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

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