Clifford module

Summary

A Clifford module is a module over a Clifford algebra. It is a vector space on which the vectors of a quadratic space act as linear maps. The square of each such map is a scalar fixed by the quadratic form: minus its value, or its value, depending on the sign convention. Spinor spaces are the irreducible Clifford modules. Bundles of Clifford modules over Riemannian manifolds carry Dirac operators, which makes Clifford modules the algebraic input of index theory. Their classification shows a periodicity of period $8$ over the reals and $2$ over the complex numbers, closely related to Bott periodicity.