nLab
higher electric background charge coupling

Context

\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Definition

Examples

Contents

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 XX be a smooth manifold of dimension dd and let nn \in \mathbb{N}. Then a degree n U(1)-gauge field on XX is a circle n-bundle with connection F^:XB nU(1) conn\hat F : X \to \mathbf{B}^n U(1)_{conn}.

For smooth currents

A background electric current for this is a circle (dn1)(d-n-1)-bundle with connection j^ el:XB dn1U(1) conn\hat j_{el} : X \to \mathbf{B}^{d-n-1} U(1)_{conn}.

The coupling action functional is

exp(iS el()):F^exp(i XF^j^) \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 δ\delta-distributed charges

The object j^ el\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 elj_{el} as being a delta distribution on the worldvolume ΣX\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 Σ\Sigma

exp(iS el()):F^hol Σ(F^). \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 AA (with dA=Fd A = F), then this is

=exp(i ΣA). \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

Created on December 21, 2011 at 01:46:06. See the history of this page for a list of all contributions to it.