  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.
