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

TL;DR

This paper develops a discrete $2$-gauge theory that generalizes $p$-form electrodynamics to non-Abelian settings by assigning edge data from a Lie $2$-group to edges and faces on a lattice. Leveraging strict $2$-categories, Lie crossed modules, and pseudo-natural transformations, the authors build configurations, local gauge symmetries, and gauge-invariant $2$-actions, yielding a partition function that generalizes Wilson's formulation. The approach recovers standard lattice gauge theory as a special case and extends to non-Abelian gerbes, with potential connections to spin foam models of quantum gravity and to lattice QCD phenomena such as centre vortices. The work outlines a hierarchy of theories, discusses technical challenges, and highlights avenues toward dual representations and topological quantum field theory constructions in higher gauge theory.

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.

Figures (4)

• Figure 1: The holonomy $g_{12}g_{23}g_{13}^{-1}$ around some triangle $(1,2,3)$. The inner triangle is labeled by a functor $F\colon{\mathcal{C}}^{V,E}\to{\mathcal{G}}^G$, the outer triangle by $\widetilde{F}$. The functors $F$ and $\widetilde{F}$ are related by a natural equivalence $\eta$.
• Figure 2: (a) Horizontal composition of faces is denoted by a dot ($\cdot$). (b) Vertical composition is indicated by a little circle ($\circ$) which is read from left to right in our equations. (c) Parentheses are not necessary provided the exchange law \ref{['eq_exchange']} holds.
• Figure 3: A tetrahedron with vertices labeled $1,2,3,4$. Each triangle $(i,j,k)$, $i<j<k$, is coloured as in \ref{['eq_labeled']}.
• Figure 4: The inner tetrahedron is labeled by a strict $2$-functor $F\colon{\mathcal{C}}^{V,E,F}\to {\mathcal{G}}^{G,H}$, the outer one by some strict $2$-functor $\widetilde{F}$ (notation as in \ref{['eq_labeled']}). Both $2$-functors are related by a pseudo-natural transformation $\eta\colon F\Rightarrow\widetilde{F}$. For simplicity, we have not drawn the double arrows on the faces.

Theorems & Definitions (36)

• Definition 2.1
• Example 2.2
• Example 2.3
• Definition 2.4
• Example 2.5
• Definition 2.6
• Example 2.7
• Definition 3.1
• Remark 3.2
• Lemma 3.3