The String Coffee Table

Parallel transport of points in bundles is most naturally described in terms of parallel transport functors. In fact, bundles with connection can be entirely encoded in a functor from some path category to some transport category.

Here we are interested in an analogous statement for categorifed abelian gauge theory and the parallel transport of abelian strings.

Our main result is that pre-trivializations of smooth 2-functors

from 2-paths in a smooth space to a smooth sub-2-category of the the monoidal category of 1-dimensional vector space (regarded as a 2-category with a single object) are in bijection with abelian bundle gerbes with connective structure. We expect that this extends to an equivalence of the respective 2-categories.

Our constructions entirely follow those in [13,15], the only difference being that we are not dealing with principal2-bundles as defined in [8] but with something we call line-2-bundles.

The fiber of a principal 2-bundle is defined to be a 2-torsor for its structure 2-group. The 2-torsor condition on the fibers turns out to be too rigid to describe general gerbes globally. Our line-2-bundles with 2-connection locally look like principal $G_2$-2-bundles with 2-connection, for $G_2$ given by the crossed module $G_2 = (U(1)\to 1)$.

Segal proposed [1] that ‘string connections’ of the kind we are interested in should be 1-functors from 2-dimensional cobordisms to $\mathbf{Vect}$. A description of parallel transport of string in abelian gerbes in terms of 1-functors on 2-cobordisms has been discussed in [10]. However, Stolz and Teichner observed [2] that 1-functors are too coarse a tool to capture all aspects of string connections. They instead pass to 2-functors from a sort of 2-paths into some 2-category, such that Segal’s picture is obtained as a special case when these 2-paths form cobordisms.

In this sense, following [13,15], our definition of parallel transport in line-2-bundles given below is more along the lines of Stolz and Teichner’s string connections, than Segal’s cobordism 1-functors.

In fact, there is a natural motivation of our definition of 2-connection in line-2-bundles obtained from imagining a string as a continuous family of objects in a 1-cobordism category. This is discussed in detail in section 2.2 after some technical preliminaries in section 2.1.

In section 2.3 we then state our definition of a line-2-bundle with 2-connection and discuss how 1-automorphisms of line-2-bundles with 2-connection are related to ordinary line bundles with connection.

In section 2.4 this is used to obtain the concept of an abelian bundle gerbe from a line-2-bundle. A ‘pre-trivialization’ of a line 2-bundle is defined to be an operation where the line-2-bundle is locally identified with trivial line-2-bundles which are related by transition 1-morphisms of line-2-bundles. As mentioned above, these 1-morphisms turn out to correspond to ordinary line bundles with connection and are in fact the bundles appearing in the notion of a bundle gerbe.

A full trivialization of a line-2-bundle would then be obtained by further trivializing these transition bundles. Fully trivializing a line-2-bundle yields a locally trivialized $(U(1)\to 1)$-2-bundle along the lines of [13,15].