Chern-Simons terms and the Three Notions of Charge

TL;DR

The work investigates how Chern-Simons terms and modified Bianchi identities in gauge theories with branes alter the notion of electric or magnetic charge. It introduces and contrasts three notions—brane source charge (localized but non-conserved and non-quantized), Maxwell charge (gauge-invariant and conserved but delocalized), and Page charge (localized, conserved, and quantized but gauge-dependent under large transformations)—and elucidates their interrelations via the modified Bianchi identity $d\,\tilde{F}_4 + F_2 \wedge H_3 = 0$ and KK reduction to higher dimensions. The Page charge emerges as the natural, higher-dimensionally grounded quantized quantity, with its quantization tied to 11D Dirac quantization and T-duality, while the brane-source and Maxwell charges illuminate brane-ending dynamics (e.g., Hanany-Witten effects) and bulk-field contributions, respectively. This framework provides a coherent language for addressing charge quantization puzzles (as raised by Bachas, Douglas, and Schweigert) and clarifies how higher-dimensional theory constrains lower-dimensional charges in brane systems.

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.