Spin spaces, Lipschitz groups, and spinor bundles

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.