# $T \bar{T},J \bar T$, $T \bar{J}$ Partition Sums From String Theory

## TL;DR

This paper computes the torus partition sum of 2D conformal field theories deformed by specific irrelevant operators using string theory methods, ensuring modular invariance and matching known spectra.

## Contribution

It introduces a string theory-based integral transform approach to calculate partition sums of deformed 2D CFTs with conserved currents and chemical potentials.

## Key findings

- Derived a modular-invariant partition function expression.
- Reproduced the known spectrum of deformed models.
- Provided a formula for the partition function with chemical potentials.

## Abstract

We calculate the torus partition sum of a general $CFT_2$ with left and right moving conserved currents $J$ and $\bar J$, perturbed by a combination of the irrelevant operators $T\bar T$, $J\bar T$ and $T\bar J$. We use string theory techniques to write it as an integral transform of the partition sum of the unperturbed CFT with chemical potentials for the left and right moving conserved charges. The resulting expression transforms in the right way under the modular group, and reproduces the known spectrum of these models. We also derive a formula for the partition function of deformed $CFT_2$ with non-vanishing chemical potential.