Floer cohomology of -equivariant Lagrangian branes

Lekili, Yankı; Pascaleff, James

doi:10.1112/s0010437x1500771x

Cited by 9 publications

(7 citation statements)

References 40 publications

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“…The BV differential satisfies ∆(Z n i ξ i ) = nZ n i and is trivial on all other components. This answer agrees with the symplectic cohomology of the punctured surface [26], giving strong evidence that the two kinds of Hochschild cohomology of the matrix factorization category coincide. However, the multiplicative structures of the compactly supported and symplectic cohomologies are generally not the same.…”

Section: Hochschild Cohomology Of the Category Of Matrix Factorizationssupporting

confidence: 76%

“…In particular, if f = x n ηi , then an induction argument yields (26) ∆(x n+1 ηi θ v ) = (n + 1)x n ηi ψ i,j ∀ n ≥ 0. Theorem 4.19.…”

Section: But Then the Pathmentioning

confidence: 99%

“…Proposition 4.21 and the identities ( 8) and (14) determine ∆ on the parts of HH 2 (J(Q)) and HH 3 (J(Q)) that are generated by HH 1 (J(Q)) and HH 0 (J(Q)). As for the remainder, the exact sequence (24) and formula (26) give…”

Section: But Then the Pathmentioning

confidence: 99%

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Dimer Models and Hochschild Cohomology

Wong¹

2019

Preprint

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Dimer models provide a method of constructing noncommutative crepant resolutions of affine toric Gorenstein threefolds. In homological mirror symmetry, they can also be used to describe noncommutative Landau-Ginzburg models dual to punctured Riemann surfaces. For a zigzag consistent dimer embedded in a torus, we explicitly compute the Hochschild cohomology of its Jacobi algebra in terms of dimer combinatorics. This includes a full characterization of the Batalin-Vilkovisky structure induced by the Calabi-Yau structure of the Jacobi algebra. We then compute the compactly supported Hochschild cohomology of the category of matrix factorizations for the Jacobi algebra with its canonical potential. Contents 1. Introduction 1 2. Preliminaries 4 3. Hochschild cohomology of a central localization 15 4. Hochschild cohomology of the Jacobi algebra 22 5. Hochschild cohomology of the category of matrix factorizations 38 References 45

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Section: Hochschild Cohomology Of the Category Of Matrix Factorizationssupporting

confidence: 76%

“…In particular, if f = x n ηi , then an induction argument yields (26) ∆(x n+1 ηi θ v ) = (n + 1)x n ηi ψ i,j ∀ n ≥ 0. Theorem 4.19.…”

Section: But Then the Pathmentioning

confidence: 99%

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Dimer Models and Hochschild Cohomology

Wong¹

2019

Preprint

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“…More recent results can be learned from [12]. See also [27,Sec 2.1] for a fast review of our sign and grading conventions. In particular, our conventions are cohomological and the unit lives in degree zero!…”

Section: Symplectic Cohomology For Invertible Polynomials 21 Symplect...mentioning

confidence: 99%

Symplectic cohomology of compound Du Val singularities

Evans¹,

Lekili²

2021

Preprint

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We compute symplectic cohomology for Milnor fibres of certain compound Du Val singularities that admit small resolution by using homological mirror symmetry. Our computations suggest a new conjecture that the existence of a small resolution has strong implications for the symplectic cohomology and conversely. We also use our computations to give a contact invariant of the link of the singularities and thereby distinguish many contact structures on connected sums of S 2 × S 3 .1 For a classification, see Wall [47].

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“…The first example may have been the bigrading on the Floer cohomology of certain Lagrangian spheres in the Milnor fibres of type (A) hypersurface singularities in C n , described in [12] (it turned out later [21,Section 20] that this is compatible with the A ∞ -structure of the Fukaya category only if n 3). The geometric origin of such symmetries (on the infinitesimal level) has been studied in [25], with applications in [1,15,24]. A roadmap is provided (via mirror symmetry) by the theory of equivariant coherent sheaves, and its applications in algebraic geometry and geometric representation theory; the relevant literature is too vast to survey properly, but [8,18] have been influential for the developments presented here.…”

Section: Introductionmentioning

confidence: 99%

Picard–Lefschetz theory and dilating ℂ*-actions

Seidel¹

2015

J Topology

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Abstract. We consider C * -actions on Fukaya categories of exact symplectic manifolds. Such actions can be constructed by dimensional induction, going from the fibre of a Lefschetz fibration to its total space. We explore applications to the topology of Lagrangian submanifolds, with an emphasis on ease of computation. IntroductionIt has been gradually recognized that certain classes of non-closed symplectic manifolds admit symmetries of a new kind. These symmetries are not given by groups acting on the manifold, but instead appear as extra structure on Floer cohomology, or more properly on the Fukaya category. The first example may have been the bigrading on the Floer cohomology of certain Lagrangian spheres in the Milnor fibres of type (A) hypersurface singularities in C n , described in [12] (it turned out later [21, Section 20] that this is compatible with the A ∞ -structure of the Fukaya category only if n ≥ 3). The geometric origin of such symmetries (on the infinitesimal level) has been studied in [25], with applications in [24,15,1]. A roadmap is provided (via mirror symmetry) by the theory of equivariant coherent sheaves, and its applications in algebraic geometry and geometric representation theory; the relevant literature is too vast to survey properly, but [18,8] have been influential for the developments presented here.In [22], the example from [12] was used as a test case for talking about such symmetries in an algebraic language of A ∞ -categories with C * -actions. Here, we generalize that approach, and combine it with the symplectic version of Picard-Lefschetz theory [21]. Recall that classical Picard-Lefschetz theory provides (among other things) a way of computing the intersection pairing in the middle-dimensional homology of an affine algebraic variety, by induction on the dimension. As one consequence of our construction, one gets a similar machinery for defining and computing algebraic analogues of the q-intersection numbers from [25]. Aside from their intrinsic interest, these q-intersection numbers have implications for the topology of Lagrangian submanifolds. These are similar in spirit to those derived in [24], but benefit from the more rigid setup of C * -actions, as well as the greater ease of doing computations in a purely algebraic framework. In particular, Example 1.20 would be out of reach of the methods in [24], since those only involved the first derivative in the equivariant parameter q around q = 1, whereas (1.80) vanishes at least to order 2 at that point. Another noteworthy comparison is [14], which contains an example of a

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Floer cohomology of -equivariant Lagrangian branes

Cited by 9 publications

References 40 publications

Dimer Models and Hochschild Cohomology

Dimer Models and Hochschild Cohomology

Symplectic cohomology of compound Du Val singularities

Picard–Lefschetz theory and dilating ℂ*-actions

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