Kasteleyn operators from mirror symmetry

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.