nLab
higher U(1)-gauge theory

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

The action functional of the bare theory is given by

exp(iS()):F^exp(i XF*F), \exp(i S(-)) : \hat F \mapsto \exp(i \int_X F\wedge \ast F) \,,

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

The presence of background electric charge on XX is modeled by a fixed circle (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} \,,

where dd is the dimension of XX, and adding to the action the higher electric background charge coupling term

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 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 nn-bundles with connection to twisted circle nn-bundles with connection (…)

Examples

References

Created on December 21, 2011 01:32:39 by Urs Schreiber (83.91.122.110)