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This question concerns topological string theory.

It was known sice its outset, that the BRST-cohomology ("observables") of the weakly coupled topological string B-model on a Calabi-Yau threefold $X$ is identified as the ordinary Dolbeault cohomology ring of $X$.

My trouble:

I'm reading the famous Witten's paper Perturbative Gauge Theory As A String Theory In Twistor Space . The basic construction of this paper involves the B-model on the Calabi-Yau space $\mathbb{CP}^{3|4}$. I find the paper wonderful, but I'm struggling with what I should understand by "Dolbeault supercohomology" or "superhomology" in this and the broader context of supermanifolds. To be specific, I want to learn about this in order to rigorously and systematically compute (if possible) the general allowed observables in the aforementioned context.

Questions:

  1. I'm asking for your kind help to find a an introductory text, notes or papers to adquire basic practical knowledge on the field of supergeometry, specifically, on the analogue of the de Rham "homology" of a supermanifold.

  2. Does anyone know if is it possible to explicitly compute the allowed branes in the case of the B-model over $\mathbb{CP}^{3|4}$ by means of homological/cohomological computations?

Observation: I'm aware of the fact that B-model branes must be strictly defined as elements of the derived bounded category of coherent sheaves on $X$ as was firstly pointed out by Douglas. But I will be happy just with a resource recommendation for "superhomology". In any case, and if possible, I would also sincerely appreciate any comment about the analogues of sheaf cohomology, Fukaya categories or stacks in the realm of supergeometry. I really like this field, it would be great to have references for the future.

Any comment or suggestion is helpful.

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    $\begingroup$ I found Witten's paper arxiv.org/abs/1209.2459 to be fairly readable. $\endgroup$
    – user108998
    Commented Jul 28, 2020 at 18:57
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    $\begingroup$ Further it might help to compute dolbeault cohom groups in some simple cases. I'll think of dolb cohom as holom sheaf cohom with values in sheaves of diff forms. Let's take superspace $X$ the odd total space of line bundle $\mathcal{L}$ on curve $\Sigma$. Super sheaf of one forms on $X$ is pullback from $\Sigma$ of super sheaf $\Omega\oplus \mathcal{L}$ where $\mathcal{L} $ is odd, so everything is quite explicitly computable. Nb that hodge decomp fails now, unless I'm missing something, indeed dR cohom itself should be isom to dR cohom of the bulk manifold (reduced space in math terms). $\endgroup$
    – user108998
    Commented Jul 28, 2020 at 19:07

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