# nLab higher U(1)-gauge theory

### Context

#### Differential cohomology

differential cohomology

## Ingredients

• cohomology

• differential geometry

• ## Connections on bundles

• connection on a bundle

• curvature

• Chern-Weil theory

• ## Higher abelian differential cohomology

• differential function complex

• differential orientation

• ordinary differential cohomology

• differential K-theory

• differential elliptic cohomology

• differential cobordism cohomology

• ## Higher nonabelian differential cohomology

• Chern-Weil theory in Smooth∞Grpd

• ∞-Chern-Simons theory

• ## Fiber integration

• higher holonomy

• fiber integration in differential cohomology

• ## Application to gauge theory

• gauge theory

• gauge field

• quantum anomaly

• # Contents

## Idea

The higher gauge theory analog of electromagnetism, including in degree 2 the B-field, in degree 3 the C-field, and so on.

## Definition

Over a spacetime $X$, a field configuration of order $n$ $U(1)$-gauge theory is a circle n-bundle with connection $\hat F : X \to \mathbf{B}^n U(1)_{conn}$.

The action functional of the bare theory is given by

$\exp(i S(-)) : \hat F \mapsto \exp(i \int_X F\wedge \ast F) \,,$

where $F \in \Omega^{n+1}_{cl}(X)$ is the field strength/curvature of $\hat F$, and where $\ast$ denotes the Hodge star operator.

The presence of background electric charge on $X$ is modeled by a fixed circle (d-n-1)-bundle with connection

$\hat j_{el} : X \to \mathbf{B}^{d-n-1} U(1)_{conn} \,,$

where $d$ is the dimension of $X$, and adding to the action the higher electric background charge coupling term

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

given by the Beilinson-Deligne cup product of the higher electromagnetic field with the background electric current, followed by fiber integration in ordinary differential cohomology.

The presence of background magnetic charge, on the other hand, is modeled by changing the configurations from circle $n$-bundles with connection to twisted circle $n$-bundles with connection (…)

## References

Created on December 21, 2011 at 01:32:39. See the history of this page for a list of all contributions to it.