Given a consistent bipartite graph in T 2 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 (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 T 2 of, 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 (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.
|Original language||English (US)|
|State||Published - Oct 14 2018|
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