Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 251

Clifford structures and spinor bundles.

Th. Friedrich, A. Trautman

Abstract: It is shown that every bundle $varSigma o M$ of complex, irreducible and faithful modules over the Clifford bundle of an even-dimensional Riemannian space $(M,g)$ with local model $(V,h)$ is associated with a cpin (``Clifford") structure on $M$, this being an extension of the $SO(h)$-bundle of orthonormal frames on $M$ to the Clifford group $Cpin(h)= (Bbb C^{ imes} imes Spin(h))/Bbb Z_{2}$. An explicit construction is given of the total space of the $Cpin(h)$-bundle defining such a structure. A canonical line bundle on a cpin manifold, associated with the spinor norm homomorphism, is identified with a subbundle of $Hom (varSigma, varSigma^{*})$. The cpin structure restricts to a spin structure iff this line bundle is trivial.}

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