  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.
