Matrix factorizations and higher residue pairings

Shklyarov, Dmytro

doi:10.1016/j.aim.2016.01.014

Cited by 15 publications

(15 citation statements)

References 22 publications

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“…For G trivial, D. Shklyarov had shown in [22] that the Gauss-Manin system via similar non-commutative methods is equivalent to that given by Saito's singularity theory [21]. Denote by BŴ = C[z, w] z n +zw 2 a curved polynomial algebra with a curvatureŴ = z n + zw 2 and consider its deformation as (2) 2k +α (4) 2k + · · · , β =β (2) +β (4)…”

Section: Computation Of D-typementioning

confidence: 99%

Unfolding of orbifold $\mathrm{LG}$ $\mathrm{B}$-models: a case study

2018

Pure and Applied Mathematics Quarterly

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In this note we explore the variation of Hodge structures associated to the orbifold Landau-Ginzburg B-model whose superpotential has two variables. We extend the Getzler-Gauss-Manin connection to Hochschild chains twisted by group action. As an application, we provide explicit computations for the Getzler-Gauss-Manin connection on the universal (noncommutative) unfolding of Z 2 -orbifolding of A-type singularities.The result verifies an example of deformed version of Mckay correspondence.surfaces, a theory known as Fan-Jarvis-Ruan-Witten (FJRW) theory [7] for Landau-Ginzburg A-models. However, Saito's construction only involves 'un-orbifold' cases (A, W, G = 1 ), while the full mirror symmetry between Landau-Ginzburg models asks for all orbifold groups. This requires the construction and computation of Frobenius manifold structure on the aforementioned moduli space M. Barannikov [1,2] and introduced the important notion of (polarized) variation of semi-infinite Hodge structures (VSHS), generalizing Saito's framework to many other geometric contexts and non-commutative world [1,14]. Following this route, we shall consider the period cyclic homology of a deformed algebra of A W [G], with a Hodge filtration induced by the cyclic parameter u and the flat Gauss-Manin connection constructed by Getzler [10]. They give rise to a flat bundle over the moduli space M, carrying important data of Hodge filtration. In this note, we establish a version of the Getzler-Gauss-Manin connection via operations of G-twisted cochainsThis encodes the same information as the Getzler-Gauss-Manin connection on the deformation space of the algebra A [G], but is easier to compute in practice. As an application, we perform a case study for orbifold A-type singularity (A 2n−1 , Z 2 ). We find (see Theorem 4.1), Theorem. Consider an orbifold LG B-model (A, W, G) with A = C[x, y], W invertible and finite G ⊂ SL(2, C) acting diagonally on C 2 . The moduli space M of miniversal deformations of A W [G] is smooth, equipped with a variation of semi-infinite Hodge structures (VSHS) given by a flat vector bundle of period cyclic homologies. In this fashion, there is an isomorphism between the moduli spaces associated to (C[x, y], x 2n + y 2 , Z 2 ) and (C[z, w], z n + zw 2 , 1 ), which is compatible with the VSHS's on them.It can be seen as an example of Mckay correspondence for LG models [20], but involves the deformation data. Here, (C[x, y], x 2n + y 2 , Z 2 ) is associated to the A 2n−1singularity W = x 2n + y 2 on an orbifold X = C 2 /Z 2 and (C[z, w], z n + zw 2 , 1 ) isThere are three directions of generalizations of such a correspondence. One is for more general triples (A = O(X), W, G) as long as the crepant resolution of X/G exists and the lifting superpotential W has good Hodge theoretical properties (see [17] for a recent discussion on this model and references therein). The second is to establish the correspondence between VSHS's via crepant resolutions and related mirror symmetry. This involves a combination of LG/CY correspondence and ...

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Section: Computation Of D-typementioning

confidence: 99%

Unfolding of orbifold $\mathrm{LG}$ $\mathrm{B}$-models: a case study

2018

Pure and Applied Mathematics Quarterly

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“…We expect that the pairing η on HH c (A W , A W g) and HH c (A W , A W g −1 ) is the same as the residue pairing in Jac(W g ) as long as W g has an isolated singularity. Alternatively, there is categorical construction of pairing (Mukai pairing) from the dg-category of matrix factorization [5,44,45]. The Mukai pairing of G-equivariant case is explicitly computed in [37].…”

Section: )mentioning

confidence: 99%

“…Definition 3.9. For g ∈ G, we define the first order quantum differential operators ∂ g i ∈ C 1 (A, Ag (i) ) by 45) and the second order quantum differential operators…”

Section: Quantum Differential Operator and Cup Productmentioning

confidence: 99%

G-twisted Braces and Orbifold Landau–Ginzburg Models

2020

Commun. Math. Phys.

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Given an algebra with group G-action, we construct brace structures for its G-twisted Hochschild cochains. An an application, we construct G-Frobenius algebras for orbifold Landau-Ginzburg Bmodels and present explicit orbifold cup product formula for all invertible polynomials. be the Hochschild differential. Let HH • (A, A[G]) be the Hochschild cohomology with a natural cup product ∪. We show that ∪ satisfies a G-twisted commutativity relation (see also [46]) and the G-invariant

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“…(5) If C is proper and admits an n-dimensional weak proper Calabi-Yau structure, then (HC − • (C), ∇ GGM ) admits a natural polarization ·, · res of dimension n, given by Shklyarov's higher residue pairing [Shk13]. Remark 1.2 Let us comment on the originality of Theorem A.…”

Section: Hochschild Invariants Of a ∞ Categoriesmentioning

confidence: 99%

“…We recall the definition of the higher residue pairing given in [Shk13]. We recall that Shklyarov's definition of cyclic homology for dg categories is compatible with our convention for A ∞ categories (see Remark 3.20).…”

Section: Higher Residue Pairing On Dg Categoriesmentioning

confidence: 99%

Formulae in noncommutative Hodge theory

Sheridan

2019

J. Homotopy Relat. Struct.

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This paper is about noncommutative Hodge theory, in the sense of Kontsevich. We use Barannikov's language of 'variations of semi-infinite Hodge structures', abbreviated VSHS [Bar01]. We introduce various degenerate notions of VSHS. Using that language, our main theorem says that: to any A ∞ category C, one can associate its negative cyclic homology HC − • (C); Getzler's Gauss-Manin connection [Get93] endows HC − • (C) with the structure of an unpolarized pre-VSHS; if C is furthermore proper, then Shklyarov's higher residue pairing [Shk13] defines a polarization; and these structures are Morita invariant. If the category is smooth and the noncommutative Hodge-to-de Rham spectral sequence degenerates, then this polarized pre-VSHS is in fact a polarized VSHS. Much of this is well-known: the main point of the paper is to present complete proofs, and also formulae for all of the relevant structures in the A ∞ setting, with unified conventions. Explicit formulae are necessary as part of a project of Ganatra, Perutz and the author, to realize Barannikov's and Kontsevich's ideas and extract information about Gromov-Witten invariants from the Fukaya category [GPS15].All of these structures are Morita invariant. If C is proper, smooth and the noncommutative Hodge-to-de Rham spectral sequence degenerates, then the polarized pre-VSHS (HC − • (C), ∇, ·, · res ) is in fact a polarized VSHS.Remark 1.1 We recall that a conjecture of Kontsevich and Soibelmann [KS08, Conjecture 9.1.2] says that the noncommutative Hodge-to-de Rham spectral sequence degenerates for any proper, smooth C.

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Matrix factorizations and higher residue pairings

Cited by 15 publications

References 22 publications

Unfolding of orbifold $\mathrm{LG}$ $\mathrm{B}$-models: a case study

Unfolding of orbifold $\mathrm{LG}$ $\mathrm{B}$-models: a case study

G-twisted Braces and Orbifold Landau–Ginzburg Models

Formulae in noncommutative Hodge theory

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