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

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 $g_{\mu \nu}$ replaced by a unit-determinant metric $h_{\mu \nu} ( F )$ which is a function of the field strength $F_{\mu \nu}$, 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 $h_{\mu \nu} ( F )$ 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 $j^\mu$ which are in one-to-one correspondence with $2$-forms that are harmonic with respect to the field-dependent metric $h_{\mu \nu} ( F )$.