Effective Action, Boundary Conditions, and Virasoro Algebra for AdS$_3$

TL;DR

Addresses the origin of the Virasoro symmetry at the $AdS_3$ boundary and its central charge within a purely gravitational framework. It derives the second-order effective action for infinitesimal diffeomorphisms $h_{\mu\nu}=\mathcal{L}_\xi g_{\mu\nu}$ and enforces finiteness of boundary terms to identify admissible asymptotic transformations, reproducing two copies of the Virasoro algebra with $c=\frac{3l}{2G}$, even under relaxed boundary conditions. The work further constructs new asymptotically $AdS_3$ spacetimes with the same ASG and central charge and outlines extensions to Kerr/CFT and hidden conformal symmetry in Kerr. Altogether, the finite-action criterion provides a unified, a priori-free route to boundary conformal symmetry and holographic entropy counting via Cardy.

Abstract

We construct, to second order, the effective action of General Relativity for small excitations generated by a vector field and use it to study conformal symmetry in the boundary of AdS$_3$. By requiring finiteness of the boundary effective action(s) for certain asymptotic transformations, we derive the well-known Virasoro algebra and central charge associated with the boundary of AdS$_3$. The bulk action for these transformations can be arbitrarily small.