Kodaira-Spencer map, Lagrangian Floer theory and orbifold Jacobian algebras

Cho, Cheol-Hyun; Lee, Sangwook

doi:10.48550/arxiv.2007.11732

Cited by 1 publication

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“…By taking a tensor product of Hom C H (K , K ) with the group ring C[H * ], and extending the A ∞ -operation suitably, we obtain a semi-direct product A ∞ -category C H H * , which can be canonically identified with C . More precisely, the component Hom C H (K , K ) ⊗ χ in the morphism space can be identified with the χ-eigenspace of H -action on C (see [CL20] for an explicit identification). Geometrically, it means that A ∞ -operations in C H actually come from C .…”

Section: Choose a Cutoff Function ψ With ψ(Tmentioning

confidence: 99%

“…Since our Milnor fiber is non-compact, we need to consider symplectic cohomology instead of quantum cohomology ( [Sei06]). Following the general definition proposed in [CL20], it is natural to define Kodaira-Spencer map in our case as y, z], m b,b 1 ) where we consider closed-open map with boundary on (L, b). Here, the target of the map is nothing by Koszul complex of W L .…”

Section: Example 93 Let Us Look At the Explicit Lagrangian L For Ferm...mentioning

confidence: 99%

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Fukaya category for Landau-Ginzburg orbifolds and Berglund-Hübsch conjecture for invertible curve singularities

Cho¹,

Choa²,

Jeong³

2020

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For a weighted homogeneous polynomial and a choice of a diagonal symmetry group, we define a new Fukaya category based on the wrapped Fukaya category of its Milnor fiber together with monodromy information. It is analogous to the variation operator in singularity theory. As an application, we formulate a full version of Berglund-Hübsch homological mirror symmetry and prove it for the case of two variables. Namely, given one of the polynomials W = x p + y q , x p + x y q , x p y + x y q and a symmetry group G, we use Floer theoretic construction to obtain the transpose polynomial W T with the transpose symmetry group G T as well as derived equivalence between the new Fukaya category of (W,G) and the matrix factorization category of (W T ,G T ). In this case, monodromy corresponds to the restriction of LG model to a hypersurface in the mirror. For ADE singularities, Auslander-Reiten quivers for indecomposable matrix factorizations were known from 80's, and we find the corresponding Lagrangians as well as surgery exact triangles. CONTENTS 1. Introduction 1 2. Quantum cap action and the new A ∞ -category 8 3. Mirror counterpart: Hypersurface restriction 16 4. A ∞ -category for a weighted homogeneous polynomial with a symmetry group 20 5. Invertible singularities and Berglund-Hübsch conjecture 27 6. Equivariant topology of Milnor fibers for invertible curve singularities 29 7. Floer theory for Milnor fiber quotients and localized mirror functor 36 8. Homological mirror symmetry for Milnor fibers (without monodromy action) 41 9. Berglund-Hübsch homological mirror symmetry for invertible curve singularities 49 10. Relation to Auslander-Reiten theory of Cohen-Macaulay modules 60 Appendix A. Moduli space of pseudo-holomorphic curves and perturbations 78 Appendix B. Compactifications of popsicle moduli spaces 82 Appendix C. Determination of Auslander-Reiten sequence 85 References 87

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Section: Choose a Cutoff Function ψ With ψ(Tmentioning

confidence: 99%

Section: Example 93 Let Us Look At the Explicit Lagrangian L For Ferm...mentioning

confidence: 99%