A general definition of $JT_a$ -- deformed QFTs

TL;DR

This work provides a non-perturbative path-integral framework for JT_a deformations of two-dimensional QFTs by coupling the original theory to a flat U(1) gauge field and half of a flat dynamical vielbein. The construction yields a simple geometric flow equation for the torus partition function and a precise energy-level flow that matches known results, establishing UV completeness and intrinsic non-locality of the deformed theories. It further shows that the deformed S-matrix is universally dressed by a phase dependent on particle charges and momenta, and in integrable cases this phase reproduces the finite-size spectrum shift via the Thermodynamic Bethe Ansatz. These results reinforce a coherent picture linking the path-integral definition, spectral data, and S-matrix structure of JT_a-deformed QFTs, and point to avenues for studying correlation functions and broader generalizations.

Abstract

We propose a general path-integral definition of two-dimensional quantum field theories deformed by an integrable, irrelevant vector operator constructed from the components of the stress tensor and those of a $U(1)$ current. The deformed theory is obtained by coupling the original QFT to a flat dynamical gauge field and "half" a flat dynamical vielbein. The resulting partition function is shown to satisfy a geometric flow equation, which perfectly reproduces the flow equations for the deformed energy levels that were previously derived in the literature. The S-matrix of the deformed QFT differs from the original S-matrix only by an overall phase factor that depends on the charges and momenta of the external particles, thus supporting the conjecture that such QFTs are UV complete, although intrinsically non-local. For the special case of an integrable QFT, we check that this phase factor precisely reproduces the change in the finite-size spectrum via the Thermodynamic Bethe Ansatz equations.