nLab field strength

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

• electromagnetic field strength

Related concepts

gauge field: models and components

physics	differential geometry	differential cohomology
gauge field	connection on a bundle	cocycle in differential cohomology
instanton/charge sector	principal bundle	cocycle in underlying cohomology
gauge potential	local connection differential form	local connection differential form
field strength	curvature	underlying cocycle in de Rham cohomology
gauge transformation	equivalence	coboundary
minimal coupling	covariant derivative	twisted cohomology
BRST complex	Lie algebroid of moduli stack	Lie algebroid of moduli stack
extended Lagrangian	universal Chern-Simons n-bundle	universal characteristic map