nLab field strength

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Differential cohomology

differential cohomology

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Idea

In gauge theory cocycles in differential cohomology model gauge fields.

By definition, every differential cohomology theory $\Gamma^\bullet(-)$ comes with a characteristic curvature form morphism

$F : \bar \Gamma^\bullet(X) \to \Omega^\bullet(X)\otimes \pi_*\Gamma \otimes \mathbb{R} \,,$

the (generalized) Chern character.

For $c \in \Gamma^\bullet(X)$ a cocycle representing a gauge field in gauge theory, its image $F(c) \in \Omega^\bullet(X)$ 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 $F \in \Omega^2(X)$ whose components are the electric and the magnetic fields.

Examples

Last revised on January 7, 2013 at 21:53:42. See the history of this page for a list of all contributions to it.