nLab
higher U(1)-gauge theory

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

The action functional of the bare theory is given by

$$\exp (iS(-)):\hat{F}\mapsto \exp (i{\int }_{X}F\wedge *F),$$\exp(i S(-)) : \hat F \mapsto \exp(i \int_X F\wedge \ast F) \,,

where $F\in {\Omega }_{\mathrm{cl}}^{n+1}(X)$ is the field strength/curvature of $\hat{F}$, and where $*$ 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

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

$$\exp (i{S}_{\mathrm{el}}(-)):\hat{F}\mapsto \exp (i{\int }_X\hat{F}\cup \hat{j}),$$\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 (…)

Examples

• electromagnetic field

• B2-field

• C3-field

• B6-field

• C6-field

Related concepts

• self-dual higher gauge theory

References

• Dan Freed, Dirac charge quantization and generalized differential cohomology

• Dan Freed, Greg Moore, Graeme Segal, The uncertainty of fluxes Commun.Math.Phys.271:247-274 (2007) (arXiv:hep-th/0605198)

• Dan Freed, Greg Moore, Graeme Segal, Heisenberg Groups and Noncommutative Fluxes AnnalsPhys.322:236-285 (2007) (arXiv:hep-th/0605200)