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 $\bar{\psi} \gamma^{a_1} \gamma^{a_2}...\gamma^{a_p}\chi$.
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 co-homology 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.