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HMS symmetries of toric boundary divisors

Špela Špenko, Michel Van den Bergh

Abstract

Let $X$ be a projective crepant resolution of a Gorenstein affine toric variety and let $((\mathbb{C}^*)^k,f)$ be the LG-model which is the Hori-Vafa mirror dual of $X$. Let ${D}$ be a generic fiber of $f$ equipped with the restriction of the standard Liouville form on $(\mathbb{C}^*)^k$. Let $\mathcal{K}_A$ be the so-called "stringy Kähler moduli space" of $X$. We show that $π_1(\mathcal{K}_A)$ acts on the wrapped Fukaya category of $D$. Using results by Gammage - Shende and Zhou, this result implies that $π_1(\mathcal{K}_A)$ acts on $D^b(\operatorname{coh}(\partial X))$ where $\partial X$ is the toric boundary divisor of $X$. We show that the induced action of $π_1(\mathcal{K}_A)$ on $K_0(\operatorname{coh}(\partial X))$ may be extended in a natural way to an action on $K_0(X)$ which corresponds to a GKZ system.

HMS symmetries of toric boundary divisors

Abstract

Let be a projective crepant resolution of a Gorenstein affine toric variety and let be the LG-model which is the Hori-Vafa mirror dual of . Let be a generic fiber of equipped with the restriction of the standard Liouville form on . Let be the so-called "stringy Kähler moduli space" of . We show that acts on the wrapped Fukaya category of . Using results by Gammage - Shende and Zhou, this result implies that acts on where is the toric boundary divisor of . We show that the induced action of on may be extended in a natural way to an action on which corresponds to a GKZ system.
Paper Structure (94 sections, 114 theorems, 231 equations)

This paper contains 94 sections, 114 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. $A$-notions
  4. The principal $A$-determinant
  5. Polyhedral decompositions of ${\mathbb R}^k$
  6. Tropical amoeba from Laurent polynomials
  7. Tailoring
  8. The $A$-toric variety
  9. Stacky fans and star-shaped triangulations
  10. Stacky fans and the FLTZ-skeleton
  11. Some symplectic geometry
  12. Liouville domains and manifolds
  13. Taming the Liouville vector field
  14. Kähler geometry
  15. General conventions
  16. ...and 79 more sections

Key Result

Theorem 1.1

There exists a natural action of $\pi_1({\mathcal{K}}_A)$ on ${\mathcal{W}}{\mathcal{F}}(D)$.

Theorems & Definitions (246)

  • Theorem 1.1: Theorem \ref{['th:mainth1']}
  • Theorem 1.2: Theorem \ref{['th:mainth1']}, Theorem \ref{['prop:lastmile']}, GammageShende
  • Remark 1.3
  • Theorem 1.4: Proposition \ref{['prop:Lazarev']}
  • Remark 1.5
  • Proposition 1.6: Proposition \ref{['prop:GKZ1_']}, \ref{['eq:sesH_']}
  • Theorem 1.7: Theorem \ref{['eq:K0boundary']}
  • Corollary 1.8: Corollary \ref{['cor:GKZ']}
  • Remark 1.9
  • Proposition 3.1
  • ...and 236 more