Table of Contents
Projects
This serves as a reminder that I need to do work, rather than just sitting around.
- HMS of $A_n$-type cluster variety (w/ Linhui and Zhe)
- HMS for toric complete intersections (w/ Nicolò and Hayato)
- Sheaf Kernel (w/ Jesse and Danny)
- Wall crossing and window skeleton (w/ Jesse)
- Gluing Categories (w/ Danny)
- cKLRW and disks (w/ Mina, Vivek, Ivan and Yixuan)
- Commutators of two Bergman projectors (w/ Robert)
- VGIT on toric LG, and windows
- Mutation of Skein algebra
Ideas
Torus acting on a Toric Variety
Let $X$ be an complex $n$ dimension smooth toric variety, and let $T = U(1)^n$ the compact torus acting on $X$. Assume $X$ is equipped with a $T$-invariant symplectic form $\omega$, and $\Delta_X$ is the moment polytope of $X$.
Theorem $$ Loc(X)^{Loc(T)} \cong \mu sh(\wt \Delta_X) $$ where $\wt \Delta_X \to \Delta_X$ is certain FLTZ Lagrangian skeleton standing over $\Delta$, with $n-k$ dimension torus fiber over the $k$-dimensional boundary strata.
Hypertoric, upstairs and downstairs
$\gdef\CS{ {\C^*}}$ $\gdef\v{\vee}$ $\gdef\ycal{\mathcal{Y}}$ $\gdef\zcal{\mathcal{Z}}$
VGIT on toric LG
No, I don't want to consider complete intersection, I just want to consider A-model where I compactify $(\C^*)^n$ somehow.
For example, $X_A = (\C^*)^3 \cup D_{1,1,1}$, and $W_A = x+y+z$. And the mirror is $X_B = \C^3$, with $W_B = xyz$.
Suppose I take products of these type. I get HMS still.
Then, I consider some torus action on the B-side, and fibration to dual torus on the A-side. We have $$ Coh([X_B / T], W) = Coh(X_B,W_B)^T \cong Fuk(X_A, W_A)^T \cong Fuk(\wt X_A, \wt W_A) $$
Now, here is the interesting thing: if we take GIT quotient, that corresponds to localization by the unstable loci.
Stanley-Reisner theory in terms of FLTZ
Jay Yang, Washington University in St. Louis
ask him about it
Categorical "integrate out"
Suppose you had some infinite dimensional path integral, and you want to get some finite dimensional approximation. Physicist told you to get some effective theory with a cut-off, but with interaction coefficients depending on the cut-off scale.
Vivek says, can we do the samething. When ever you see the action $S$, you do Fukaya category with $S: (Fields) \to \C$. Can you compare this with $S_{eff}: (low modes fields) \to \C$?