Homological mirror symmetry for punctured spheres

Abouzaid, Mohammed; Auroux, Denis; Efimov, Alexander I.; Katzarkov, Ludmil; Orlov, Dmitri

doi:10.1090/s0894-0347-2013-00770-5

Cited by 57 publications

(156 citation statements)

References 44 publications

(99 reference statements)

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“…To show that it is also surjective we can induction on k. If k = 2 we can use the same argument as in lemma 4.4 in [2] to show that in each case µ(γ 1 , γ 2 ) = τ there is also corresponding ribbon tree map.…”

Section: Equality Holds If and Only Ifmentioning

confidence: 99%

“…As is explained in [2], in the case of surfaces each such immersed convex polygon gives rise to a ribbon tree map. …”

Section: Equality Holds If and Only Ifmentioning

confidence: 99%

“…Secondly we show that matrix factorizations of a consistent dimer model also give rise to such an A ∞ -structure and finally we tie the two sides together by constructing an explicit duality on dimer models. We end with a discussion about the connection between the commutative results in [2] and the noncommutative geometry we employed. In view of the readability of the paper, we deferred the proof of the classification of A ∞ -structures to an appendix.…”

Section: Commutative Versionmentioning

confidence: 99%

“…In [2] Abouzaid, Auroux, Efimov, Katzarkov and Orlov proved an instance of mirror symmetry between such objects. On the symplectic side they considered a sphere with k punctures and on the algebraic side they considered a special Landau Ginzburg model on a certain toric quasiprojective noncompact Calabi Yau threefold and they proved an equivalence between the derived wrapped Fukaya category of the former and the derived category of matrix factorizations of the latter.…”

Section: Introductionmentioning

confidence: 97%

“…We illustrate how our viewpoint and the approach in [2] fit together. The simplest example gives us an equivalence between the sphere with 3 punctures S…”

Section: Introductionmentioning

confidence: 99%

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Noncommutative mirror symmetry for punctured surfaces

Bocklandt

2015

Trans. Amer. Math. Soc.

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ABSTRACT. In [2] Abouzaid, Auroux, Efimov, Katzarkov and Orlov showed that the wrapped Fukaya Categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities. We generalize this result to other punctured Riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models. In particular, given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is A∞-isomorphic to a subcategory of the wrapped Fukaya category of a punctured Riemann surface. The connection between the dimer model and the punctured Riemann surface then has a nice interpretation in terms of a duality on dimer models. INTRODUCTIONOriginally homological mirror symmetry was developed by Kontsevich [31] as a framework to explain the similarities between the symplectic geometry and algebraic geometry of certain Calabi-Yau manifolds. More precisely its main conjecture states that for any compact Calabi-Yau manifold with a complex structure X, one can find a mirror CalabiYau manifold X ′ equipped with a symplectic structure such that the derived category of coherent sheaves over X is equivalent to the zeroth homology of the triangulated envelop of the split closure of the Fukaya category of X ′ . The latter is a category that represents the intersection theory of Lagrangian submanifolds of X ′ .Over the years it turned out that this conjecture is part of a set of equivalences which are much broader than the compact Calabi-Yau setting [27,21,1,6,7]. Removing the compactness or Calabi-Yau condition often makes the mirror a singular object, which physicists call a Landau-Ginzburg model [37,38]. A Landau-Ginzburg model (X, W ) is a pair of a smooth space X and a complex valued function W : X → C, which is called the potential. On the algebraic side we associate to it the dg-category of matrix factorizations MF(X, W ). Its objects are diagrams P 0 G G P 1 o o where P i are vector bundles and the composition of the maps results in multiplication with W . The morphisms are morphisms between these vector bundles equipped with a natural differential.On the other hand if X ′ is noncompact we need to tweak the notion of the Fukaya category, by imposing certain conditions on the behaviour of the Lagrangians near infinity and using a Hamiltonian flow to adjust the intersection theory. This gives us the notion of a wrapped Fukaya category [3].In [2] Abouzaid, Auroux, Efimov, Katzarkov and Orlov proved an instance of mirror symmetry between such objects. On the symplectic side they considered a sphere with k punctures and on the algebraic side they considered a special Landau Ginzburg model on a certain toric quasiprojective noncompact Calabi Yau threefold and they proved an equivalence between the derived wrapped Fukaya category of the former and the derived category of matrix factorizations of the lat...

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Section: Equality Holds If and Only Ifmentioning

confidence: 99%

“…As is explained in [2], in the case of surfaces each such immersed convex polygon gives rise to a ribbon tree map. …”

Section: Equality Holds If and Only Ifmentioning

confidence: 99%

Section: Commutative Versionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

“…We illustrate how our viewpoint and the approach in [2] fit together. The simplest example gives us an equivalence between the sphere with 3 punctures S…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Noncommutative mirror symmetry for punctured surfaces

Bocklandt

2015

Trans. Amer. Math. Soc.

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Action-Angle and Complex Coordinates on Toric Manifolds

Azam

Cannizzo

Lee

2021

Association for Women in Mathematics Series

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A Beginner’s Introduction to Fukaya Categories

Auroux

2014

Bolyai Society Mathematical Studies

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103

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. The goal of these notes is to give a short introduction to Fukaya categories and some of their applications. The first half of the text is devoted to a brief review of Lagrangian Floer (co)homology and product structures. Then we introduce the Fukaya category (informally and without a lot of the necessary technical detail), and briefly discuss algebraic concepts such as exact triangles and generators. Finally, we mention wrapped Fukaya categories and outline a few applications to symplectic topology, mirror symmetry and low-dimensional topology.These notes are in no way a comprehensive text on the subject; however we hope that they will provide a useful introduction to Paul Seidel's book [40] and other texts on Floer homology, Fukaya categories, and their applications. We assume that the reader is generally familiar with the basics of symplectic geometry, and some prior exposure to pseudo-holomorphic curves is also helpful; the reader is referred to [28,29] for background material.Acknowledgements. The author wishes to thank the organizers of the Nantes Trimester on Contact and Symplectic Topology for the pleasant atmosphere at the Summer School, and Ailsa Keating for providing a copy of the excellent notes she took during the lectures. Much of the material presented here I initially learned from Paul Seidel and Mohammed Abouzaid, whom I thank for their superbly written papers and their patient explanations. Finally, the author was partially supported by an NSF grant (DMS-1007177).1. Lagrangian Floer (co)homology 1.1. Motivation. Lagrangian Floer homology was introduced by Floer in the late 1980s in order to study the intersection properties of compact Lagrangian submanifolds in symplectic manifolds and prove an important case of Arnold's conjecture concerning intersections between Hamiltonian isotopic Lagrangian submanifolds [12].Specifically, let (M, ω) be a symplectic manifold (compact, or satisfying a "bounded geometry" assumption), and let L be a compact Lagrangian submanifold of M . Let ψ ∈ Ham(M, ω) be a Hamiltonian diffeomorphism. (Recall that a time-dependentThe author was partially supported by NSF grant DMS-1007177. Note that, by Stokes' theorem, since ω |L = 0, the symplectic area of a disc with boundary on L only depends on its class in the relative homotopy group π 2 (M, L).The bound given by Theorem 1.1 is stronger than what one could expect from purely topological considerations. The assumptions that the diffeomorphism ψ is Hamiltonian, and that L does not bound discs of positive symplectic area, are both essential (though the latter can be slightly relaxed in various ways). Example 1.2. Consider the cylinder M = R × S 1 , with the standard area form, and a simple closed curve L that goes around the cylinder once: then ψ(L) is also a simple closed curve going around the cylinder once, and the assumption that ψ ∈ Ham(M ) means that the total signed area of the 2-chain bounded by L and ψ(L) is zero. It is then an elementary fact that |ψ(L) ∩ L| ≥ 2, as claimed by Theorem 1.1; see Figure 1 left. O...

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Homological mirror symmetry for punctured spheres

Cited by 57 publications

References 44 publications

Noncommutative mirror symmetry for punctured surfaces

Noncommutative mirror symmetry for punctured surfaces

Action-Angle and Complex Coordinates on Toric Manifolds

A Beginner’s Introduction to Fukaya Categories

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