A Path Integral Realization of Joint $J\bar T$, $T\bar J$ and $T\bar T$ Flows

TL;DR

This paper presents a path integral approach to understanding joint $Jar{T}$, $Tar{J}$, and $Tar{T}$ deformations, connecting them to topological gravity and gauge theory, and verifies the method through classical and quantum tests.

Contribution

It introduces a novel path integral kernel framework that geometrizes these deformations and reproduces known results for free theories and energy spectra.

Findings

Partition function satisfies a diffusion-like flow equation.

Classical deformed actions are recovered from the kernel.

Torus path integral localizes to a finite-dimensional integral.

Abstract

We recast the joint $J\bar{T}$, $T\bar{J}$ and $T\bar{T}$ deformations as coupling the original theory to a mixture of topological gravity and gauge theory. This geometrizes the general flow triggered by irrelevant deformations built out of conserved currents and the stress-energy tensor, by means of a path integral kernel. The partition function of the deformed theory satisfies a diffusion-like flow equation similar to that found in the pure $T\bar{T}$ case. Our proposal passes two stringent tests. Firstly, we recover the classical deformed actions from the kernel, reproducing the known expressions for the free boson and fermion. Secondly, we explicitly compute the torus path integral along the flow and show it localizes to a finite-dimensional, one-loop exact integral over base space torus moduli. The dressed energy levels so obtained match exactly onto those previously reported in the literature.