The Dirac quantisation condition for fluxes on four-manifolds

TL;DR

This paper systematically analyzes the Dirac quantisation condition for electromagnetic fluxes on complex four-manifolds, linking topological invariants with quantum wave functions and duality properties, and exploring implications for higher-dimensional theories.

Contribution

It provides a comprehensive treatment of flux quantisation on four-manifolds, incorporating spin structures, Stiefel-Whitney classes, and a quantum Stokes' theorem, connecting topology with quantum field theory.

Findings

Derived a quantum Stokes' theorem for Wilson loops.

Clarified the role of spinor vs scalar wave functions in flux quantisation.

Connected topological invariants with electromagnetic duality and index theorems.

Abstract

A systematic treatment is given of the Dirac quantisation condition for electromagnetic fluxes through two-cycles on a four-manifold space-time which can be very complicated topologically, provided only that it is connected, compact, oriented and smooth. This is sufficient for the quantised Maxwell theory on it to satisfy electromagnetic duality properties. The results depend upon whether the complex wave function needed for the argument is scalar or spinorial in nature. An essential step is the derivation of a "quantum Stokes' theorem" for the integral of the gauge potential around a closed loop on the manifold. This can only be done for an exponentiated version of the line integral (the "Wilson loop") and the result again depends on the nature of the complex wave functions, through the appearance of what is known as a Stiefel-Whitney cohomology class in the spinor case. A nice picture emerges providing a physical interpretation, in terms of quantised fluxes and wave functions, of mathematical concepts such as spin structures, spin-C structures, the Stiefel-Whitney class and Wu's formula. Relations appear between these, electromagnetic duality and the Atiyah-Singer index theorem. Possible generalisations to higher dimensions of space-time in the presence of branes is mentioned.