nLab higher electric background charge coupling

Idea

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.

Definition

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 $\hat F : X \to \mathbf{B}^n U(1)_{conn}$.

For smooth currents

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

$$\exp(i S_{el}(-)) : \hat F \mapsto \exp(i \int_X \hat F \cup \hat j)$$

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.

For $\delta$-distributed charges

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 $\Sigma$

$$\exp(i S_{el}(-)) : \hat F \mapsto hol_\Sigma(\hat F) \,.$$

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

$$\cdots = \exp(i \int_\Sigma A) \,.$$

In the form of this simple special case the higher electric background charge coupling is often presented in physics texts.

References

• Dan Freed, Dirac charge quantization and generalized differential cohomology