Connes fusion of spinors on loop space

TL;DR

This paper constructs a fusion product on the spinor bundle over loop space of a string manifold using Connes fusion of von Neumann bimodules, linking string structures, loop fusion, and index theory.

Contribution

It introduces a novel fiberwise fusion product on the spinor bundle on loop space via Connes fusion, connecting string geometry with operator algebra techniques.

Findings

Established a relation between string structures and Connes fusion.

Constructed a fiberwise fusion product on the spinor bundle.

Linked loop fusion with higher-categorical bundles and index theory.

Abstract

The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the description of 2-dimensional sigma models as the bundle of states over the configuration space of the superstring. We construct a product on this bundle covering the fusion of loops, i.e., the merging of two loops along a common segment. For this purpose, we exhibit it as a bundle of bimodules over a certain von Neumann algebra bundle, and realize our product fibrewise using the Connes fusion of von Neumann bimodules. Our main technique is to establish a novel relation between string structures, loop fusion, and the Connes fusion of Fock spaces. The fusion product on the spinor bundle on loop space was proposed by Stolz and Teichner as a part of a programme to explore the relation between generalized cohomology theories, functorial field theories, and index theory. It is related to the pair of pants worldsheet of the superstring, to the extension of the corresponding smooth functorial field theory down to the point, and to a higher-categorical bundle on the underlying string manifold, the stringor bundle.