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

TL;DR

This work develops a dimension-spanning framework unifying 4D duality-invariant nonlinear electrodynamics with 2D integrable sigma models through a Courant–Hilbert generating function and a new auxiliary-field formalism. Introducing two deformation parameters, $λ$ (irrelevant) and $γ$ (marginal), it identifies a universal root-$Tar{T}$ flow that preserves both duality and integrability across a broad class of theories. It systematically constructs generalized Born–Infeld, logarithmic, and $q$-deformed models, as well as a novel closed-form integrable theory, with explicit Lagrangians and perturbative expansions. The framework provides a unified, dimension-agnostic mechanism for incorporating marginal and irrelevant deformations while maintaining exact solvability, and it extends to new integrable structures via auxiliary-field dynamics. Collectively, the results reveal a deep geometric-principled link between 4D self-dual electrodynamics and 2D integrable sigma models, governed by a common deformation paradigm.

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, $λ$ (irrelevant) and $γ$ (marginal), we identify a universal class of $γ$-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 ($λ$,$γ$) while preserving the root-$T\bar{T}$ flow condition for all $γ$-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.