On boundary conditions in three-dimensional AdS gravity

TL;DR

The paper introduces a finite action principle for three-dimensional AdS gravity by imposing a boundary condition on the extrinsic curvature and adding a boundary term equal to half the Gibbons-Hawking term. This yields an extremum of the action, finite quasilocal charges, and a finite Euclidean action without Dirichlet counterterms, with the boundary term decomposing into standard counterterms plus a topological invariant. The construction is shown to be compatible with the Dirichlet problem in AdS gravity and the Chern-Simons formulation, and it appears naturally extensible to higher odd dimensions. In 3D, the formalism recovers the correct BTZ thermodynamics and carries over to the CS framework, offering a geometrically motivated regularization that connects holographic renormalization, boundary dynamics, and topological invariants.

Abstract

A finite action principle for three-dimensional gravity with negative cosmological constant, based on a boundary condition for the asymptotic extrinsic curvature, is considered. The bulk action appears naturally supplemented by a boundary term that is one half the Gibbons-Hawking term, that makes the Euclidean action and the Noether charges finite without additional Dirichlet counterterms. The consistency of this boundary condition with the Dirichlet problem in AdS gravity and the Chern-Simons formulation in three dimensions, and its suitability for the higher odd-dimensional case, are also discussed.