This last week, a bunch of us met at AIM for the second meeting of a square. The first meeting was where we proved “augmentations are sheaves [NRSSZ]“. This time we talked about some related things, and also began a dialogue about what the contact/symplectic geometry has to say about the cluster charts structure on the moduli of augmentations. I’m writing down some of the things we did while they’re fresh in my head. (Probably I have forgotten some already, and wasn’t present for all the discussions, so by all means write about the ones I have left out.)
Counting augmentations.
In these three posts, I explained how one should count objects in the augmentation category, explained how our “augmentation counting conjecture” would follow from a certain count of the number of objects isomorphic to a given one, and proved this for the line algebra. Steven noted a consequence here: any double Legendrian slice knot has the same DGA as the unknot. Anyway, at the square, Dan showed how the line algebra argument just worked in the general case, based on the “dga homotopy = isomorphism in the augmentation category” result in [NRSSZ].
Now we’re writing that down, but still have to sort out how to make the above arguments work for the 2-periodic category.
Higher rank.
[NRSSZ] dealt with microlocal rank one sheaves versus maps of the DGA to a field. We always thought that there should be a microlocal rank n sheaves versus rank n representations of the DGA correspondence, but previous efforts to work with noncommutative coefficients in the augmentation category were unsuccessful.
This time we picked back up with the following picture in mind. Dan and Lenny had studied the relationship of satellites to higher rank augmentations before.
Given a pattern $P \subset J^1(S^1)$, say drawn as a front on the cylinder $S^1 \times \mathbb{R}$, and any Legendrian knot $K$, one can form the satellite $K^P$ by replacing a neighborhood of $K$ with $P \subset J^1(S^1)$. Note the Reeb vector field encodes the framing.
On the sheaf side, one can consider sheaves on the cylinder with rank zero at $S^1 \times -\infty$ and microlocal rank $1$ along $P$. For now let us restrict $P$ to be a braid with $n$ strands and Maslov potential identically zero. Then restriction to $S^1 \times \infty$ gives a rank $n$ locally constant sheaf.
Taking $K \subset \mathbb{R}^3 = J^1(\mathbb{R})$ (or $J^1(S^1)$ would work too) there are evident functors
$$Sh_{P,1}(S^1 \times \mathbb{R}) \times_{Loc_n(S^1)} Sh_{K,n}(\mathbb{R}^2) \to Sh_{K^P,1}(\mathbb{R}^2) \to Sh_{K, n}(\mathbb{R}^2)$$
Steven calls the second one “blurring” because it blurs the lines of $P$ together, and the first one “squinting” because one squints to see blurred things properly. Indeed, fixing a sheaf $X$ on the pattern, and a matching microrank n sheaf $F$ on the knot, the composition of the two functors takes on objects $(X, F) \mapsto F$. Moreover, it is easy to show that the second functor is essentially surjective when $P$ is the full twist. However, we do not know a good description of what happens to the morphisms under these functors.
Dan and Lenny’s result amounts (I think) to the existence of these functors at the level of maps of sets for augmentations. That is, on objects, on has
$$Aug_{P,1} \times_{Loc_n(S^1)} Aug_{K,n} \to Aug_{K^P,1} \to Aug_{K, n}$$
As usual, the analogue of the microlocal monodromy is the image of the basepoint.
Anyway, this suggested (to Dan) a way to define Hom in $Aug_{K, n}$: take the definition for $Aug_{K^P,1}$, and kill all the generators corresponding to Reeb chords of the pattern. This seems to work; Dan is writing it down.
Proving augmentations = sheaves can probably be done along the same lines as before, but there is some hope for a simpler argument by reduction to rank one. Recall that the really hard part was essential surjectivity. But, assuming one defines $Aug \to Sh$ in a way commuting with the sequences above, we can deduce $Aug_{K, n} \twoheadrightarrow Sh_{K, n}(\mathbb{R}^2)$ from the result we already have, $Aug_{K^P,1} \twoheadrightarrow Sh_{K^P,1}(\mathbb{R}^2)$. It would be really nice to get morphisms the same way; my guess is that “P = full twist${}^\infty$” might help.
(We know one full twist is not enough: cabling the unknot with a two-strand full twist gives an unlink, from which it follows that the corresponding squinting is not fully faithful.)
Another option is to define the map using the F.L. functor, which will obviously make the maps commute, and I think by a general argument is fully faithful.
Anyway, we are writing this down as well. One idea to keep kicking around is a question of Steven: find a knot which admits no rank 1 or 2 sheaves, but does admit sheaves of some finite rank.
The two-variable ruling polynomial.
The “counting augmentations” result, plus previous work of Brad and Dan, implies that the augmentation category categorifies the ruling polynomial in the sense that counting objects in the category gives this polynomial. There’s a corresponding two-variable refinement of the augmentation polynomial (which however doesn’t need the counting result): take the mixed Hodge polynomial of the augmentation variety. Really it’s better to say: just take its cohomology, and remember the weight filtration. As usual, this goes down to the original ruling polynomial by taking the “weight polynomial” — i.e., keeping *only* the weights and taking Euler characteristic with respect to the cohomological grading.
In the 2-periodic case, I conjecture that there is a spectral sequence from the appropriate piece of the HOMFLY homology to the cohomology of this variety. (I know how to prove this for positive braids.)
The “moduli of augmentations are like character varieties” yoga says that studying these cohomologies should be analogous to the work of Hausel and Rodriguez-Villegas. In particular, note that, because of the relation to the ruling polynomial, all these varieties are polynomial count and moreover have the $q \mapsto q^{-1}$ symmetry on their point counts, suggesting that the “curious Poincare duality” or “curious Hard Lefschetz” conjectures may hold for these varieties as well. Most speculatively, the [H-RV] symmetry conjectures were “explained” by the P=W conjecture, the formulation of which required that the variety looked like an integrable system in some other complex structure. It would be very interesting to know whether any such thing is going on here as well.
We didn’t get very far with this subject, but computed some examples. To compute more, we need to understand how the stratification of Brad and Dan behaves when strata meet.
Higher dimensions and the search for Q.
One application of the contact techniques has been to topological knots. This works as follows. Contact infinity of $\mathbb{R}^3$ the contact manifold $S^2 \times \mathbb{R}^3$, and a knot in $\mathbb{R}^3$ has a conormal bundle, which meets contact infinity in a 2-torus. One can then do contact homology in this setup. For various reasons, it is useful to view $S^2 \times \mathbb{R}^3$ as $J^1(S^2)$, since in particular it is for such jet bundles that LCH is most readily defined.
An preliminary indication that the sheaf point of view has something interesting to say is the following. Chris Cornwell showed that certain representations of the fundamental group of the knot complement give rise to “Q=1” augmentations of the DGA. Such representations of the fundamental group turn out to correspond (more or less tautologically) to sheaves on $\mathbb{R}^3$ with rank one microsupport along the conormal bundle of the knot.
A nagging problem with this has been: to match the contact story — even to have the same front projections — and especially if we want to use the the F.L. functor — we want to work not in $J^1(S^2) = T^\infty \mathbb{R}^3$, but instead in $J^1(S^2) = T^{\infty, -} (S^2 \times \mathbb{R})$. So we want sheaves on $S^2 \times \mathbb{R}$, not sheaves on $\mathbb{R}^3$. This time we understood, or anyway Eric and Lenny explained to me, that this can be done by a sort of Radon transform. One pulls the sheaf on $\mathbb{R}^3$ back to a sheaf on $\mathbb{R}^3 \times S^2$. Viewing the $S^2$ as parameterizing the unit cosphere, one gets a family of functions $\mathbb{R}^3 \to \mathbb{R}$, parameterized by $S^2$, just by applying the given covector to the $\mathbb{R}^3$. One now applies the sub-level set construction (as in my paper on generating families) to the sheaf on $\mathbb{R}^3$, and gets a sheaf on $S^2 \times \mathbb{R}$. This transform preserves microsupports for formal reasons, so a sheaf microsupported on the conormal to the knot in $\mathbb{R}^3$ will end up microsupported on the lift of the front projection of the 2-torus to $S^2 \times \mathbb{R}$.
This raises a question which we should eventually answer: the Cornwell correspondence matches sheaves on $\mathbb{R}^3$ to DGA representations; the F. L. functor takes DGA representations to sheaves on $S^2 \times \mathbb{R}$, and the Radon transform takes sheaves on $\mathbb{R}^3$ to sheaves on $S^2 \times \mathbb{R}$. The diagram should commute.
A confusion we have had for a year now has to do with the fact that the DGA has three parameters, $\lambda, \mu, Q$ corresponding to generators of the group ring $\mathbb{Z}[H_2(J^1(S^2), T^2)]$. Two of these parameters, namely $\lambda, \mu$, we can see easily on the sheaf side: they are the monodromies around the 2-torus of the microlocal monodromy of the sheaf. But $Q$ was a mystery.
An additional mystery was that $Q$ in some sense counts holomorphic disks, but the sheaf category has to do only with exact Lagrangians, which have no real truck with such things.
One final mystery was that the F.L. functor really should be a fully faithful map from the augmentation category to the sheaf category, and therefore would record $Q$ — but how could this happen, if the sheaf category couldn’t see $Q$?
Anyway, David Nadler showed up on the last day of the square and resolved these mysteries for us. From the point of view of the F.L. functor (as apparently I was always worried about), the issue is that it’s *not* a map from augmentations to sheaves, it’s a map from augmentations to a category which, locally on X (the X of $J^1(X)$), is the category of sheaves. In other words, it’s a map to the category of sheaves on a gerbe over $X$. Such gerbes are parameterized by $H^2(X, \mathbb{G}_m)$, i.e., exactly the thing which keeps track of where $Q$ is sent under an augmentation.
In fact, we had tossed around the word gerbe last time around, but were discouraged from using it for the following reason. Consider local systems on $S^2$. If you try and make them on a non-trivial (non-integral) gerbe, you are doing something which (see the next paragraph) amounts to trying to find a local system on $S^3$ with nontrivial monodromy around the Hopf fibres; no such thing exists. But apparently we gave up too soon: what must be happening is that the sheaves with microsupport on interesting fronts in $S^2 \times \mathbb{R}$ do deform in the gerby direction. Presumably there is a cohomology calculation which would bear this out, which, assuming augmentations = sheaves, can be made by computing linearized contact homology. […In fact, one doesn’t even have to do this: assuming Hom in the augmentation category = Hom in the sheaf category, then the fact that the augmentation variety is smooth should say that the LCH calculation will reveal the tangent space is what it has to be to see the Q deformation, hence the sheaf will necessarily admit the gerby deformation…]
Apparently there is some standard way to understand constructible sheaves on gerbes over $X$ as some sort of sheaves on circle bundles over $X$, but I don’t exactly understand what it is. (I also have no real concept of what a gerbe is supposed to be.) Whatever it is, it suggests trying the same construction on the contact geometry side. That is, rather than thinking about a front projection of a 2-torus in $J^1(S^2)$, just take the Hopf bundles and pull back the front. Now you have the front of a 3-torus in $J^1(S^3)$. We have an identification of parameters $H_2(J^1(S^3), T^3) = H_2(J^1(S^2), T^2)$, but now all the parameters play symmetric roles (!!!)
Question @ physicists in the audience: is that the M-theory circle?
That’s more or less where things left off. From my perspective, we need to work out the details of the F.L. functor asap, since it seems to be making a lot of tantalizing promises.
On the technical side, Lenny understands, maybe modulo analytic details, how to extend the definition / invariance of the augmentation category to higher dimensions.
Cluster transformations and Surgery
We understand that moduli of augmentations have algebraic torus charts coming from exact Lagrangian fillings. We also have explicit charts coming from building Lagrangians from the bicolored graphs on the surface used by Postnikov etc. If two of these graphs are related by “the square move”, the corresponding fillings are a family of Lagrangians which pass through a nodal singularity, and we can show (by sheaf calculations) that the corresponding
It would be much more satisfactory to have a general proof that, for any family of Lagrangians passing through a nodal singularity, the charts are related by the same cluster transformation. This ultimately comes down to the calculation of a certain Hom in the Fukaya category. David T. thinks he understands the Fukaya category of the neighborhood of a nodal Lagrangian well enough to just compute this Hom, maybe up to analytic difficulties. I think that in fact no additional computation is necessary: if the Fukaya category of the neighborhood can be understood in any universal way, then the desired equation is checked by the above-mentioned calculation for the square move. (Also, Seidel calculated the desired thing in some case…)
More interestingly, there are some cluster transformations which no-one knows how to do in the context of graphs on surfaces, e.g. the “hexagon move”. We sort of hope that the reason is: “on the other side of the move” sits a Lagrangian which does not come from a bipartide graph. (Open question: how can you show that a Lagrangian does not come from a bipartide graph?)
But, the above discussion suggests how to think about what Lagrangian you get. There is a certain disk which “looks like the vanishing cycle” of the hexagon move, namely the boundary of the hexagon. To find out what is on the other side, make a 4-ball around that disk. The 3-sphere boundary intersects the corresponding Lagrangian in a Legendrian link. Now ask: what fillings does this link have? One of them we see, and if there are others, then we can do a surgery and get a new chart!!
One meta-realization I came to is that we should in the [STWZ] paper have a section of “precise questions for symplectic geometers”.
Satellites and Fillings.
There was some discussion about how to carry fillings across the satellite construction. For some reason, I find this subject totally mind-killing, so was absent for it — maybe someone else can write what happened.