Let $E$ be a Z_2 graded vector bundle on a Riemannian manifold. We say that E is a bundle of Clifford modules if there is a bundle map
c : T^* M \longrightarrow End (E) such that c(u) c (v) + c (v) c (u ) = - 2 <u, v> id c swaps the \pm parts of the bundle.
In other words, the fiber of E is a graded module for the Clifford algebra of the tangent fiber. If M is even dimensional, the Cliffor algebra is simple and we obtain the decomposition: End (E) \simeq C (M) \otimes End_{C(M)} (E).
If M is a spin manifold, then the module C(M) itself can be written as End (S) for a certain spinor bundle S.
Polarization
editA polarization of V is a subspace in V \otimes \mathbb C which is isotropic and coisotropic, ie such that V \otimes C = P \oplus \bar P.
The exterior algebra of P is then a spinor module for V.