Thermal partition function of $J_3{\bar{J}}_3$ deformed $Ad{S}_3$

We derive a compact formula for the one-loop, bosonic string partition function of Euclideanized $J_3{\bar{J}}_3$ deformed $Ad{S}_3$ with periodic Euclidean time as an integral transform of the partition function of the undeformed Euclideanized $Ad{S}_3$. Such a deformation is interpretable as an irrelevant “single-trace $T\bar{T}$ deformation” of the boundary. We will do this by first establishing a formal procedure to compute a worldsheet torus zero point function for an exactly marginal $J\bar{J}$ deformation of a sigma model with $U(1{)}_L\times U(1{)}_R$ global symmetry. We then describe how this procedure is implemented on $SL(2,R)$ sigma model and its Euclidean continuation. Finally, we describe the embedding of the deformed $SL(2,R)$ torus amplitude into critical string theory and interpret the result as the leading perturbative contribution to the thermal partition function of the deformed theory.