nLab field strength

Surveys, textbooks and lecture notes

Differential cohomology

differential cohomology

Contents

Idea

In gauge theory cocycles in differential cohomology model gauge fields.

By definition, every differential cohomology theory ${\Gamma }^{•}\left(-\right)$ comes with a characteristic curvature form morphism

$F:{\overline{\Gamma }}^{•}\left(X\right)\to {\Omega }^{•}\left(X\right)\otimes {\pi }_{*}\Gamma \otimes ℝ\phantom{\rule{thinmathspace}{0ex}},$F : \bar \Gamma^\bullet(X) \to \Omega^\bullet(X)\otimes \pi_*\Gamma \otimes \mathbb{R} \,,

the (generalized) Chern character.

For $c\in {\Gamma }^{•}\left(X\right)$ a cocycle representing a gauge field in gauge theory, its image $F\left(c\right)\in {\Omega }^{•}\left(X\right)$ is the field strength of the gauge field. If we think of this cocycle as being (a generalization of) a connection on a bundle, this is essentially the curvature of that connection.

Often gauge fields are named after their field strength. For instance the field strength of the electromagnetic field is the $2$-form $F\in {\Omega }^{2}\left(X\right)$ whose components are the electric and the magnetic fields.

Examples

gauge field: models and components

Revised on January 7, 2013 21:53:42 by Urs Schreiber (89.204.154.29)