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 $M$ …

# Author: David T.

# Bezrukavnikov-Kapranov

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

# F-fields

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 …

# Nadler-Zaslow without Floer theory

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 …

# 3-dimensional example

Let $V$ be a two-dimensional vector space and let $L_0,\ldots,L_{n-1}$ be $n$ lines in $V$ obeying $L_i \neq L_{i+1}$ for all $i \in \mathbf{Z}/n$. Suppose $n$ is odd. If $g \in \mathrm{GL}(V)$ has $gL_0 = L_1,gL_1 = L_2,\ldots,gL_{n-1} = L_0$, then $g^n$ must be a scalar matrix and the ratio of the two eigenvalues of …

# Thorny character varieties 2

Chamber decomposition of $\mathrm{GL}_n^{\mathit{rss}}/\mathrm{GL}_n$. Let $\mathbf{z}$ be an $n$-element subset of $\mathbf{C}^*$, e.g. the eigenvalues of a regular semisimple matrix. For generic $\mathbf{z}$, associate a reduced word decomposition of the long element $w_0 \in S_n$, in the following way. Each element $z_k \in \mathbf{z}$ determines a stretched-and-shifted cosine graph, $\mathrm{Re}(z_k \exp(i\theta))$ over $\theta \in [0,\pi]$. The …

# Positroid varieties

Let $\Phi(p,q)$ be a rainbow-style front projection of the Legendrian $(p,q)$-torus knot. Let $\mathcal{M}_1(p,q)$ be its moduli of constructible sheaves of microlocal rank one. Proposition. $\mathcal{M}_1(p,q)$ is naturally identified with an “open positroid variety” in $\mathrm{Gr}_p(\mathbf{C}^{p+q})$, divided by its action of $(\mathbf{C}^*)^{p+q}$ To explain why, exchange a sheaf on a rainbow front, such as for …

# Thorny character varieties

Let $M$ be a one-dimensional manifold. A thorn on $M$ is a ray in $T^* M$, or equivalently a point of $T^{\infty} M$. Thorns $S \subset T^{\infty} M$ are in “general position” if the projection $\pi:S \to M$ is injective. If $S$ is in general position then we define a quiver $Q_S$: 1. The vertices of $Q_S$ are …

# More regular singularities

“More” because the discussion here is also about regular singularities. Let $X,Y,Z \in \mathrm{GL}_2$ have $XYZ = 1$. They determine a rank 2 local system on a 3-times punctured $\mathbf{P}^1$ — $X$ the monodromy around $0$, $Y$ the monodromy around $1$, $Z$ the monodromy around $\infty$. The simultaneous conjugacy class of the triple $(X,Y,Z)$ is determined …

# Perverse sheaves, Fourier transform

Here is some information about perverse sheaves on Riemann surfaces. At the end is a point I’m confused about. A perverse sheaf is a constructible chain complex of sheaves \(P\) that obeys the following conditions: 1. except for finitely many points \(x\), the stalk of \(P\) at \(x\) is concentrated in degree zero, 2. at …