Nonlinear self-duality for arbitrary spin, superspin, and supersymmetry type

TL;DR

This work develops a unified, four-dimensional framework for nonlinear self-duality applicable to fields of arbitrary (super)spin and ${\cal N}$-extended supersymmetry by exploiting duality rotations within conformal (super)space. It shows that any ${\mathsf{U}}(1)$ duality-invariant model is self-dual under Legendre transformation and extends this property to higher-spin and complex superconformal gauge fields, using both the Ivanov-Zupnik auxiliary-field approach and an auxiliary-variable (rho) formulation. The ModMax theory and higher-spin Born-Infeld/ModMax generalizations illustrate concrete self-dual conformal models, while duality can be enhanced to ${\mathsf{SL}}(2,\mathbb{R})$ via dilaton/axion couplings. The framework is systematically extended to ${\cal N}$-extended superconformal multiplets, with explicit discussion of self-dual models for the ${\cal N}=1$ and ${\cal N}=2$ vector and gravitino multiplets and their Legendre-duality properties. Overall, the work provides a comprehensive toolkit for constructing and analyzing self-dual nonlinear theories across spins and supersymmetries, with implications for higher-spin dynamics in conformal backgrounds and their supersymmetric extensions.

Abstract

We review the general formalism of duality rotations for $\cal N$-extended (super)conformal gauge multiplets of arbitrary (super)spin in four dimensions, with ${\cal N} \geq 0$. Self-dual models for a vector field (${\cal N}=0$) and for ${\cal N}=1$ and ${\cal N}=2$ vector supermultiplets are naturally formulated on general (super)gravity backgrounds. For all other (super)spin values, the corresponding self-dual systems are realised on arbitrary conformally flat backgrounds. Every $\mathsf{U}(1)$ duality-invariant model is demonstrated to be self-dual with respect to a Legendre transformation. Methods are proposed to generate such self-dual models including superconformal ones. We show that every model for self-dual nonlinear electrodynamics admits a higher-spin extension. Throughout the review, we make use of the formalism of conformal (super)space, that is the geometric setting to describe the gauge theory of the (super)conformal group.