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Closed-string mirror symmetry for dimer models

Dahye Cho, Hansol Hong, Hyeongjun Jin, Sangwook Lee

TL;DR

This work establishes a closed-string mirror symmetry for dimer-model–based settings by proving a ring isomorphism KS: SH^*(X) ≅ HH^*_c(mf(W)) between the symplectic cohomology of a punctured Riemann surface X and the compactly supported Hochschild cohomology of the matrix factorization category of the noncommutative LG model (J,W). Central to the approach is a noncommutative Maurer–Cartan deformation of a zigzag Lagrangian L in the dual dimer, yielding (J,W) as the NC mirror, with HH^*_c(mf(W)) computable both algebraically (via a Koszul resolution) and FLOER-theoretically (via CF^*((L,b),(L,b))). The paper analyzes how the KS map interacts with even/odd degree components and singularities of the toric base Y_𝒬, identifying extra classes Ψ_{i,j} and Θ_v that account for codimension-two strata and their effect on the mirror correspondence. Consequently, the authors not only prove closed-string mirror symmetry in this NC setting but also illuminate how singularities of Y_𝒬 manifest on the A-model side through the Floer theory of zigzag Lagrangians and their associated Hochschild invariants, offering a robust framework for relating A- and B-model closed-string data in NC mirror symmetry.

Abstract

For all punctured Riemann surfaces arising as mirror curves of toric Calabi--Yau threefolds, we show that their symplectic cohomology is isomorphic to the compactly supported Hochschild cohomology of the noncommutative Landau--Ginzburg model defined on the NCCR of the associated toric Gorenstein singularities. This mirror correspondence is established by analyzing the closed-open map with boundaries on certain combinatorially defined immersed Lagrangians in the Riemann surface, yielding a ring isomorphism. We give a detailed examination of the properties of this isomorphism, emphasizing its relationship to the singularity structure.

Closed-string mirror symmetry for dimer models

TL;DR

This work establishes a closed-string mirror symmetry for dimer-model–based settings by proving a ring isomorphism KS: SH^*(X) ≅ HH^*_c(mf(W)) between the symplectic cohomology of a punctured Riemann surface X and the compactly supported Hochschild cohomology of the matrix factorization category of the noncommutative LG model (J,W). Central to the approach is a noncommutative Maurer–Cartan deformation of a zigzag Lagrangian L in the dual dimer, yielding (J,W) as the NC mirror, with HH^*_c(mf(W)) computable both algebraically (via a Koszul resolution) and FLOER-theoretically (via CF^*((L,b),(L,b))). The paper analyzes how the KS map interacts with even/odd degree components and singularities of the toric base Y_𝒬, identifying extra classes Ψ_{i,j} and Θ_v that account for codimension-two strata and their effect on the mirror correspondence. Consequently, the authors not only prove closed-string mirror symmetry in this NC setting but also illuminate how singularities of Y_𝒬 manifest on the A-model side through the Floer theory of zigzag Lagrangians and their associated Hochschild invariants, offering a robust framework for relating A- and B-model closed-string data in NC mirror symmetry.

Abstract

For all punctured Riemann surfaces arising as mirror curves of toric Calabi--Yau threefolds, we show that their symplectic cohomology is isomorphic to the compactly supported Hochschild cohomology of the noncommutative Landau--Ginzburg model defined on the NCCR of the associated toric Gorenstein singularities. This mirror correspondence is established by analyzing the closed-open map with boundaries on certain combinatorially defined immersed Lagrangians in the Riemann surface, yielding a ring isomorphism. We give a detailed examination of the properties of this isomorphism, emphasizing its relationship to the singularity structure.

Paper Structure

This paper contains 32 sections, 32 theorems, 232 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Floer theory preliminaries: noncommutative mirror construction
  3. Noncommutative deformation of $CF^*(\mathbb{L},\mathbb{L})$ by (weak) bounding cochains
  4. Dimer models and nc mirror symmetry for punctured Riemanns surfaces
  5. Dimer models and noncommutative LG mirror
  6. Zigzag paths and consistency
  7. Perfect matchings
  8. Zigzag Lagrangians in the dual dimer
  9. Closed-string B-model invariants for $(J, W)$
  10. The compactly supported Hochschild cohomology $HH^*_c(\mathrm{mf}(W))$
  11. Calabi-Yau algebras and BV-structure on $HH^*_c(J,W)$
  12. The Koszul resolution of $J$
  13. Hochschild invariants from the Koszul resolution
  14. BV operators in the Hochschild cohomology in view of different resolutions
  15. Explicit calculation for $HH^\ast(J,J)$ and $d_W$
  16. ...and 17 more sections

Key Result

Theorem 1.1

Bock16 Let $\mathcal{Q}$ be a consistent dimer which admits a perfect matching (see subsec:pmpmpm), and $X_{\mathcal{Q}^\vee}:=\Sigma^\vee \setminus \mathcal{Q}^\vee_0$.Then there is a fully faithful embedding where the right hand side is the matrix factorization category of the noncommutative Landau-Ginzburg model $(J,W)=(\mathrm{Jac}\,(\mathcal{Q}),W)$.

Figures (16)

  • Figure 1: A punctured Riemann surface, the mirror toric Gorenstein singularity, and its toric crepant resolution
  • Figure 2: Zigs and zags
  • Figure 3: A zigzag cycle $Z$ and its two anti-zigzags $\mathcal{O}^\pm(Z)$
  • Figure 4: A suspended pinchpoint $\mathcal{Q}$ and its matching polytope $MP\left(\mathcal{Q}\right)$
  • Figure 5: Neighboring faces near the vertex $v_Z \in \mathcal{Q}^\vee_0$ dual to the zigzag path $Z$ in $\mathcal{Q}$
  • ...and 11 more figures

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1: CHL21
  • Definition 2.2: CHL21
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • ...and 60 more