Ribbon graphs and mirror symmetry

Sibilla, Nicolò; Treumann, David; Zaslow, Eric

doi:10.1007/s00029-014-0149-7

Cited by 25 publications

(39 citation statements)

References 13 publications

(13 reference statements)

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“…This is an idea that to our knowledge originated with Kontsevich [25], and was subsequently developed by various other authors (see e.g. [45,6,47,30]); the ultimate goal being to bypass the analysis of pseudo-holomorphic curves in favor of algebraic and topological methods. It is too early to tell how successful these approaches will be, but it is entirely possible that they will ultimately supplant the techniques we have described in this text.…”

Section: Cotangent Bundlesmentioning

confidence: 99%

A Beginner’s Introduction to Fukaya Categories

Auroux

2014

Bolyai Society Mathematical Studies

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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Section: Cotangent Bundlesmentioning

confidence: 99%

A Beginner’s Introduction to Fukaya Categories

Auroux

2014

Bolyai Society Mathematical Studies

103

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“…Thus we are reduced to checking property (2). In order to do this, it is useful to use an explicit model for the fiber product of dg-categories, which can be found for instance in Proposition 2.2 of [29]. The category C 1 × C 3 C 2 has…”

Section: Preordered Semi-orthogonal Decompositionsmentioning

confidence: 99%

Gluing semi-orthogonal decompositions

2020

Self Cite

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We introduce preordered semi-orthogonal decompositions (psod-s) of dg-categories. We show that homotopy limits of dg-categories equipped with compatible psod-s carry a natural psod. This gives a way to glue semi-orthogonal decompositions along faithfully-flat covers, extending some results of [4]. As applications we will:(1) construct semi-orthogonal decompositions for root stacks of log pairs (X, D) where D is a (not necessarily simple) normal crossing divisors, generalizing results from [17] and [3], (2) compute the Kummer flat K-theory of general log pairs (X, D), generalizing earlier results of Hagihara and Nizio l in the simple normal crossing case [15], [23].

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“…[35]). The authors of [52] conjecture that their model is quasi-equivalent to the D π F(T 0 ). Clause (i) of Theorem B implies this conjecture.…”

Section: Introductionmentioning

confidence: 99%

Arithmetic mirror symmetry for genus 1 curves with n marked points

Lekili

Polishchuk

2016

Sel. Math. New Ser.

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Abstract. We establish a Z[[t 1 , . . . , t n ]]-linear derived equivalence between the relative Fukaya category of the 2-torus with n distinct marked points and the derived category of perfect complexes on the n-Tate curve. Specialising to t 1 = . . . = t n = 0 gives a Z-linear derived equivalence between the Fukaya category of the n-punctured torus and the derived category of perfect complexes on the standard (Néron) n-gon. We prove that this equivalence extends to a Z-linear derived equivalence between the wrapped Fukaya category of the n-punctured torus and the derived category of coherent sheaves on the standard n-gon. The corresponding results for n = 1 were established in [31]. IntroductionOver the past few decades, Kontsevich's Homological Mirror Symmetry (HMS) conjecture ([28]), which predicts a remarkable equivalence between an A-model category associated to a symplectic manifold and a B -model category associated to a complex manifold, has been studied extensively in many instances: abelian varieties, toric varieties, hypersurfaces in projective spaces, etc. From the beginning, the case of the symplectic 2-torus played a special role as this is the simplest instance of a Calabi-Yau manifold where concrete computations in the Fukaya category can be made, and the non-trivial nature of the HMS conjecture becomes manifest. Calculations provided in [38]

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Ribbon graphs and mirror symmetry

Cited by 25 publications

References 13 publications

A Beginner’s Introduction to Fukaya Categories

A Beginner’s Introduction to Fukaya Categories

Gluing semi-orthogonal decompositions

Arithmetic mirror symmetry for genus 1 curves with n marked points

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