Global Charges in Chern-Simons theory and the 2+1 black hole

TL;DR

The paper develops a Regge-Teitelboim framework to define global charges in Chern-Simons theory on manifolds with boundaries, showing how boundary terms generate affine and Virasoro algebras as the physical charges and identifying conditions under which a classical central charge arises in the Virasoro algebra. It clarifies how gauge fixing leaves boundary degrees of freedom that realize the residual symmetry as a central-extended algebra, with an explicit Sugawara-type relation linking affine currents to Virasoro generators. The authors apply this to $SO(2,2)$ (AdS$_{2+1}$) gravity, demonstrating two commuting Virasoro algebras with central charges arising from boundary conditions, and they relate the BTZ black hole mass and angular momentum to zero modes of these algebras. They also address the quantization of the resulting algebras, giving explicit expressions for the quantum Virasoro generators, central charges, and energy bounds, thereby connecting boundary dynamics, black hole thermodynamics, and possible microscopic state counting via boundary degrees of freedom.

Abstract

We use the Regge-Teitelboim method to treat surface integrals in gauge theories to find global charges in Chern-Simons theory. We derive the affine and Virasoro generators as global charges associated with symmetries of the boundary. The role of boundary conditions is clarified. We prove that for diffeomorphisms that do not preserve the boundary there is a classical contribution to the central charge in the Virasoro algebra. The example of anti-de Sitter 2+1 gravity is considered in detail.