Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics

Abstract

In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of the connection which can be calculated from the holonomy around (infinitesimal) loops. For Abelian symmetry groups, say G = U (1), there exists a generalization, known as p-form electrodynamics, in which (p − 1)-dimensional charged objects can be propagated along p-surfaces and in which the Lagrangian depends on a generalized curvature associated with (infinitesimal) closed p-surfaces. In this article, we use Lie 2-groups and ideas from higher category theory in order to formulate a discrete gauge theory which generalizes these models at the level p = 2 to possibly non-Abelian symmetry groups. An important feature of our model is that it involves both parallel transports along paths and generalized transports along surfaces with a non-trivial interplay of these two types of variables. Our main result is the geometric picture, namely the assignment of non-Abelian quantities to geometrical objects in a coordinate free way. We construct the precise assignment of variables to the curves and surfaces, the generalized local symmetries and gauge invariant actions and we clarify which structures can be non-Abelian and which others are always Abelian. A discrete version of connections on non-Abelian gerbes is a special case of our construction. Even though the motivation sketched so far suggests applications mainly in string theory, the model presented here is also related to spin foam models of quantum gravity and may in addition provide some insight into the role of centre monopoles and vortices in lattice QCD.

An invitation to higher gauge theory

“…This work was deeply inspired by the ideas of Breen and Messing [28,29], who considered a special class of 2-groups, and omitted the equation t(B) = d A + A ∧ A, since their sort of connection did not assign holonomies to surfaces. One should also compare the closely related work of Mackaay, Martins, and Picken [65,67,68], and the work of Pfeiffer and Girelli [76,56].…”

Topological higher gauge theory: From BF to BFCG theory

“…This expression is independent of the choice of the base edge (jm) because δ H is a gauge invariant function [7]. Similarly, exploiting the local gauge symmetry of higher gauge theory [7,8], it is not difficult to show that the partition function (15) does not depend on the ordering of the vertices.…”

“…In particular, there is a local gauge symmetry which makes sure that surface-ordered products are independent of the base point and of the source curve of the surface [7,8].…”

“…In this section, we present a combinatorial description of such a model for any triangulation of any smooth manifold of dimension d ∈ {3, 4}, using the integral formulation of higher gauge theory [7]. For d = 3, this is precisely the Yetter model [10] whereas for d = 4 it coincides with the Porter's TQFT [11] for d = 4 and n = 2.…”

“…In this article, we study generalizations of d-dimensional BF -theory, d ∈ {3, 4}, to the context of higher gauge theory [6,7,8,9].…”