Relating auxiliary field formulations of $4d$ duality-invariant and $2d$ integrable field theories

TL;DR

This paper explores the connections between auxiliary field formulations in 4D duality-invariant electrodynamics and 2D integrable sigma models, revealing how Legendre transformations relate different models and preserve key properties like duality and integrability.

Contribution

It clarifies the relations between various auxiliary field formulations in 2D and 4D, establishing new correspondences and extending integrable deformation families using Legendre transformations and field redefinitions.

Findings

Established a correspondence between Russo-Townsend and Ivanov-Zupnik formalisms in 4D.

Developed the $$-frame for 2D sigma models to generate new integrable deformations.

Showed how auxiliary field deformations relate to $Tar{T}$-like deformations and higher-spin generalizations.

Abstract

Auxiliary field techniques have recently gained interest in four-dimensional non-linear electrodynamics and two-dimensional integrable sigma models. In these settings, coupling a suitable ``seed'' theory to auxiliary fields provides a powerful mechanism to generate infinite families of models while preserving key dynamical properties, such as electromagnetic duality invariance in four dimensions and classical integrability in two dimensions. Deformations induced through auxiliary fields are closely related to $T\bar{T}$-like deformations and, in two dimensions, also to their higher-spin generalisations. In this paper, we analyse and clarify the relations between different auxiliary field formulations in two and four dimensions, showing how they are governed by Legendre transformations of the interaction functions combined with appropriate field redefinitions. In four-dimensional electrodynamics, we establish a correspondence between the auxiliary field model of Russo and Townsend and the Ivanov--Zupnik formalism. In two dimensions, we develop the analogue of the Ivanov--Zupnik $\mu$-frame to deform Principal Chiral, symmetric-space, non-Abelian T-dual, and (bi-)Yang-Baxter sigma models. We discuss how integrability is preserved and use properties of the $\mu$-frame to further extend known families of integrable deformations.