This is a very naive question, but I have been wondering about the role of spin geometry and spinor structures in the context of algebraic geometry. I know the definition of spin structures and associated spinor bundles on manifolds and CW-complexes. The question is: is there analog notions of the following classical objects:
Bundle of Clifford algebras
Spinor bundle/bundle of Clifford modules
in the context of schemes (over the complex numbers) and which can be developed in a purely algebraic language? For example, is there a notion of "spin structure" or sheaf of Clifford algebras on a scheme?