$T\bar{T}$-flow effects on torus partition functions

TL;DR

This paper studies how $Tar{T}$ deformations affect the partition functions of 2D conformal field theories on a torus using perturbative quantum field theory, confirming consistency with previous Hamiltonian results.

Contribution

It provides explicit perturbative calculations of $Tar{T}$-deformed partition functions for free theories and discusses associated counterterms, extending understanding of these deformations.

Findings

First- and second-order deformed partition functions derived

Results consistent with Hamiltonian formalism in literature

First-order quantum KdV charge VEV obtained

Abstract

In this paper, we investigate the partition functions of conformal field theories (CFTs) with the $T\bar{T}$ deformation on a torus in terms of the perturbative QFT approach. In Lagrangian path integral formalism, the first- and second-order deformations to the partition functions of 2D free bosons, free Dirac fermions, and free Majorana fermions on a torus are obtained. The corresponding Lagrangian counterterms in these theories are also discussed. The first two orders of the deformed partition functions and the first-order vacuum expectation value (VEV) of the first quantum KdV charge obtained by the perturbative QFT approach are consistent with results obtained by the Hamiltonian formalism in literature.