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

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. Starting from the Banados geometry, we obtained the $T\bar{T}$ deformed Chern-Simons gauge fields which are parametrized by two classes of independent charges. With help of the mixed boundary condition, the residual gauge symmetries of deformed gauge fields and the associated surface charges can be obtained respectively. These surface charges satisfy a non-linear deformed Virasoro algebra, which is obtained differently in the literature. Finally, the time-independent charges can be systematically constructed from these surface charges and they satisfy the field-dependent Virasoro algebra.