Parallel 2-transport and 2-group torsors

Abstract

We provide a new perspective on parallel 2-transport and principal 2-group bundles with 2connection. We define parallel 2-transport as a 2-functor from the thin fundamental 2-groupoid to the 2-category of 2-group torsors. The definition of the 2-category of 2-group torsors is new, and we develop the tools necessary for computations in this 2-category. We prove a version of the non-Abelian Stokes Theorem and the Ambrose-Singer Theorem for 2-transport. This definition motivated by the fact that principal G-bundles with connection are equivalent to functors from the thin fundamental groupoid to the category of G-torsors. In the same lines we deduce a notion of principal 2-bundle with 2-connection, and show it is equivalent to our notion 2-transport functors. This gives a stricter notion than appears in the literature, which is more concrete. It allows for computations of 2-holonomy which will be exploited in a companion paper to define Wilson surface observables. Furthermore this notion can be generalized to a concrete but strict notion of n-transport for arbitrary n.

A Global Geometric Approach to Parallel Transport of Strings in Gauge Theory

“…It is different from that in previous work derived from category theory (see for instance [SZ15] and [Voo18]). We realize that the reason is due to some choices under local trivializations, so at the end of Section 2, we restrict our attention to local computations of surface holonomy.…”

“…Since the dependence of surface holonomy on a smooth homotopy of bigons is controlled by the relationship between higher connection (fake curvature B) and higher curvature (2-curvature F B ), a kind of non-abelian Stokes's theorem in three dimension is necessary. Generally, in the established higher gauge theory, there are lots of ways accessing the proof (see for instance [FP10], [FP11a], [SZ15] and [Voo18]). Since we are considering squares swept out by free strings, the involved non-abelian Stokes's formula needs some modification.…”

“…When written out by local trivializations and restricted to the case of bigons, the last two properties provide the motivation to define parallel transport on bigons as a 2functor between certain 2-groupoids (see [SW11] and [Par15] for more details). However, it should be careful to notice that our horizontal composition formula is a little bit different from the original one (see for instance [SZ15] and [Voo18]). The dissimilarity is caused by a choice related to local sections, which will be explained in the end of this section.…”

“…Since we are considering a reformulated surface holonomy for squares with free edges, next theorem is a little bit different from its original form in higher gauge theory. Previous approaches to non-abelian Stokes' theorem can be seen in [FP10], [FP11a], [SZ15] and [Voo18]. We attempt to make a new proof in the following case.…”

“…After figuring out how to compute surface holonomy locally, we are in a position to clarify the reason that horizontal composition formula claimed in Proposition 2.8(c) is different from the classical one: tra(Σ 2 • Σ 1 ) = tra(Σ 1 ) • α g −1 tra(Σ 2 ) (see [SZ15] and [Voo18] for more details). Suppose squares Σ 1 : γ 0 ⇒ γ 1 and Σ 2 : γ ′ 0 ⇒ γ ′ 1 are smoothly compatible in the horizontal direction.…”