I want to riff a bit on Vivek’s “microlocal test curves” and in particular try to synthesize it with David’s calculation from earlier in the summer that the trivialization of a cone over a disk looks like a cluster variable. TLDR: monodromy trivialization homotopies are exactly the data of “wrong-way parallel transport” across a negative …
Hi, I really enjoyed Vivek’s talk and our discussions, which prompted me to come back here and discover a half-finished post from a while ago; let me finish it now. Let $C$ be a Riemann surface. Let $\Sigma \subset T^* C$ be a complex curve (we think of it as a spectral curve for a Hitchin system on $C$.) …
Recall that given a collection of curves $C_i$ on a surface $\Sigma$, we construct a 4-manifold $W$ by Weinstein handle attachment of disks to $T^* \Sigma$ along the Legendrian lifts of the $C_i$. It contains a Lagrangian skeleton $\mathbb{L}$, formed as the union of $\Sigma$, the positive conormals to the $C_i$ and the 2-d disks $D_i$ …
The unmentioned context for David T.’s previous post is the following conjecture of mine. Construction. Let $S$ be a surface, and $C_i$ be oriented simple curves on it. Lift the $C_i$ to Legendrians on the cocircle bundle by taking the positive conormal. Form a symplectic 4-manifold $M$ by Weinstein handle attachment to the lifts of the …
Let $S$ be a surface, let $\Lambda = \{\lambda_i\}_{i \in I}$ be a Legendrian link in $T^{\infty} S$ with components $\lambda_i$. Write $\phi_i \subset S$ for the front projection of $\lambda_i$. Write $\underline{\Lambda} \subset T^* S$ for the conic Lagrangian obtained by coning off $\Lambda$ and taking the union with the zero section. Let $M$ …
They’ve discovered what Eric and Nicolo and I call “the constructible plumbing model”, for Riemann surfaces glued transversely. In the first and last sections they seem to envision many of the same applications. I haven’t read the middle yet. arXiv:1506.07050
This is a follow-up post regarding discussions I had with Harold, David, and Steven at the SQuaRE. (Edit: some of this is now obsolete—see new material below.) Let’s try to count exact Lagrangian fillings of the standard Legendrian $(2,n)$ torus knot for $n$ odd, up to Hamiltonian (Lagrangian?) isotopy. From the cluster picture, there are …
I’ve been trying to write down the construction for the Poisson structure of the knot character varieties, and it got me thinking about the Poisson geometry of the Grothendieck resolution and how it ties with our rank 1 and higher rank microlocal support conditions. Let’s consider two trivial cases of our knot character varieties. Consider …
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 …
Definition. Fix a prime $p$. Let $M$ be a manifold. An $F$-field on $M$ is a locally constant sheaf of perfect rings of characteristic $p$. Each element of $H^1(M,\mathbf{Z})$ determines an isomorphism class of $F$-fields. The $F$-field $\mathfrak{f}$ corresponding to the cocycle $z$ has one fiber that is $\overline{\mathbf{F}_p}$, and each loop in $M$ acts …
