$T\bar{T}$ deformation of chiral bosons and Chern-Simons AdS$_3$ gravity

We study the $T\bar{T}$ deformation of the chiral bosons and show the equivalence between the chiral bosons of opposite chiralities and the scalar fields at the Hamiltonian level under the deformation. We also derive the deformed Lagrangian of more generic theories which contain an arbitrary number of chiral bosons to all orders. By using these results, we derive the $T\bar{T}$ deformed boundary action of the AdS$_3$ gravity theory in the Chern-Simons formulation. We compute the deformed one-loop torus partition function, which satisfies the $T\bar{T}$ flow equation up to the one-loop order. Finally, we calculate the deformed stress tensor of a solution describing a BTZ black hole in the boundary theory, which coincides with the boundary stress tensor derived from the BTZ black hole with a finite cutoff.