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There is a well-established notion of "supermanifold", and in the world of supergeometry it makes sense to talk about symplectic structures. Actually, there are various kinds of symplectic structures: "even", "odd", "inhomogeneous", ... and all show up in applications. I care about the "even" kind.

Has anyone tried to build a "Fukaya category" for symplectic supermanifolds? An example? Suggestions of how such a category should look? String-theory mumbo jumbo about topological super sigma models?

The major difference between the super world and the non-super ("Muggle"?) world is that symplectic supermanifolds need not admit any Lagrangians at all, so presumably the appropriate Fukaya category is defined entirely in terms of coisotropic A-branes rather than just Lagrangian ones.

Googling doesn't turn up anything and I don't know where to begin looking, so this is a general-purpose reference request.

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    $\begingroup$ An immediate subquestion is whether there is Morse theory for supermanifolds. $\endgroup$
    – Chris Gerig
    Commented Jan 29, 2016 at 1:28
  • $\begingroup$ Maybe mimic Poisson sigma model with $T[1]\Sigma \to T^*[1] S$ with $S$ a Poisson supermanifold now? $\endgroup$
    – AHusain
    Commented Jan 29, 2016 at 1:44
  • $\begingroup$ @AHusain Given my intended interests, I should revisit that case. And come to think of it, I did claim in math.northwestern.edu/~theojf/FloerTheoryNotes.pdf that A-model / Fukaya category arises from a particular gauge-fixing of PSM. But note that that particular discussion requires the branes to be Lagrangian. $\endgroup$
    – Theo Johnson-Freyd
    Commented Jan 29, 2016 at 4:25
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    $\begingroup$ @ChrisGerig You will have trouble writing down a function with isolated critical points, I think, because the odd directions on a supermanifold are really really small --- too small to have very many functions. For some applications the odd directions feel noncompact, in which case you'd have different Morse complexes depending on the boundary conditions of your Morse function, but for many applications the odd directions feel compact (see "really really small" above). Probably you can construct a Morse--Bott theory. Note that all supermanifolds deformation retract onto their even cores. $\endgroup$
    – Theo Johnson-Freyd
    Commented Jan 29, 2016 at 4:30
  • $\begingroup$ Where did the change happen from $N^*[1]C$ with $C$ coisotropic to $N^*[1]L$? $\endgroup$
    – AHusain
    Commented Jan 31, 2016 at 2:23

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