Thermal partition function of \( {J}_3{\overline{J}}_3 \) deformed AdS₃

We derive a compact formula for the one-loop, bosonic string partition function of Euclideanized $J_3{\overline{J}}_3$ \( {J}_3{\overline{J}}_3 \) deformed AdS₃ with periodic Euclidean time as an integral transform of the partition function of the undeformed Euclideanized AdS₃. Such a deformation is interpretable as an irrelevant “single-trace $T\overline{T}$ \( T\overline{T} \) 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 $J\overline{J}$ \( J\overline{J} \) deformation of a sigma model with U(1)_L × 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.