In Conjecture 7.5 of [STZ], we hypothesized that, for $\Lambda$ a Legendrian knot in the standard 3-space, the following were equal over any finite field: (1) The groupoid cardinality of the augmentation category (2) The groupoid cardinality of the category of rank 1 sheaves on $\Lambda$ (3) The raw number of augmentations, times some fudge factor …
The following is a comment to David’s post “Nadler-Zaslow without Floer theory”. There was some error posting it as a comment probably because it is too long, so I turn it into a post. Based on David’s comment about specializations, I think the following categorical calculations of Floer cohomology (and higher $A_\infty$-structures) should work, though …
After my talk at the Berkeley math/physics meeting, I talked a little with Davide Gaiotto. This resulted in the following compelling pictures which I do not actually understand: Say you have a spectral network and a spectral curve. There’s an exact Lagrangian naturally associated to this data: each line of the spectral network comes labelled with …
Wrote this a week or two ago but didn’t post, it’s mostly to try to get my own head straight about previous correspondences on the issues of asking whether the functor of MR1 sheaves is “really” a derived stack, and asking whether it is represented by an algebraic stack. The main point is A) these …
A question from Laura Schaposnik: If you have a 3-manifold $M$ with boundary in a surface $S$, you get a holomorphic Lagrangian $Loc(M) \subset Loc(S)$. Translate over to the Hitchin system, you have a usual Lagrangian inside $T^* Bun_G(S)$. Now forget the Lagrangian and just remember its asymptotics, $\Lambda$. What’s the meaning of $Sh_{\Lambda}(Bun_g(S))$? I …
Theorem. (Nadler-Zaslow) Let $L \subset T^* M$ be an exact Lagrangians, whose brane obstructions are trivial, and that is subject to certain tameness conditions. Then there is a functor from local systems of $\mathbf{C}$-vector spaces on $L$ to constructible sheaves on $M$. The functor is well-defined up to a shift and signs, coming from the …
Investigating the monodromy of the rank-two sheaf microlocal dual to the complex Lagrangian brane $L = \{y = \pm \sqrt{x}\} \subset T^*\mathbb C_x$, we follow the family $hom_{Fuk}(L,T^*_{x = e^{2\pi i t}}\mathbb C)$ of cochain complexes as $t$ goes from $0$ to $1.$ To compute, we turn our heads and consider that $L = \{ …
Summary of discussion with David the other night – let me know if anything sounds wrong: Claim: Let $\Sigma \subset N_\mathbb{R} \cong \mathbb{R}^2$ be a complete fan, $\Lambda_\Sigma \subset T^* (M_\mathbb{R}/M)$ the associated Lagrangian, and $n \in N_\mathbb{R}$. Let $\mu_n: Sh_{cc}(M_\mathbb{R}/M, \Lambda_\Sigma) \to dg-Vect$ take a sheaf to its microstalk at $(M,n)$. If $-n$ is …
A quasi-summary (to be taken with a grain of salt) of my understanding of the paper Eric alluded to as well as the paper of Smith on the $A_1$ Hitchin 3-folds, mostly to draw out some subtle differences between the two settings that might reward reflection. 3-folds from Toric Spectral Curves Start with a complete …
Let $X$ be a manifold. I am going to sketch some axiomatics the big augmentation category $\mathcal{A}(X)$ of Legendrians in the jet space $J^1(X)$. These will certainly be satisfied by the “straightforward generalization” of the category we defined in the “augmentations are sheaves” paper. I’ll show such a category always has a morphism to the sheaf category, and reduce showing …
