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 01:46:06 by Urs Schreiber (83.91.122.110)