nLab Hodge-Maxwell theorem

Idea

The Hodge theorem in the language of electromagnetism. Over a Riemann surface this may be regarded as simple case of the Narasimhan-Seshadri theorem.

Statement

Let $(X,g)$ be a compact oriented Riemannian manifold of dimension $n$. Write $\star$ for the corresponding Hodge star operator.

Then for every exact differential n-form $j$ of degree $n-k-1$ there is in each de Rham cohomology class of degree k a unique representative closed k-form $F$

$$\mathbf{d} F = 0$$

such that

$$\mathbf{d}\star F = j \,.$$

Reading this as Maxwell's equations on $(X,g)$ then $g$ is the field of gravity, $F$ is the Faraday tensor measuring the field strength of the electromagnetic field and $j$ is the electric current.

References

The term “Hodge-Maxwell theorem” in the above form appears in

• Jonathan Evans, Aspects of Yang-Mills theory, lecture 4 pdf