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
for higher abelian targets
for symplectic Lie n-algebroid targets
A background electric current for this is a circle -bundle with connection .
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 above models the electric current of a smooth density of charged electric (n-1)-branes. If we think of the current form as being a delta distribution? on the worldvolume 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
In the form of this simple special case the higher electric background charge coupling is often presented in physics texts.