Relative Calabi–Yau structures

Brav, Christopher; Dyckerhoff, Tobias

doi:10.1112/s0010437x19007024

Cited by 34 publications

(95 citation statements)

References 57 publications

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“…thought of as an A ∞ -category containing A, contains all the information on the noncommutative divisor. Indeed, this was the definition of noncommutative divisor used in [34, Section 2f] (and which, under a different name, already appears in [31]; for related notions, see [19,6]).…”

Section: Basic Notions: Algebramentioning

confidence: 99%

Fukaya 𝐴_{∞}-structures associated to Lefschetz fibrations. IV

Seidel¹

2019

Proceedings of Symposia in Pure Mathematics

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To a symplectic Lefschetz pencil on a monotone symplectic manifold, we associate an algebraic structure, which is a pencil of categories in the sense of noncommutative geometry. One fibre of this "noncommutative pencil" is related to the Fukaya category of the open (meaning, with the base locus removed, and hence exact symplectic) fibre of the original Lefschetz pencil; the other fibres are newly constructed kinds of Fukaya categories.

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Section: Basic Notions: Algebramentioning

confidence: 99%

Fukaya 𝐴_{∞}-structures associated to Lefschetz fibrations. IV

Seidel¹

2019

Proceedings of Symposia in Pure Mathematics

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“…On one hand, it is known that the relative Picard variety of a family of smooth curves in a toric surface has a natural Poisson structure in which the forgetful map to the Hilbert scheme is a Lagrangian fibration (this is a special case of [DM96, Section 8]). On the other hand, it is understood by work of [BD16], pursued in the present context in [ST16], that the Poisson structure on this space is essentially intrinsic to the underlying category whose moduli we are considering. Thus in our example the natural Poisson structures from the coherent and constructible descriptions of the category should coincide, and a Lagrangian fibration for one is a Lagrangian fibration for the other.…”

Section: Isotopies and Integrabilitymentioning

confidence: 99%

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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“…What we call a Calabi-Yau structure follows the terminology of [4], though this notion has also been called a proper Calabi-Yau structure [7], a right Calabi-Yau structure [3], and a "symplectic structure on a formal pointed dg-manifold" [13]. Example 4.2.…”

Section: Calabi-yau Structure On Fmentioning

confidence: 99%

“…) is zero for the following reason: m 3 F is non-zero only when m 3 F outputs an element of degree ≥ n + p + 1. Thus, by Lemma 3.9(3), g 2 (1 ⊗ m 3 F ) and g 2 (m 3 F ⊗ 1) are both identically zero.…”

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confidence: 99%

Odd sphere bundles, symplectic manifolds, and their intersection theory

Tanaka

Tseng

2018

Cambridge Journal of Mathematics

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We introduce and investigate a new perspective which relates invariants of a symplectic manifold to topological invariants of certain odd-dimensional sphere bundles over the symplectic manifold. Specifically, we show that the novel symplectic A ∞ -algebras of differential forms recently found by Tsai-Tseng-Yau are in fact equivalent to the standard de Rham differential graded algebra of the odd sphere bundles when the cohomology class of the symplectic form is integral. As applications of this equivalence, we deduce for a closed symplectic manifold that Tsai-Tseng-Yau's symplectic A ∞ -algebras satisfy the Calabi-Yau property and argue that they can be used to define an intersection theory for cycles built from coisotropic/isotropic submanifolds. We further demonstrate that these symplectic A ∞ -algebras satisfy several functorial properties and lay the groundwork for addressing Weinstein functoriality.Hiro Lee Tanaka and Li-Sheng Tseng 5.2 Homology of primitive currents 246 5.3 Intersection theory via lifts to E 248 5.4 Example: Kodaira-Thurston four-fold. 249 6 Functoriality of F 251

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Relative Calabi–Yau structures

Cited by 34 publications

References 57 publications

Fukaya 𝐴_{∞}-structures associated to Lefschetz fibrations. IV

Fukaya 𝐴_{∞}-structures associated to Lefschetz fibrations. IV

Kasteleyn operators from mirror symmetry

Odd sphere bundles, symplectic manifolds, and their intersection theory

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