Chern-Simons terms and the Three Notions of Charge

TL;DR

This paper discusses three different notions of electric and magnetic charge in theories with Chern-Simons terms, analyzing their properties and implications for charge quantization.

Contribution

It introduces a unified language to describe three types of charges and clarifies their gauge invariance, conservation, localization, and quantization properties.

Findings

Brane source charge is gauge invariant and localized but not conserved or quantized.

Maxwell charge is gauge invariant and conserved but not localized or quantized.

Page charge is conserved, localized, and quantized but not gauge invariant.

Abstract

In theories with Chern-Simons terms or modified Bianchi identities, it is useful to define three notions of either electric or magnetic charge associated with a given gauge field. A language for discussing these charges is introduced and the properties of each charge are described. `Brane source charge' is gauge invariant and localized but not conserved or quantized, `Maxwell charge' is gauge invariant and conserved but not localized or quantized, while `Page charge' conserved, localized, and quantized but not gauge invariant. This provides a further perspective on the issue of charge quantization recently raised by Bachas, Douglas, and Schweigert.