$T\bar T$-deformations in closed form

TL;DR

The paper develops a general method to integrate TT̄ deformations exactly in two dimensions, recasting the flow as a local PDE for the Lagrangian that often reduces to the Burgers equation, and demonstrates closed-form deformations for a variety of 2D theories including free and interacting scalars, sigma-models, WZW, and massive Thirring. It then extends the framework to higher dimensions via Cardy’s det T deformations, obtaining explicit solutions in several cases and highlighting the continued utility of Burgers-type methods. The results provide concrete, computable deformations across multiple models, offer insights into curvature couplings and higher-spin generalisations, and identify open questions that motivate further work in holography and beyond. Overall, the work broadens the class of theories amenable to exact TT̄-type deformations and clarifies the structure of these flows in both two and higher dimensions.

Abstract

We consider the problem of exact integration of the $T\bar{T}$-deformation of two dimensional quantum field theories, as well as some higher dimensional extensions in the form of $\det T$-deformations. When the action can be shown to only depend algebraically on the background metric the solution of the deformation equation on the Lagrangian can be given in closed form in terms of solutions of the (extended) Burgers' equation. We present such examples in two and higher dimensions.