the transformation. This is in contrast to the usual quantum central charge which arises from the fact that the measure of the path integral is not invariant under the relevant transformation. Moreover, we discuss the implications of our analysis in relation to the black hole entropy.