Generalised Born-Infeld models, Lax operators and the $\textrm{T} \bar{\textrm{T}}$ perturbation

TL;DR

The paper investigates classical aspects of the $T\bar{T}$ deformation in two-dimensional field theories, showing that deformed bosonic Lagrangians adopt Born-Infeld–type forms and that the deformation is governed by a Burgers-type flow equation for the energy spectrum. It proves that the $T\bar{T}$–deformed sine-Gordon model remains classically integrable by constructing a Lax pair with a single spectral parameter and derives explicit kink-like solutions. It further extends the TTbar framework to four-dimensional Maxwell–Born–Infeld electrodynamics, interpreting the deformation as driven by $\sqrt{\det[T^{\text{MBI}}]}$ and providing the corresponding Lagrangian and Hamiltonian structures, including interacting potentials. Finally, it discusses TTbar deformations of two-dimensional Yang–Mills via a dressed heat kernel, establishing consistency with diffusion-type relations and highlighting the geometric interpretation of TTbar in gauge theories, with implications for AdS/CFT connections and higher-dimensional generalizations.

Abstract

Surprising links between the deformation of 2D quantum field theories induced by the composite $\textrm{T} \bar{\textrm{T}}$ operator, effective string models and the $AdS/$CFT correspondence, have recently emerged. The purpose of this article is to discuss various classical aspects related to the deformation of 2D interacting field theories. Special attention is given to the sin(h)-Gordon model, for which we were able to construct the $\textrm{T} \bar{\textrm{T}}$-deformed Lax pair. We consider the Lax pair formulation to be the first essential step toward a more satisfactory geometrical interpretation of this deformation within the integrable model framework. Furthermore, it is shown that the 4D Maxwell-Born-Infeld theory, possibly with the addition of a mass term or a derivative-independent potential, corresponds to a natural extension of the 2D examples. Finally, we briefly comment on 2D Yang-Mills theory and propose a modification of the heat kernel, for a generic surface with genus $p$ and $n$ boundaries, which fully accounts for the $\textrm{T} \bar{\textrm{T}}$ contribution.

Figures (2)

• Figure 1: The $\textsc{T} \bar{\textsc{T}}$-deformed stationary kink solution (\ref{['eq:kinksol']}) ($\alpha=1,\beta=0$) for different values of the perturbation parameter $\tau$. The critical value $\tau=1/8$ (c) corresponds to a shock wave singularity.
• Figure 2: The general solution (\ref{['eq:elsolu']}) for the undeformed (a) and the deformed (b) theory, for small values of $\kappa$.