Spin spaces, Lipschitz groups, and spinor bundles

TL;DR

This paper establishes a link between spinor bundles over Riemannian manifolds and Lipschitz structures, providing explicit constructions and conditions for their existence, especially in odd dimensions where orientability is crucial.

Contribution

It introduces the concept of Lipschitz structures on manifolds and relates them to spinor bundles, extending the understanding of spin geometry and providing explicit bundle constructions.

Findings

Lipschitz structures correspond to certain reductions of orthonormal frame bundles.

In even dimensions, Lipschitz groups match the complex Clifford group, enabling reduction to pin^c structures.

Topological conditions for Lipschitz structures in odd dimensions are derived and exemplified.

Abstract

It is shown that every bundle $\varSigma\to M$ of complex spinor modules over the Clifford bundle $\Cl(g)$ of a Riemannian space $(M,g)$ with local model $(V,h)$ is associated with an lpin ("Lipschitz") structure on $M$, this being a reduction of the ${\Ort}(h)$-bundle of all orthonormal frames on M to the Lipschitz group $\Lpin(h)$ of all automorphisms of a suitably defined spin space. An explicit construction is given of the total space of the $\Lpin(h)$-bundle defining such a structure. If the dimension m of M is even, then the Lipschitz group coincides with the complex Clifford group and the lpin structure can be reduced to a pin$^{c}$ structure. If m=2n-1, then a spinor module $\varSigma$ on M is of the Cartan type: its fibres are 2^n-dimensional and decomposable at every point of M, but the homomorphism of bundles of algebras $\Cl(g)\to\End\varSigma$ globally decomposes if, and only if, M is orientable. Examples of such bundles are given. The topological condition for the existence of an lpin structure on an odd-dimensional Riemannian manifold is derived and illustrated by the example of a manifold admitting such a structure, but no pin^c structure.