T T¯-flow effects on torus partition functions

In this paper, we investigate the partition functions of the conformal field theories (CFTs) with the $T\bar{T}$ deformation on torus in terms of perturbative QFT approach. In path integral formalism, the first and second order deformations to the partition functions of the 2D free boson, free Dirac fermion and free Majornana fermion on torus are obtained. In the free fermion, we find that the first two orders of the deformed partition functions are consistent with results obtained by the operator formalism. In the free boson, the first order of the deformation to the partition function is the same as the deformed partition function given by the operator formalism, however, the second order correction to the partition function contains additional contribution from the $T\bar{T}-$flow effect in the path integral formalism