The stringor bundle

TL;DR

This paper develops a rigorous framework for 2-Hilbert bundles to define the stringor bundle, linking higher differential geometry with string structures and their representations, analogous to spin geometry.

Contribution

It introduces a formal framework for 2-Hilbert bundles, constructs the stringor bundle, and proves its canonical isomorphism with a bundle derived from string structures and representations.

Findings

Established a framework for 2-Hilbert bundles.

Constructed the stringor bundle within this framework.

Proved the canonical isomorphism to a bundle from string structures.

Abstract

We set up a framework of 2-Hilbert bundles, which allows to rigorously define the "stringor bundle", a higher differential geometric object anticipated by Stolz and Teichner in an unpublished preprint about 20 years ago. Our framework includes an associated bundle construction, allowing us to associate a 2-Hilbert bundle with a principal 2-bundle and a unitary representation of its structure 2-group. We prove that the Stolz-Teichner stringor bundle is canonically isomorphic to the 2-Hilbert bundle obtained from applying our associated bundle construction to a string structure on a manifold and the stringor representation of the string 2-group that we discovered in earlier work. This establishes a perfect analogy to spin manifolds, representations of the spin groups, and spinor bundles.