I just suffered a moment of terrifying clarity.
Theorem: an augmentation determines a sheaf
Proof: work on $J^1(X)$; assume we’ve defined the augmentation category and localized it over $X$ (I think the fact that the localization can be done follows formally from M. Sullivan’s paper on Morse trees). To an augmentation $\alpha$ of the DGA, I am supposed to give you a sheaf on $X \times \mathbb{R}$. In other words, I should say what its stalk is at a point $(x, t)$ in a sufficiently functorial way that I end up defining a sheaf.
So, recall that Eric observed that we have a “universal” version of the augmentation category, which allows us to define homs between augmentations on different Legendrians. Look in a small neighborhood, $U \ni x$, I consider the Legendrian whose front projection is $$\Phi = U \times -N \bigcup U \times t$$
and equip it with the augmentation $A_t$ corresponding to the sheaf which occupies $U \times (-N, t]$. Here, $N \gg 0$ is a number large enough to have no relevance. Anyway, now I define a sheaf by
$$(\mathcal{F}(\alpha))_{(x,t)} = \mathrm{Hom}_+(A_t, \alpha|_{U})$$
QED.
… this moment of clarity brought to you by a meditation on the action as a generating function, which is still in process and should hopefully yield further insights…
p.s. I have not though super carefully about whether I wrote down exactly the right Legendrian + augmentation above, but anyway the point is that once we have a universal augmentation category, some augmentation will serve as a “probe brane”. Also, I view it as possible (what is Q?) that this construction will suffer from global anomalies. Finally, maybe the above + a devissage of arbitrary knots to unknots (i.e. like David Nadler uses to prove “microlocal branes are sheaves”) will prove “augmentations are sheaves”.
Hi Vivek,
Awesome! This sounds like a very plausible sketch to me. Though some things, like the localization of the augmentation category, still need to be developed (and it will be interesting to do so).
I think the probing Legendrian could just be half of what you suggest. Delete the component at $ z = -N$ and just take the front diagram to be $\Phi_t = U \times \{t\}$.
Here’s another way to see that what you have sounds correct from the point of view of generating family homology. (Not thinking about the action functional, although this perspective is clearly very useful.)
Suppose you believe that the (mixed hom-space version of) theorem of Fuchs and myself holds in all dimensions. That is,
“Theorem”
For any Legendrians $L_1, L_2 \subset J^1(X)$ with respective (say linear at infinity) generating families $F_1: X \times \mathbb{R}^{N_1} \rightarrow \mathbb{R}$ and $F_2: X\times \mathbb{R}^{N_2} \rightarrow \mathbb{R}$, there exist augmentations $\epsilon_1$ and $\epsilon_2$ of the LCH DGAs of $L_1$ and $L_2$ such that
\[
H_*(Hom(\epsilon_1, \epsilon_2)) = \mathcal{G}H_*(F_1, F_2).
\]
Recall that $\mathcal{G}H_*(F_1, F_2) := H_{*+ ?}(F_1 -F_2 \geq 0, F_1-F_2 = 0)$ where the grading shift may be determined, for instance, by comparing the given Maslov potentials on $L_1$ and $L_2$ with the Maslov potentials arising from the generating families (as the Morse index of critical points). [Here, $F_1-F_2 = F_1(x, e_1)-F_2(x,e_2)$ is actually defined on the fiber product $X \times \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$ as usual.]
So, suppose that $\epsilon_2$ is related to $F_2$ as in the “Theorem”. Then, for testing out your suggested construction, it would be reasonable to hope that the sheaf you construct, $\mathcal{F}(\epsilon_2)$, agrees with the sheaf $\mathcal{F}(F_2)$ constructed from the generating family $F_2$ as in gfacs.pdf. Recall that $\mathcal{F}(F_2)$ has stalks
\[
\mathcal{F}(F_2)_{(x,t)} = H_*(F_2(x, \cdot) \leq t, F_2(x, \cdot) \leq -N) \cong H^*( F_2(x, \cdot) \leq t, F_2(x, \cdot) = t)
\]
where $N$ is huge. (I’m ignoring the grading, and it’s possible that I’m messing up cohomology vs. homology and/or relative vs. absolute.)
Alright, now with $\Phi_t = U \times \{t\}$ we have a very simple generating family,
\[
H_t: X \times \mathbb{R}^0 \rightarrow \mathbb{R}, \quad H_t(x, 0) = t,
\]
that should correspond to what is called $A_t$ in your note. Then, we compute that the stalk that you define satisfies
\[
(\mathcal{F}(\epsilon_2))_{(x,t)} := Hom_+(A_t, \epsilon_2|_{U}) = \mathcal{G}H^*(H_t, F_2) =
\]
\[
H^*( H_t- F_2 \geq 0, H_t -F_2 =0) = H^*( t \geq F_2|_{\{x\}}, t = F_2) = \mathcal{F}(F_2)_{(x,t)},
\]
where the second equality is the “Theorem”. That is, everything checks out!
(Among other inaccuracies, I’m probably missing taking a limit, $\lim_U$.)
So I guess an alternate route to stumble onto your construction would be:
1. Realize that the stalks in the gfacs.pdf construction can be described in terms of certain Hom spaces in the universal (all Legendrians) version of the generating family category.
2. Suppose that the generating family and augmentation categories must be the same.
3. Use this alternate description of the stalks from gfacs.pdf as the definition for the augmentation category.
Anyway, I’m not sure whether anyone but me will find this convincing, but for me it is a good evidence for the correctness of your construction (or something close to it).
Dan
Hi Dan,
That’s a nice alternate derivation / sanity check. I agree the bottom strand isn’t so important.
Or maybe it is — on the train ride just now, I was thinking that perhaps I can avoid the localization altogether, by cusping off the legendrian I suggested very close to -N. I think the microlocal morse lemma on the sheaf side, and whatever corresponding calculation on the augmentation side, should say this is the same as the hom I was describing above.
That is, I’m proposing to pick up the stalk by hom from an unknot. Actually, yet another way to stumble upon this description is to meditate on the original Nadler-Zaslow prescription. This said: suppose we secretly believed Fuk = Sh. Then to take the stalk of a sheaf on the sheaf side, I compute
$$\mathcal{F}_x = Hom_{Sh}(\mathbb{Q}_{Ball(x)} , \mathcal{F}) $$
but if I secretly believe Fuk = Sh, and want to know what sheaf corresponds to a given element $L \in Fuk$, then I only need to know which element $\overline{x}$ of Fuk gives $\mathbb{Q}_{Ball(x)}$. Because, then I can define
$$\Xi(L)_x = Hom_{Fuk}(\overline{x}, L)$$
and if there is any functor $\Xi: Fuk \to Sh$ which takes $\overline{x} \to \mathbb{Q}_{Ball(x)}$, then it must necessarily have this property, just by being a functor!
So now we are doing the same thing, except with a slightly different probe than the constant sheaf on a ball (because the singular support of that doesn’t just live at minus infinity…)
Vivek
Love this line of reasoning even though it seems a mystery that you can use this in place of a ball as a probe sheaf. Or maybe there’s a general statement that goes: Suppose $F$ is a constructible sheaf on $X\times R$ with SS in $T^-$, where here $T^-$ means anything of the form (something) $dx$ + (something negative) $dt.$ Then for a sufficiently small neighborhood $U$ of $(x,t_0)$ we have $hom(k_U,F) = hom(k_{\{t<t_0\},F)$, i.e. the stalk is computable as proposed.
Perhaps this is just a rephrasing (or a vertabim copy) of something you said above.
this fact or something like it is what I meant by “microlocal morse lemma”, compare e.g. page 15 of LKACS. the other useful related thing is what Kashiwara-Schapira call “non-characteristic pullback”…