The higher gauge theory analog of electromagnetism, including in degree 2 the B-field, in degree 3 the C-field, and so on.
Over a spacetime $X$, a field configuration of order $n$ $U(1)$-gauge theory is a circle n-bundle with connection $\hat F : X \to \mathbf{B}^n U(1)_{conn}$.
The action functional of the bare theory is given by
where $F \in \Omega^{n+1}_{cl}(X)$ is the field strength/curvature of $\hat F$, and where $\ast$ denotes the Hodge star operator.
The presence of background electric charge on $X$ is modeled by a fixed circle (d-n-1)-bundle with connection
where $d$ is the dimension of $X$, and adding to the action the higher electric background charge coupling term
given by the Beilinson-Deligne cup product of the higher electromagnetic field with the background electric current, followed by fiber integration in ordinary differential cohomology.
The presence of background magnetic charge, on the other hand, is modeled by changing the configurations from circle $n$-bundles with connection to twisted circle $n$-bundles with connection (…)
Dan Freed, Dirac charge quantization and generalized differential cohomology
Dan Freed, Greg Moore, Graeme Segal, The uncertainty of fluxes Commun.Math.Phys.271:247-274 (2007) (arXiv:hep-th/0605198)
Dan Freed, Greg Moore, Graeme Segal, Heisenberg Groups and Noncommutative Fluxes AnnalsPhys.322:236-285 (2007) (arXiv:hep-th/0605200)