Kasteleyn operators from mirror symmetry

TL;DR

This paper establishes a deep connection between Kasteleyn operators on bipartite graphs in a torus and mirror symmetry, showing they produce equivalent spectral sheaves within a sheaf-theoretic framework.

Contribution

It demonstrates the equivalence of spectral transforms of Kasteleyn operators and mirror sheaves of conjugate Lagrangians, advancing the understanding of toric mirror symmetry via sheaf theory.

Findings

Kasteleyn spectral transform matches mirror sheaf construction.

Tensoring with line bundles corresponds to Legendrian autoisotopies.

Provides a sheaf-theoretic model of mirror symmetry for bipartite graphs.

Abstract

Given a consistent bipartite graph $\Gamma$ in $T^2$ with a complex-valued edge weighting $\mathcal{E}$ we show the following two constructions are the same. The first is to form the Kasteleyn operator of $(\Gamma, \mathcal{E})$ and pass to its spectral transform, a coherent sheaf supported on a spectral curve in $(\mathbb{C}^\times)^2$. The second is to form the conjugate Lagrangian $L \subset T^* T^2$ of $\Gamma$, equip it with a brane structure prescribed by $\mathcal{E}$, and pass to its mirror coherent sheaf. This lives on a stacky toric compactification of $(\mathbb{C}^\times)^2$ determined by the Legendrian link which lifts the zig-zag paths of $\Gamma$ (and to which the noncompact Lagrangian $L$ is asymptotic). We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry. We also show that tensoring with line bundles on the compactification is mirror to certain Legendrian autoisotopies of the asymptotic boundary of $L$.