Holonomy and Parallel Transport for Abelian Gerbes

Abstract

In this paper we establish a one-to-one correspondence between U (1)-gerbes with connections, on the one hand, and their holonomies, for simply-connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with * presently working as a postdoc at the University of Nottingham, UK 1 group U (1) on a simply-connected manifold M is a group morphism from the thin second homotopy group to U (1), satisfying a smoothness condition, where a homotopy between maps from [0, 1] 2 to M is thin when its derivative is of rank ≤ 2. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two special Lie groupoids, which we call Lie 2-groups. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply-connected case.

Thin loop groups

“…(1) Ω(X) ։ π 1 1 (X) ։ π 1 (X) This space plays an important role in studying holonomy, connections and transport. As shown in [2,24], and even earlier in [20,33], there is a bijective correspondence between homomorphisms π 1 1 (X) h / / G with G a compact Lie group, up to conjugation, and principal G-bundles with connections on X, unique up to equivalence. A technical condition is put on the topology of loops so that there is the notion of a smooth family of loops which has a smoothly varying holonomy image in G ( [2], §2).…”

“…In this final section we treat the case of maps of higher dimensional spheres (cf. [24]). Here X is a simplicial complex as before, S n the unit sphere in R n+1 .…”

Abelian BF theory and Turaev-Viro invariant

“…Deligne-Beilinson Cohomology [1,2] takes its roots in Algebraic Geometry, and more specifically in the theory of Regulators and L-Functions as well as in K-Theory [3]. It is also effective in the study of flat vector bundles [4,5] or in the classification of abelian Gerbes with connections [6]. Alternative descriptions are provided by Cheeger-Simons Differential Characters [7,8,9], Hopkins-Singer Differential Cohomology [10] and Harvey-Lawson Sparks [11], all these notions being themselves equivalent in some sense [12].…”

Two-dimensional algebra in lattice gauge theory

“…Higher form symmetries have also been of recent interest in high energy physics and condensed matter in the exploration of surface operators and charges for higher-dimensional excitations [GKSW15]. However, the forms in the latter are strictly abelian and the proper mathematical framework for describing them is provided by abelian gerbes (aka higher bundles) [MP02], [TWZ12] and Deligne cohomology. Higher non-abelian forms appear in many other contexts in physics, such as in a stack of D-branes in string theory [Mye99], in the ABJM model [PS12], and in the quantum field theory on the M5-brane [FSS14].…”