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.

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    $\begingroup$ Can you please transfer the question to physics stack exchange so I can try to provide you with an answer from a more physical point of view $\endgroup$ – David Bar Moshe Jun 18 '15 at 13:27
  • $\begingroup$ @DavidBarMoshe I doubt that the question would be welcome on Physics SE, as too mathematical questions are often closed there. But it can also be found here, where answers from a physical point of view are appropriate. $\endgroup$ – Dilaton Jun 18 '15 at 19:01
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    $\begingroup$ @DavidBarMoshe Thanks for you interest. I posted it here. physics.stackexchange.com/questions/190222/… $\endgroup$ – Prathyush Jun 19 '15 at 6:04
  • $\begingroup$ Crossposted to mathoverflow.net/q/209604/13917 $\endgroup$ – Qmechanic Jun 19 '15 at 6:53
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    $\begingroup$ There is an answer at physicsoverflow.org/32072 $\endgroup$ – Arnold Neumaier Jun 28 '15 at 13:28

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