Kasteleyn operators from mirror symmetry

Given a consistent bipartite graph Γ in T² with a complex-valued edge weighting E we show the following two constructions are the same. The first is to form the Kasteleyn operator of (Γ , E) and pass to its spectral transform, a coherent sheaf supported on a spectral curve in (C×)2. The second is to form the conjugate Lagrangian L⊂ T^∗T² of Γ , equip it with a brane structure prescribed by E, and pass to its mirror coherent sheaf. This lives on a stacky toric compactification of (C×)2 determined by the Legendrian link which lifts the zig-zag paths of Γ (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.