Surface charges in Chern-Simons gravity with $T\bar{T}$ deformation

TL;DR

This paper investigates the surface charges and their algebra in $Tar{T}$ deformed AdS$_3$ gravity using Chern-Simons formalism, revealing a non-linear deformed Virasoro algebra structure.

Contribution

It constructs surface charges and their algebra in $Tar{T}$ deformed theories via Chern-Simons formalism, providing a new perspective on boundary symmetries.

Findings

Surface charges form a non-linear deformed Virasoro algebra.

Residual gauge symmetries are characterized by two independent charges.

A method to construct time-independent charges satisfying a field-dependent Virasoro algebra.

Abstract

The $T\bar{T}$ deformed 2D CFTs correspond to AdS$_3$ gravity with Dirichlet boundary condition at finite cutoff or equivalently a mixed boundary condition at spatial infinity. In this work, we use the latter perspective and Chern-Simons formalism of AdS$_3$ gravity to construct the surface charges and associated algebra in $T\bar{T}$ deformed theories. Starting from the Ba\~nados geometry, we obtain the Chern-Simons gauge fields for the $T\bar{T}$ deformed geometry, which are parametrized by two independent charges. With help of the mixed boundary condition, the residual gauge symmetries of the deformed gauge fields and the associated surface charges were obtained respectively. The charge algebra turns out to be a non-linear deformed Virasoro algebra, which was obtained in different way by applying the cutoff perspective. Finally, we propose a way to construct the time-independent charges from these surface charges and they satisfy the field-dependent Virasoro algebra.