Timeline for Is there any work on "super Fukaya categories"?
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| Feb 5, 2016 at 14:27 | comment | added | user_11437 | Just a, maybe obvious, remark. There is a theorem by Rothstein ("The structure of supersymplectic supermanifolds") which says that, up to a suitable equivalence, the data of a supersymplectic supermanifold is just the underlying symplectic manifold and a metric on the odd part of the tangent bundle with a compatible connection. So, I, guess, this suggests one has to work with families (as usual in supergeometry). Maybe a category-valued functor on the cateory of superpoints? | |
| Jan 31, 2016 at 2:23 | comment | added | AHusain | Where did the change happen from $N^*[1]C$ with $C$ coisotropic to $N^*[1]L$? | |
| Jan 29, 2016 at 4:30 | comment | added | Theo Johnson-Freyd | @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. | |
| Jan 29, 2016 at 4:25 | comment | added | Theo Johnson-Freyd | @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. | |
| Jan 29, 2016 at 1:44 | comment | added | AHusain | Maybe mimic Poisson sigma model with $T[1]\Sigma \to T^*[1] S$ with $S$ a Poisson supermanifold now? | |
| Jan 29, 2016 at 1:28 | comment | added | Chris Gerig | An immediate subquestion is whether there is Morse theory for supermanifolds. | |
| Jan 29, 2016 at 1:16 | history | asked | Theo Johnson-Freyd | CC BY-SA 3.0 |