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.
ASJC Scopus subject areas
- Physics and Astronomy(all)