A symplectic prolegomenon

Smith, Ivan

doi:10.1090/s0273-0979-2015-01477-1

Cited by 17 publications

(14 citation statements)

References 92 publications

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“…Our treatment of the Fukaya category and mirror symmetry has been necessarily sketchy. The interested reader is invited to consult the survey articles of Auroux [2] and Smith [35], and to explore their extensive bibliographies, for a more detailed introduction. The monotone Fukaya category was constructed rigorously in the work of Sheridan [34] and Ritter-Smith [27].…”

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confidence: 99%

Superfiltered $A_\infty$-deformations of the exterior algebra, and local mirror symmetry

Smith

2019

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The exterior algebra E on a finite-rank free module V carries a Z{2-grading and an increasing filtration, and the Z{2-graded filtered deformations of E as an associative algebra are the familiar Clifford algebras, classified by quadratic forms on V . We extend this result to A8-algebra deformations A, showing that they are classified by formal functions on V . The proof translates the problem into the language of matrix factorisations, using the localised mirror functor construction of Cho-Hong-Lau, and works over an arbitrary ground ring. We also compute the Hochschild cohomology algebras of such A.By applying these ideas to a related construction of Cho-Hong-Lau we prove a local form of homological mirror symmetry: the Floer A8-algebra of a monotone Lagrangian torus is quasiisomorphic to the endomorphism algebra of the expected matrix factorisation of its superpotential.

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confidence: 99%

Superfiltered $A_\infty$-deformations of the exterior algebra, and local mirror symmetry

Smith

2019

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“…The most famous example is that of the Fukaya category Fuk(M ) of a symplectic manifold M (with additional technical assumptions), which is an A ∞ -category whose higher multiplications are defined by counting moduli spaces of pseudo-holomorphic disks with Lagrangian boundary conditions and n+1 marked points on their boundary. We refer for instance to [Smi15] and [Aur14] for introductions to the subject.…”

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confidence: 99%

Higher algebra of $A_\infty$ and $ΩB As$-algebras in Morse theory I

Mazuir¹

2021

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Elaborating on works by Abouzaid and Mescher, we prove that for a Morse function on a smooth compact manifold, its Morse cochain complex can be endowed with an ΩBAs-algebra structure by counting moduli spaces of perturbed Morse gradient trees. This rich structure descends to its already known A∞-algebra structure. We then introduce the notion of ΩBAs-morphism between two ΩBAs-algebras and prove that given two Morse functions, one can construct an ΩBAs-morphism between their associated ΩBAs-algebras by counting moduli spaces of two-colored perturbed Morse gradient trees. This morphism induces a standard A∞-morphism between the induced A∞-algebras. We work with integer coefficients, and provide to this extent a detailed account on the sign conventions for A∞ (resp. ΩBAs)-algebras and A∞ (resp. ΩBAs)-morphisms, using polytopes (resp. moduli spaces) which explicitly realize the dg-operadic objects encoding them. Our proofs also involve building at the level of polytopes an explicit functor from the category of ΩBAs-algebras to the category of A∞-algebras, drawing from a result by Markl and Shnider. This paper is adressed to people acquainted with either symplectic topology or algebraic operads, and written in a way to be hopefully understood by both communities. It comes in particular with a detailed survey on operads, A∞-algebras and A∞-morphisms, the associahedra and the multiplihedra, as well as some details on the usual techniques used in symplectic topology to define algebraic structures on geometrical (co)chain complexes. It moreover lays the basis for a second article in which we solve the problem of finding a satisfactory homotopic notion of higher morphisms between A∞-algebras and between ΩBAs-algebras, and show how this higher algebra of A∞ and ΩBAs-algebras naturally arises in the context of Morse theory.The associahedron K 4 and the multiplihedron J 3 ...

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“…(Prerequisites) While we have endeavoured to survey most of the previous works directly aimed at the Thomas-Yau conjecture, there is a very extensive literature on geometric measure theory and symplectic geometry in the background. We do not assume expertise on these matters, but some previous exposures such as F. Morgan's introductory book [60], and the excellent surveys of Auroux [9] and Smith [73] would be useful. The most important background facts for our main purpose are also recalled in section 5.1 and the Appendix on the Fukaya category.…”

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confidence: 99%

Thomas-Yau conjecture and holomorphic curves

Li¹

2022

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The main theme of this paper is the Thomas-Yau conjecture, primarily in the setting of exact, (quantitatively) almost calibrated, unobstructed Lagrangian branes inside Calabi-Yau Stein manifolds. In our interpretation, the conjecture is that Thomas-Yau semistability is equivalent to the existence of special Lagrangian representatives. We clarify how holomorphic curves enter this conjectural picture, through the construction of bordism currents between Lagrangians, and in the definition of the Solomon functional. Under some extra hypothesis, we shall prove Floer theoretic obstructions to the existence of special Lagrangians, using the technique of integration over moduli spaces. In the converse direction, we set up a variational framework with the goal of finding special Lagrangians under the Thomas-Yau semistability asumption, and we shall make sufficient progress to pinpoint the outstanding technical difficulties, both in Floer theory and in geometric measure theory.

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A symplectic prolegomenon

Cited by 17 publications

References 92 publications

Superfiltered $A_\infty$-deformations of the exterior algebra, and local mirror symmetry

Superfiltered $A_\infty$-deformations of the exterior algebra, and local mirror symmetry

Higher algebra of $A_\infty$ and $ΩB As$-algebras in Morse theory I

Thomas-Yau conjecture and holomorphic curves

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