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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
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.
gauge field: models and components