Electromagnetic duality and central charge

TL;DR

This paper extends Maxwell's theory to include boundary edge modes, enabling a full realization of electromagnetic duality, revealing a boundary central charge, and linking electric charge quantization to topological properties.

Contribution

It introduces an extended phase space with boundary edge modes to realize electromagnetic duality and uncovers a boundary central charge in the symmetry algebra.

Findings

Boundary extension allows well-defined electric and magnetic symmetry generators.

Electric charge quantization is derived from boundary topological properties.

A boundary central charge appears in the symmetry algebra.

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.