Consider a (pseudo-)Riemannian manifold $(M,g)$ and the corresponding Clifford bundle $Cl_g(T^*M)$. Let $R$ be the algebra of sections of $CL_g(T^*M)$, with point-wise multiplication. What are the simple finitely generated projective modules over $R$ (I realized that simple modules may not have a nice interpretation in terms of bundles)? If $S$ is such a module, is it projective as a module over $C^\infty(M)\subset R$? And if so, what is the structure of the vector bundle on $M$ that $S$ corresponds to (Serre-Swan)?

I believe its pretty straight forward that given a spin structure on $(M,g)$ (that is, a $\mathrm{Spin}_g$-principal bundle on $M$), an associated bundle of an irreducible spinor representation of the $\mathrm{Spin}_g$ group defines precisely a finitely generated projective $R$-module. But what about the case when $(M,g)$ has no spin structures?


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