Field-Dependent Metrics and Higher-Form Symmetries in Duality-Invariant Theories of Non-Linear Electrodynamics

TL;DR

The paper establishes an exact equivalence between duality-invariant 4d non-linear electrodynamics and Maxwell theory on a field-dependent metric $h_{\,\mu\nu}(F)$ with unit determinant, thereby geometrizing a broad class of stress-tensor deformations. It provides a constructive link between duality invariance and the metric, showing that the equations of motion reduce to Maxwell’s equations with $(h^{-1})^{\mu\nu}$ encoding the non-linearities, and proves that $|\det(h)|=1$ is both necessary and sufficient for this equivalence. The authors extend the analysis to higher-form symmetries, demonstrating that duality-invariant models carry two conserved 2-form currents and that, modulo trivial currents, one obtains a conserved 0-form current for each $h$-harmonic 2-form, with currents built from $\widetilde{F}$ and $\widetilde{G}$. They also discuss constructions of non-trivial currents in Maxwell theory via twisted self-duality, generalize to interacting theories, and comment on the role of the self-dual sector and Lipkin’s zilch, highlighting how the field-dependent metric perspective clarifies the symmetry structure of these theories. Overall, the work offers a geometric and algebraic framework to analyze duality-invariant electrodynamics and its generalized global symmetries, with implications for deformations, holography, and higher-dimensional extensions.

Abstract

We prove that a $4d$ theory of non-linear electrodynamics has equations of motion which are equivalent to those of the Maxwell theory in curved spacetime, but with the usual metric $g_{μν}$ replaced by a unit-determinant metric $h_{μν} ( F )$ which is a function of the field strength $F_{μν}$, if and only if the theory enjoys electric-magnetic duality invariance. Among duality-invariant models, the Modified Maxwell (ModMax) theory is special because the associated metric $h_{μν} ( F )$ produces identical equations of motion when it is coupled to the Maxwell theory via two different prescriptions which we describe. We use the field-dependent metric perspective to analyze the electric and magnetic $1$-form global symmetries in models of self-dual electrodynamics. This analysis suggests that any duality-invariant theory possesses a set of conserved currents $j^μ$ which are in one-to-one correspondence with $2$-forms that are harmonic with respect to the field-dependent metric $h_{μν} ( F )$.