Kasteleyn operators from mirror symmetry

Abstract

Given a consistent bipartite graph Γ 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 (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 * T 2 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.

Lagrangian submanifolds from tropical hypersurfaces

“…Sheridan and Smith [16] use Lagrangian submanifolds over tropical curves to study the Lagrangian cobordism group on fibres of a Lagrangian fibration. It also turns out that our Lagrangians are are similar to Lagrangian submanifolds in the cotangent bundle of a surface constructed in [15], see also [17] for applications to mirror symmetry.…”

Tropical Lagrangians in toric del-Pezzo surfaces

“…3.1 we construct from this collection of polytopes a Lagrangian whose valuation projection lies nearby the corresponding tropical hypersurface, and whose argument projection matches the dual dimer. Section 3.2 is a slight detour from the main focus of the paper to provide a combinatorial model for the Floer-theoretic support of a tropical Lagrangian in terms of the Kasteleyn operator (similar to the computation in [39] for microlocal sheaf theory).…”

“…We now introduce a combinatorial framework generalizing some of the ideas discussed in [26,Section 5.2], and the previous work of [11,35,39,40].…”

Tropical Lagrangian hypersurfaces are unobstructed

“…The tropical Lagrangians constructed in this paper are Hamiltonian isotopic to existing parallel constructions of tropical Lagrangians presented in [25, 26, 28]. These constructions have recently been employed in the works of [32, 35], which look at how tropical geometry can be used to obstruct the existence of certain unobstructed Lagrangian cobordisms, and how the combinatorics of dimers is related to these tropical Lagrangians.…”