Root-$T\bar{T}$ Flows Unify 4D Duality-Invariant Electrodynamics and 2D Integrable Sigma Models

TL;DR

This paper introduces a unified framework linking 4D duality-invariant electrodynamics and 2D integrable sigma models through root-$Tar{T}$ flows, encompassing new models and deformations.

Contribution

It develops a novel two-parameter deformation approach that unifies 4D and 2D theories via common generating functions and potentials, extending the root-$Tar{T}$ flow to a broader class of models.

Findings

Constructed generalized Born-Infeld and q-deformed models.

Identified a universal class of $oldsymbol{\gamma}$-flows including root-$Tar{T}$.

Established a systematic connection between 4D electrodynamics and 2D integrable systems.

Abstract

We present a unified framework that connects four-dimensional duality-invariant nonlinear electrodynamics and two-dimensional integrable sigma models via the Courant-Hilbert and new auxiliary field formulations, both governed by a common generating function and a generating potential, respectively. Introducing two commuting deformation parameters, $\lambda$ (irrelevant) and $\gamma$ (marginal), we identify a universal class of $\gamma$-flows, including the root-$T\bar{T}$ deformation and its rescaled variants. Our approach generalizes conventional single-coupling structures via novel field transformations that extend to a two-parameter space ($\lambda$,$\gamma$) while preserving the root-$T\bar{T}$ flow condition for all $\gamma$-coupled theories. We construct several integrable models, including generalized Born-Infeld, logarithmic, q-deformed, and a new closed-form theory applicable to both electrodynamics and integrable systems. This unified framework, based on the unique form of the root-$T\bar{T}$ flow, systematically spans duality-invariant nonlinear electrodynamics in 4D and their exact 2D integrable counterparts.