Electric-Magnetic Duality Rotations in Non-Linear Electrodynamics

TL;DR

The work investigates when non-linear electrodynamics in curved spacetime can preserve electric-magnetic duality. It shows that duality-invariant Lagrangians form a one-parameter (function-of-one-variable) family, with Born-Infeld theory as the canonical example, obtained from a Hamilton-Jacobi construction tied to timelike geodesics in Witten's two-dimensional black hole. When axion and dilaton fields from open string theory are included, duality persists only at lowest order and is broken by the full non-linear electromagnetic sector, illustrating how stringy couplings can obstruct duality. The paper also generalizes to higher forms in higher dimensions, analyzes the open string Lagrangian's duality structure, and demonstrates duality implications in four-dimensional black hole solutions, highlighting potential connections between duality, string scale, and black hole physics.

Abstract

We show that there is a function of one variable's worth of Lagrangians for a single Maxwell field coupled to gravity whose equations of motion admit electric-magnetic duality. Such Lagrangians are given by solutions of the Hamilton-Jacobi equation for timelike geodesics in Witten's two-dimensional black hole. Among them are the Born-Infeld Lagrangian which arises in open string theory. We investigate the effect of the axion and the dilaton in the open superstring case and we show that this theory loses its electric-magnetic duality invariance when one considers the higher order electromagnetic field terms. We discuss some implications for black holes in string theory and an extension to $2k$-forms in $4k$ spacetime dimensions.