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

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