This is to finally announce the release of a slow-burn project joint with Raymond Vozzo …. at least, Part I of it.

• DMR, Raymond F. Vozzo, Rigid models for 2-gerbes I: Chern–Simons geometry, arXiv:2209.05521, 63+5 pages.

Here’s the abstract:

Motivated by the problem of constructing explicit geometric string structures, we give a rigid model for bundle 2-gerbes, and define connective structures thereon. This model is designed to make explicit calculations easier, for instance in applications to physics. To compare to the existing definition, we give a functorial construction of a bundle 2-gerbe as in the literature from our rigid model, including with connections. As an example we prove that the Chern–Simons bundle 2-gerbe from the literature, with its connective structure, can be rigidified—it arises, up to isomorphism in the strongest possible sense, from a rigid bundle 2-gerbe with connective structure via this construction. Further, our rigid version of 2-gerbe trivialisation (with connections) gives rise to trivialisations (with connections) of bundle 2-gerbes in the usual sense, and as such can be used to describe geometric string structures.

The entire point of this project is the drive to make calculations simpler. We started out gritting our teeth and working with bundle 2-gerbes as defined by Danny Stevenson (based on earlier work by Carey, Murray and Wang), but there was a combinatorial growth in the complexity of what was going on, even though it was still only in relatively small degrees. Ultimately, through a process of refining our approach we landed on what is in the paper. The length of the paper is due to two things: keeping detail from calculations (it’s done in eg analytic number theory papers, why not here? I dislike the trend in some areas of pure maths to hide the details to make the paper look slick and conceptual, when there’s real work to be done), and the big appendix. Oh that appendix. It was a useful exercise, I think, to actually work through the (functorial) construction of a bundle 2-gerbe as in the literature from one of our rigid bundle 2-gerbes. Relying on a cohomological classification result here feels a bit too weak for my liking (and, additionally, requires building up the classification theory, here we can build things by hand).

The introduction is intended to give an overview of the main ideas, so I will point you there, but perhaps this is the best place to outline the plan for the rest of the project. The main result of Part II will be to construct a universal, diffeological, Chern–Simons-style rigid bundle 2-gerbe, rigidifying the usual universal 2-gerbe on a suitable . Moreover, we build explicit classifying spans (that is, a span of maps the left of which is shrinkable) for the basic gerbes on arbitrary suitable , taking the universal ‘basic gerbe’ to arise from a diffeological crossed module much as the basic gerbe is intimately linked to the crossed module underlying the string 2-group of . This permits us to justify focussing on a such a rigid model, in that we will then know every bundle 2-gerbe should be stably isomorphic to a bundle 2-gerbe with a rigidification in our sense. Finally, Part III will give the intended first main application of this project, namely explicit geometric string structures for a wide class of examples. There are some spinoff applications, but they are further down the line.