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Combinatorial Reid's recipe for consistent dimer models

Alastair Craw, Liana Heuberger, Jesus Tapia Amador

Abstract

Reid's recipe for a finite abelian subgroup $G\subset \text{SL}(3,\mathbb{C})$ is a combinatorial procedure that marks the toric fan of the $G$-Hilbert scheme with irreducible representations of $G$. The geometric McKay correspondence conjecture of Cautis--Logvinenko that describes certain objects in the derived category of $G\text{-Hilb}$ in terms of Reid's recipe was later proved by Logvinenko et al. We generalise Reid's recipe to any consistent dimer model by marking the toric fan of a crepant resolution of the vaccuum moduli space in a manner that is compatible with the geometric correspondence of Bocklandt--Craw--Quintero-Vélez. Our main tool generalises the jigsaw transformations of Nakamura to consistent dimer models.

Combinatorial Reid's recipe for consistent dimer models

Abstract

Reid's recipe for a finite abelian subgroup is a combinatorial procedure that marks the toric fan of the -Hilbert scheme with irreducible representations of . The geometric McKay correspondence conjecture of Cautis--Logvinenko that describes certain objects in the derived category of in terms of Reid's recipe was later proved by Logvinenko et al. We generalise Reid's recipe to any consistent dimer model by marking the toric fan of a crepant resolution of the vaccuum moduli space in a manner that is compatible with the geometric correspondence of Bocklandt--Craw--Quintero-Vélez. Our main tool generalises the jigsaw transformations of Nakamura to consistent dimer models.

Paper Structure

This paper contains 21 sections, 26 theorems, 37 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Dimer models
  4. Consistency
  5. Moduli of quiver representations
  6. Toric VGIT for consistent dimer models
  7. The (dual) tautological bundle is a tilting bundle
  8. Classical Reid's recipe
  9. Nakamura's jigsaw transformations
  10. Distinguished $\theta$-stable $A$-modules
  11. Fundamental hexagons
  12. Meandering walks
  13. Generalised Nakamura jigsaw transformations
  14. Combinatorial Reid's recipe
  15. The $0$-generated stability condition
  16. ...and 6 more sections

Key Result

Theorem \oldthetheorem

Let $Q$ be the quiver dual to a consistent dimer model and choose a vertex $0\in Q_0$. There is a combinatorial recipe that marks every internal lattice point and line segment of the fan $\Sigma_\theta$ with one or more nonzero vertices of $Q$; specifically, vertex $i\in Q_0$ marks: Moreover, this recipe agrees with Reid's original recipe for marking cones in the toric fan of $G\operatorname{-Hil

Figures (11)

  • Figure 1: (a) A dimer model $\Gamma$ and its dual quiver $Q$; (b) The characteristic polygon $\Delta(\Gamma)$.
  • Figure 2: (a) Divisors labelling arrows in $Q$; (b) Triangulation of $\Delta(\Gamma)$ defining a fan $\Sigma_\theta$.
  • Figure 3: The red dashed lines form the boundary of the fundamental hexagon.
  • Figure 4: The perfect matchings and meandering walks (see Section \ref{['sec:meandering']}) for $\operatorname{Hex}(\sigma)$.
  • Figure 5: Lifts of the meandering walk $\mathfrak{m}_\tau$ with homology class $[\mathfrak{m}_\tau]=(0,1)$.
  • ...and 6 more figures

Theorems & Definitions (69)

  • Theorem \oldthetheorem: Combinatorial Reid's recipe
  • Corollary \oldthetheorem: Compatibility with Geometric Reid's recipe
  • Example \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem: Ishii--Ueda IshiiUeda15
  • Example \oldthetheorem
  • Theorem \oldthetheorem: Logvinenko Logvinenko10
  • Lemma \oldthetheorem
  • proof
  • ...and 59 more