Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties

Hanlon, Andrew D.

doi:10.48550/arxiv.1809.06001

Cited by 3 publications

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“…Here we apply the construction of [STWZ15] in a similar spirit, but with a tropical coamoeba roughly in the role played by a tropical amoeba in the cited works. Results similar to our Proposition 1.2 were also recently obtained in [Han18], but with the role of Legendrian isotopies replaced monodromies of coefficients in Landau-Ginzburg potentials. It is plausible to us that brane brick models [FLS16b,FLS16a,FLSV17] (see also [FU14]) and their generalizations provide a combinatorial framework around which various aspects of the present work could be extended to higher dimensions.…”

supporting

confidence: 88%

Kasteleyn operators from mirror symmetry

Treumann

Williams

Zaslow

2019

Sel. Math. New Ser.

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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.

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supporting

confidence: 88%

Kasteleyn operators from mirror symmetry

Treumann

Williams

Zaslow

2019

Sel. Math. New Ser.

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“…This correspondence is based on a notion of admissibility for Lagrangians in (Y, W ), about which much has been written; see for instance [1,2,88,89]. For our purposes the notion of monomial admissibility recently introduced by Hanlon [61] seems to capture the necessary structure.…”

Section: The A-modelmentioning

confidence: 99%

Moment maps, nonlinear PDE, and stability in mirror symmetry

Collins,

Yau

2018

Preprint

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We study the deformed Hermitian-Yang-Mills (dHYM) equation, which is mirror to the special Lagrangian equation, from the variational point of view via an infinite dimensional GIT problem mirror to Thomas' GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold H mirror to Solomon's space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with C 1,α regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on collapsing manifolds with boundary. We apply these results to the infinite dimensional GIT problem for deformed Hermitian-Yang-Mills. We associate algebraic invariants to certain birational models of X × ∆, where ∆ ⊂ C is a disk. Using the existence of regular weak geodesics we prove that these invariants give rise to obstructions to the existence of solutions to the dHYM equation. Furthermore, we show that these invariants fit into a stability framework closely related to Bridgeland stability. Finally, we use a Fourier-Mukai transform on toric Kähler manifolds to describe degenerations of Lagrangian sections of SYZ torus fibrations of Landau-Ginzburg models (Y, W ). We speculate on the resulting algebraic invariants, and discuss the implications for relating Bridgeland stability to the existence of special Lagrangian sections of (Y, W ).

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“…, where the superpotential W Σ int yields a monomial admissibility condition (in the sense of [Han18])…”

Section: Homological Mirror Symmetry For Cp 2 \ Ementioning

confidence: 99%

Tropical Lagrangians in toric del-Pezzo surfaces

Hicks

2020

Preprint

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We look at how one can construct from the data of a dimer model a Lagrangian submanifold in (C * ) n whose valuation projection approximates a tropical hypersurface. Each face of the dimer corresponds to a Lagrangian disk with boundary on our tropical Lagrangian submanifold, forming a Lagrangian mutation seed. Using this we find tropical Lagrangian tori L T 2 in the complement of a smooth anticanonical divisor of a toric del-Pezzo whose wall-crossing transformations match those of monotone SYZ fibers.An example is worked out for the mirror pair (CP 2 \ E, W ), X 9111 . We find a symplectomorphism of CP 2 \E interchanging L T 2 and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier-Mukai transform on X 9111 .

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Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties

Cited by 3 publications

References 0 publications

Kasteleyn operators from mirror symmetry

Kasteleyn operators from mirror symmetry

Moment maps, nonlinear PDE, and stability in mirror symmetry

Tropical Lagrangians in toric del-Pezzo surfaces

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