nLab
Hodge-Maxwell theorem

Context

Differential cohomology

Ingredients

Connections on bundles

    • ,
    • ,

Higher abelian differential cohomology

    • ,

    • ,

Higher nonabelian differential cohomology

  • ,

  • ,

Fiber integration

Application to gauge theory

    • /

Physics

, ,

Surveys, textbooks and lecture notes

  • ,


,

, ,

    • , , , ,

      • ,

      • ,

      • ,

        • ,

      • and
    • Axiomatizations

          • ,
        • -theorem

    • Tools

      • ,

        • ,

        • ,
    • Structural phenomena

    • Types of quantum field thories

        • ,

        • , ,

        • examples

          • ,
          • ,
        • , , , ,

        • , ,

Contents

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

Then for every exact differential n-form jj of degree nk1n-k-1 there is in each de Rham cohomology ckass of degree 2 a unique representative closed 2-form FF

dF=0 \mathbf{d} F = 0

such that

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

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

References

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

Last revised on May 21, 2014 at 15:50:45. See the history of this page for a list of all contributions to it.