On the geometric origin of central charges in gravity

Abstract

Central extensions of asymptotic symmetry algebras are traditionally determined through an explicit computation of the algebra of conserved charges. We show that, within an ambiguity-free covariant phase space formalism based on the Noetherian split, they instead arise as intrinsic quantities completely determined by the Lagrangian through the geometry of the associated covariant phase space. We prove two general Charge Representation Theorems describing the canonical realization of diffeomorphism symmetries in the presence of field-dependent vector fields and Lagrangian anomalies, corresponding respectively to vanishing and integrable Noetherian flux. In both cases, we derive explicit expressions for the associated central extensions and establish the corresponding cocycle identities. This provides a constructive algorithm for determining central charges directly from the Lagrangian formulation of the theory, without requiring an explicit computation of the charge algebra. As a non-trivial application, we consider asymptotically locally AdS$_3$ spacetimes with Penrose--Brown--Henneaux boundary conditions. We recover the corresponding centrally extended asymptotic symmetry algebra, providing an explicit realization of the Charge Representation Theorems.