I have been thinking of a way to apply the derived algebraic geometry of ToenVezzosi to construct virtual fundamental cycles on moduli spaces of pseudoholomorphic curves. This seems to be the natural setting for which to define symplectic topological invariants when the relevant moduli spaces aren't necessarily cut out transversally. Indeed, Joyce adopts this philosophy in his "Dorbifolds" project. There, Joyce defines a dorbifold to be a kind of DeligneMumford $C^{\infty}$ stack equipped with a sequence of quasicoherent sheaves on this stack that should be thought of a cotangent complex. Should one expect that this "cotangent complex" is a literal cotangent complex for some derived geometric $C^{\infty}$  stack? More precisely  should there correspond, to a Dorbifold, a derived stack $\mathcal{X}^{der}$ which is a derived extension of some geometric stack $\mathcal{X}$, and where this derived structure is the analog of a perfect obstruction theory?
1 Answer
I don't know how to define a VFC for the moduli of pseudoholomorphic curves other than using Pardon's paper. Nevertheless here's the general picture (couldn't tell from your question whether you're familiar with this story).
Let $X$ be a DeligneMumford stack. Say we want to define the virtual fundamental class $[X]^{vir}$. The optimal way to proceed is to construct a quasismooth derived stack $X^{der}$ whose classical truncation is $X$. For historical and cultural reasons there are various shadows of the notion of quasismooth derived stack people came up with that are "good enough", meaning that you can still define a VFC out of that data. Perfect obstruction theories and dorbifolds are examples of such approximations. The relevant point here is that a QS derived DM stack gives you a perfect obstruction theory via the cotangent complex. Quasismooth means that the cotangent complex of $L_{X^{der}}$ is a perfect complex concentrated in degrees $1$ and $0$ (Toramplitude $[1,0]$). Therefore the canonical thickening map $i : X \to X^{der}$ induces a morphism of complexes $$ i^*(L_{X^{der}}) \to L_X $$ which is a perfect obstruction theory for $X$. The dorbifold construction is similar (presumably). Anyway then you can follow any of the usual paths to get your VFC.
The question of how to define $X^{der}$ deserves to be addressed. The stack $X$ defines some moduli problem, say $X$ is the moduli of thingamajiggies. So for every affine scheme $Spec(R)$, you have a groupoid $X(R)$ of thingamajiggies over $Spec(R)$. The way to define $X^{der}$ is to figure out how to define the space $X^{der}(R)$ of thingamajiggies over $Spec(R)$ when $R$ is a derived commutative ring. If you do this correctly (it's a "you'll know it when you see it" kinda thing) then you'll end up with something quasismooth, hopefully actually with the cotangent complex you expected, and you'll be good to go.
A relevant example of this general picture, for the moduli of stable maps (GWtheory), is worked out in the paper of Toen, Vezzosi, Schuerg in Crelle. They define a derived moduli stack classifying derived stable maps and "surprise surprise", it's quasismooth and gives the expected perfect obstruction theory and VFC.
To answer your question, it is reasonable to expect that every perfect obstruction theory arising in practice actually comes from a qs derived structure (there's no reason it shouldn't). At the same time there's also no reason to believe that an arbitrary/abstract perfect obstruction theory necessarily comes from a qs derived structure (this is highly unlikely IMO).

$\begingroup$ Thanks for your answer! I have heard variants of this story  there seem to be some serious obstacles to proving this, in practice, however. 1) How should we prove that the morphism of complexes induced by the canonical thickening map has Toramplitude [1,0]? As far as I can see the TVSC paper doesn't deal with the case of derived smooth stacks. (It does, however, deal with the case of the simplicial commutative $\mathbb{R}$algebras that come from "forgetting" the smooth structure of the $\mathbb{R}$smooth rings). $\endgroup$ May 21, 2019 at 18:33

$\begingroup$ 2) Even if it did, there doesn't exist a notion of "intrinsic normal cone" in the smooth case, as far as I know. So, why should we expect the situation to mirror that for the algebraic/complex setting of BehrendFantechi? $\endgroup$ May 21, 2019 at 18:33

$\begingroup$ @John missed that the thrust of your question was specifically about the smooth case. 1) should still be tautological from the definition of quasismooth. For 2), the intrinsic normal cone would be way overkill for the smooth case, wouldn't it? Whatever Joyce does for dorbifolds is probably way simpler. $\endgroup$ May 21, 2019 at 18:45

$\begingroup$ I don't know the smooth picture so well but the idea is that there is supposed to be (locally) a potential for which ur derived space is the derived critical locus. This implies the appropriate cotangent connectivity and vanishing cycles should compute the behrend type function. Take this with a pinch of salt! $\endgroup$– EBzMay 21, 2019 at 21:00

$\begingroup$ Btw i would advise just emailing Dominic, he's def the person to ask $\endgroup$– EBzMay 21, 2019 at 21:04