For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
The action functional for higher U(1)-gauge theory in the presence of background electric charge contains a charge-coupling term which is of infinity-Chern-Simons theory-type.
Let $X$ be a smooth manifold of dimension $d$ and let $n \in \mathbb{N}$. Then a degree n U(1)-gauge field on $X$ is a circle n-bundle with connection $\hat F : X \to \mathbf{B}^n U(1)_{conn}$.
A background electric current for this is a circle $(d-n-1)$-bundle with connection $\hat j_{el} : X \to \mathbf{B}^{d-n-1} U(1)_{conn}$.
The coupling action functional is
given by the higher holonomy/fiber integration in ordinary differential cohomology of the Beilinson-Deligne cup product of the gauge field with the higher electric background.
The object $\hat j_{el}$ above models the electric current of a smooth density of charged electric (n-1)-branes. If we think of the current form $j_{el}$ as being a delta distribution? on the worldvolume $\Sigma \to X$ of a single charged (n-1)-branes, then (one may thing of this via Poincare duality) the electric charge coupld action functional becomes the higher holonomy of the higher U(1)-gauge field over $\Sigma$
If, moreover, we restrict attention to gauge field configurations whose underlying circle n-bundle is trivial, which are given by globally defined n-forms $A$ (with $d A = F$), then this is
In the form of this simple special case the higher electric background charge coupling is often presented in physics texts.