nLab Dirac charge quantization

Surveys, textbooks and lecture notes

Differential cohomology

differential cohomology

Contents

Idea

If the field of electromagnetism serves as a background gauge field for electrically charged quantum particles it is subject to various quantization conditions. These say that outside the locus of any magnetic charge – for instance a magnetic monopole topological defect – the electromagnetic field is a circle bundle with connection and the first Chern class of the underlying $U(1)$-principal bundle is the discrete measure for the units of magnetic charge.

On the locus of the magnetic charge itself the situation is more complex. There the magnetic current is given by a cocycle in ordinary differential cohomology of degree 3 (with compact support) and now the electromagnetic field is a connection on a twisted bundle.

Reference

The name of the concept is due to

• P.A.M. Dirac Quantized Singularities in the Electromagnetic Field, Proceedings of the Royal Society, A133 (1931) pp 60–72.

Review is for instance in