General solutions in Chern-Simons gravity and $T \bar T$-deformations

TL;DR

This paper derives the most general solutions in Chern-Simons AdS3 gravity with curved boundaries, establishes a variational principle for boundary conditions, and connects these to $T ar T$-deformations.

Contribution

It provides the first comprehensive solution framework for Chern-Simons AdS3 gravity with non-trivial boundaries and links boundary conditions to $T ar T$-deformations.

Findings

Derived the most general solutions in Chern-Simons AdS3 gravity.

Established a variational principle for Dirichlet boundary conditions.

Connected boundary conditions to $T ar T$-deformations.

Abstract

We find the most general solution to Chern-Simons AdS$_3$ gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $T \bar T$-deformation.