Charges and fluxes in Maxwell theory on compact manifolds with boundary

TL;DR

This paper explores the structure of charges and fluxes in higher-order Abelian gauge theories on compact manifolds with boundary, revealing their organization via homology and cohomology, and establishing conditions for charge quantization and dyonic configurations.

Contribution

It introduces a detailed mathematical framework using homology and cohomology to analyze charges and fluxes in Maxwell theory on manifolds with boundary, including quantization and integrability conditions.

Findings

Charges and fluxes are organized via relative homology and de Rham cohomology.

All electric charges and magnetic fluxes are quantized and satisfy Dirac's condition.

Unquantized electric fluxes can exist in the presence of quantized magnetic fluxes, leading to dyonic states.

Abstract

We investigate the charges and fluxes that can occur in higher-order Abelian gauge theories defined on compact space-time manifolds with boundary. The boundary is necessary to supply a destination to the electric lines of force emanating from brane sources, thus allowing non-zero net electric charges, but it also introduces new types of electric and magnetic flux. The resulting structure of currents, charges, and fluxes is studied and expressed in the language of relative homology and de Rham cohomology and the corresponding abelian groups. These can be organised in terms of a pair of exact sequences related by the Poincar\'e-Lefschetz isomorphism and by a weaker flip symmetry exchanging the ends of the sequences. It is shown how all this structure is brought into play by the imposition of the appropriately generalised Maxwell's equations. The requirement that these equations be integrable restricts the world-volume of a permitted brane (assumed closed) to be homologous to a cycle on the boundary of space-time. All electric charges and magnetic fluxes are quantised and satisfy the Dirac quantisation condition. But through some boundary cycles there may be unquantised electric fluxes associated with quantised magnetic fluxes and so dyonic in nature.