Holonomy and parallel transport for Abelian gerbes

Abstract

In this paper we establish a one-to-one correspondence between $S^1$-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 group $S^1$ on a simply connected manifold $M$ is a group morphism from the thin second homotopy group to $S^1$, satisfying a smoothness condition, where a homotopy between maps from $[0,1]^2$ to $M$ is thin when its derivative is of rank $\leq 2$. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two monoidal Lie groupoids. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. Our approach to abelian gerbes with connections holds out prospects for generalizing to the non-abelian case via the theory of double Lie groupoids.