Higher gauge theory -- differential versus integral formulation

TL;DR

This work unifies two perspectives on higher gauge theory—the differential picture with 1- and 2-form connections and the integral picture with labeled curves and surfaces—via Lie 2-groups and Lie 2-algebras. It derives the differential formulation from the integral one, revealing a level-1 flatness condition that constrains the 2-connection and leads to Abelian 3-curvature in many cases, while showing that the integral formulation remains nonperturbatively richer (e.g., nontrivial 2-holonomies in ker t). The analysis clarifies the geometric underpinnings of BF-theory’s extended topological symmetry and discusses the limitations of a non-Abelian Yang–Mills generalization within strict Lie 2-group frameworks. The results highlight a substantial structural difference between nonperturbative (integral) and perturbative (differential) formulations of higher gauge theory and point to future work in weak or coherent 2-group generalizations and potential singularities encoded by 2-holonomies.

Abstract

The term higher gauge theory refers to the generalization of gauge theory to a theory of connections at two levels, essentially given by 1- and 2-forms. So far, there have been two approaches to this subject. The differential picture uses non-Abelian 1- and 2-forms in order to generalize the connection 1-form of a conventional gauge theory to the next level. The integral picture makes use of curves and surfaces labeled with elements of non-Abelian groups and generalizes the formulation of gauge theory in terms of parallel transports. We recall how to circumvent the classic no-go theorems in order to define non-Abelian surface ordered products in the integral picture. We then derive the differential picture from the integral formulation under the assumption that the curve and surface labels depend smoothly on the position of the curves and surfaces. We show that some aspects of the no-go theorems are still present in the differential (but not in the integral) picture. This implies a substantial structural difference between non-perturbative and perturbative approaches to higher gauge theory. We finally demonstrate that higher gauge theory provides a geometrical explanation for the extended topological symmetry of BF-theory in both pictures.

Figures (4)

• Figure 1: Our conventions for the integral formulation of $2$-gauge theory on the squares, in particular (c) the generalized holonomy \ref{['eq_holonomy']}.
• Figure 2: A flattened cube in order to read off the $2$-holonomy \ref{['2holonomy']}.
• Figure 3: The local gauge transformation of the edge labels in the integral picture, c.f.\ref{['eq_2gaugelarge']}. The bottom layer is the old configuration, the top layer the new one. In order to pass to the differential picture, we shrink the rectangle to infinitesimal width, $a\rightarrow 0$, but keep its height $\varepsilon$ fixed.
• Figure 4: (a) After shrinking the square of Figure \ref{['fig_largegauge']} to infinitesimal width $a\rightarrow 0$, its vertical edges carry a group label $g(p)=\eta_\alpha(p)\in G$ at each point $p\in M$. (b) The vertical surface label $\eta_{\mu\alpha}(p)$ would ideally yield an $H$-valued $1$-form $h_\mu(p)dx^\mu=\eta_{\mu\alpha}(p)dx^\mu$ which associates with each vector $X\in T_pM$ an element $h(X)\in H$. The linear structure of $T_pM$ here imposes a serious constraint as we explain in the text.