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

TL;DR

This work recasts the joint J̄T, T̄J, and T̄T deformations as coupling a seed 2D QFT to a topological gravity/gauge sector, yielding a path-integral kernel that maps base-space data to target-space backgrounds. The kernel, built with compensator fields, enforces flatness constraints and renders the deformations solvable by reducing the quantum partition function on the torus to a finite-dimensional, one-loop exact integral, whose saddle-point equations reproduce the known spectra. Classical checks recover the deformed actions for free bosons and fermions, while quantum checks confirm diffusion-type flow equations and exact spectra matching prior results in the literature across several deformation combinations. The framework illuminates the geometrical origin of the flows, clarifies the role of target vs. base space diffeomorphisms, and points toward extensions to curved backgrounds and correlation functions, with potential holographic implications. Overall, the paper provides a unifying, topological-geometry perspective on solvable irrelevant deformations in two dimensions, with concrete computational handles via torus partition functions and energy spectra.

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.