Electromagnetic duality and central charge

TL;DR

The paper argues that electromagnetic duality in Maxwell theory with boundaries is realized by extending the phase space with edge modes and a boundary dual gauge field, enabling well-defined electric and magnetic soft charges. It derives a boundary action and a CS-like symplectic structure that yields a nontrivial central charge in the algebra between electric and magnetic boundary charges, and shows that electric charge quantization follows from the topology of the dual U(1) bundle. The main contribution is a concrete canonical framework where EM duality is a boundary, not bulk, symmetry, with implications for soft modes, vacuum structure, and potential quantum effects such as compact QED behavior and memory phenomena. These results deepen the understanding of edge degrees of freedom in gauge theories and connect topological and quantum aspects of duality at finite boundaries.

Abstract

We provide a full realization of the electromagnetic duality at the boundary by extending the phase space of Maxwell's theory through the introduction of edge modes and their conjugate momenta. We show how such extension, which follows from a boundary action, is necessary in order to have well defined canonical generators of the boundary magnetic symmetries. In this way, both electric and magnetic soft modes are encoded in a boundary gauge field and its conjugate dual. This implementation of the electromagnetic duality has striking consequences. In particular, we show first how the electric charge quantization follows straightforwardly from the topological properties of the $U(1)$-bundle of the boundary dual potential. Moreover, having a well defined canonical action of the electric and magnetic symmetry generators on the phase space, we can compute their algebra and reveal the presence of a central charge between them. We conclude with possible implications of these results in the quantum theory.