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I am trying to see if there is any existing cohomology theory for Dirac manifolds.

For the case of poisson manifolds, we have the notion of Poisson cohomology. For a manifold $M$, one can consider the cochain complex with multi vector fields and the differential coming from Schouten-Nijenhuis bracket.

As Dirac manifolds are to be considered as a generalization of Poisson manifolds, I am trying to find if there is any notion of cohomology theory of Dirac manifolds (which boils down to Poisson cohomology if the Dirac structure is graph of poisson structure).

There is already some (non-political) opposition to poisson cohomology theory as they are not easy to compute, see Grothendieck Groups of Poisson Vector Bundles. This could be one reason why cohomology theory for Dirac manifolds did not see the light. If seen from another point of view, this should be the reason why Dirac cohomology should come int light (if it is already taken birth somewhere)

One can just focus on the Lie algebroid part of Dirac structure $L\subseteq TM\oplus T^*M\rightarrow TM$, and consider Lie algebroid cohomology, but, I am sure this Lie algebroid does not have full information about the Dirac structure and we may lose some more information if we do the cohomology of this Lie alegbroid.

Any suggestions are welcome.

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    $\begingroup$ Not really an answer, but you can find some interest in this paper about infinitesimal deformations of Dirac structures arxiv.org/abs/math/0606674 $\endgroup$
    – Nicola Ciccoli
    Commented May 19, 2023 at 17:19
  • $\begingroup$ @NicolaCiccoli thank you for the reference. O will check this.. :) $\endgroup$
    – Praphulla Koushik
    Commented May 19, 2023 at 18:44
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    $\begingroup$ Poisson cohomology literally coincides with the Lie algebroid cohomology of the cotangent Lie algebroid, what information would be lost by doing the same for Dirac manifolds? $\endgroup$
    – LisanalGaib
    Commented Apr 13 at 21:08

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