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

TL;DR

This paper explores classical aspects of the $ extrm{T} ar{ extrm{T}}$ deformation in 2D quantum field theories, constructing Lax pairs for integrable models and relating 4D Born-Infeld theory to these deformations, with implications for geometrical interpretation.

Contribution

It constructs the $ extrm{T} ar{ extrm{T}}$-deformed Lax pair for the sine-Gordon model and relates 4D Maxwell-Born-Infeld theory to 2D deformations, advancing understanding of integrable deformations.

Findings

Constructed the $ extrm{T} ar{ extrm{T}}$-deformed Lax pair for sine-Gordon model.

Linked 4D Born-Infeld theory to 2D $ extrm{T} ar{ extrm{T}}$ deformation.

Proposed a modified heat kernel to incorporate $ extrm{T} ar{ extrm{T}}$ effects in 2D Yang-Mills.

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.