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

TL;DR

The paper develops a unified view of auxiliary-field formulations linking 4d duality-invariant nonlinear electrodynamics and 2d integrable sigma models. It shows that the ν-frame IZ description, the μ-frame IZ reformulation, and the RT scalar-auxiliary approach are related by Legendre transforms, with a precise mapping H(β) and Ω(y) linking the μ-frame to RT and CH data. In 2d, the μ-frame provides a simplification where a single scalar auxiliary field encodes integrable deformations (AFSM) via Lax connections and a Maillet Poisson structure; extending to a two-parameter $(oldsymbol{ extmu},oldsymbol{ ho})$-frame reveals both the potential and limits of maintaining integrability. Extending the μ-frame to non-PCM classes (T-dual, Yang–Baxter, symmetric spaces) demonstrates a broad, frame-translatable structure for integrable, stress-tensor–driven deformations, with implications for constructing new solvable 2d theories and understanding how duality and integrability mirror each other across dimensions.

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 $μ$-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 $μ$-frame to further extend known families of integrable deformations.