Is there a generalization of De Rham cohomology for spinors fields?

I can see that one can construct p form fields out of spinor field by contraction of the type $\bar{\psi} \gamma^{a_1} \gamma^{a_2}...\gamma^{a_p}\gamma$.

Now we can consider the integral of the p form fields on p-cycles. There is a natural derivative like operation acting on spinor fields $\gamma \cdot \partial$. Does this map have the the desired properties to make a cohomology in some way. If I cannot use this map can I use some other operator to suitably generalize the exterior derivative operator.

I have a background in physics and not in mathematics, please keep that in view when you write your answer.

formswith values in any vector bundle, and then try to compute its homology. You may also want to check the Atiah--Singer theorem concerning the relation between analytic and topological invariants. $\endgroup$ – Alex Degtyarev Jun 18 '15 at 9:05