Path integral derivation of the Brown-Henneaux central charge - art. no. 064016

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