Genus zero Gopakumar–Vafa type invariants for Calabi–Yau 4-folds

Cao, Yalong; Maulik, Davesh; Toda, Yukinobu

doi:10.1016/j.aim.2018.08.013

Cited by 44 publications

(77 citation statements)

References 66 publications

(181 reference statements)

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“…For instance, we can use DT 4 invariants count one dimensional stable sheaves supported on conics inside X, as well as Gromov-Witten invariants count stable maps to conics in X. In this section, we compare them and verify a conjectural relation [4] between GW invariants and DT 4 invariants for one dimensional stable sheaves in this setting.…”

Section: Counting Conics On Sextic 4-foldsmentioning

confidence: 99%

“…The following conjecture is proposed in [4] as an interpretation of Klemm-Pandharipande's Gopakumar-Vafa type invariants [10] on CY 4-folds in terms of DT 4 invariants of one dimensional stable sheaves.…”

Section: 1mentioning

confidence: 99%

“…The proof of the above result relies on the study of the Hilbert scheme of conics on X. By the deformation invariance of DT 4 and GW invariants, we may assume X to be a generic hypersurface in P 5 , so that M 2l is smooth of expected dimension whose closed point is the structure sheaf of a smooth conic or a pair of distinct intersecting lines (i.e. there is no double line) (ref.…”

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Counting conics on sextic $4$-folds

Cao

2019

Mathematical Research Letters

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We study rational curves of degree two on a smooth sextic 4-fold and their counting invariant defined using Donaldson-Thomas theory of Calabi-Yau 4-folds. By comparing it with the corresponding Gromov-Witten invariant, we verify a conjectural relation between them proposed by the author, Maulik and Toda. IntroductionLet X be a smooth sextic 4-fold in P 5 and M β be the moduli scheme of one dimensional stable sheaves on X with Chern character (0, 0, 0, β, 1). We are interested in the counting invariant of M β defined using Donaldson-Thomas theory of Calabi-Yau 4-folds, introduced in [1, 2]. In particular, there exists a virtual classAnd we may use insertions to define counting invariants: for a class γ ∈ H 4 (X, Z), let, where π X , π M are projections from X ×M β onto corresponding factors and ch 3 (E) is the Poincaré dual to the fundamental class of the universal sheaf E.The degree matches and we define DT 4 invariants as followsSince the definition of the virtual class involves a choice of orientation on certain (real) line bundle over M β , the invariant will also depend on that (see Sect. 2.1 for more detail). Another obvious way to enumerating curves on X is by GW theory. More specifically, for γ ∈ H 4 (X, Z), the genus 0 Gromov-Witten invariant of X iswhere ev : M 0,1 (X, β) → X is the evaluation map.In a previous work [4], the author, Maulik and Toda proposed a conjectural relation between DT 4 invariants for one dimensional stable sheaves on X and genus zero GW invariants of X (see Conjecture 2.3 for details). The main result of this note is to verify this conjecture for degree two curve class on a smooth sextic 4-fold.Theorem 0.1. (Theorem 2.4) Conjecture 2.3 is true for degree one and two classes of a smooth sextic 4-fold X ⊆ P 5 , i.e. for the line class l ∈ H 2 (X, Z) and any γ ∈ H 4 (X), we have GW 0,l (γ) = DT 4 (l | γ), GW 0,2l (γ) = DT 4 (2l | γ) + 1 4 · DT 4 (l | γ),for certain choice of orientation in defining the RHS.

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Section: Counting Conics On Sextic 4-foldsmentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

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Counting conics on sextic $4$-folds

Cao

2019

Mathematical Research Letters

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“…Note that there is a selection rule n i=1 p i = dim X 1 + n − 3 to realize non-trivial genus-0 n-point correlators O h p 1 · · · O h pn P 1 arising from the index theorem. The number n d (h 1 , h k , h n−r−k−2 ) in (3.12) is an integer and enumerates the number of degree d holomorphic maps intersecting with the cycles dual to h, h k , and h n−r−k−2 [30,31,33] (see also [34,35]).…”

Section: Complete Intersections In P N−1mentioning

confidence: 99%

Determinantal Calabi-Yau varieties in Grassmannians and the Givental I-functions

Honma

Manabe

2018

J. High Energ. Phys.

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We examine a class of Calabi-Yau varieties of the determinantal type in Grassmannians and clarify what kind of examples can be constructed explicitly. We also demonstrate how to compute their genus-0 Gromov-Witten invariants from the analysis of the Givental Ifunctions. By constructing I-functions from the supersymmetric localization formula for the two dimensional gauged linear sigma models, we describe an algorithm to evaluate the genus-0 A-model correlation functions appropriately. We also check that our results for the Gromov-Witten invariants are consistent with previous results for known examples included in our construction. † yhonma@law.meijigakuin.ac.jp ‡

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“…By Corollary 2.7, we are reduced to prove the Katz's conjecture for K S . This is done by combining DT/PT/GW correspondence and geometric vanishing in [5, Corollary A.7. ].Remark 2.9.…”

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Genus zero Gopakumar–Vafa type invariants for Calabi–Yau 4-folds II: Fano 3-folds

Cao

2019

Commun. Contemp. Math.

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In analogy with the Gopakumar-Vafa (GV) conjecture on Calabi-Yau (CY) 3folds, Klemm and Pandharipande defined GV type invariants on Calabi-Yau 4-folds using Gromov-Witten theory and conjectured their integrality. In a joint work with Maulik and Toda, the author conjectured their genus zero invariants are DT 4 invariants of one dimensional stable sheaves. In this paper, we study this conjecture on the total space of canonical bundle of a Fano 3-fold Y , which reduces to a relation between twisted GW and DT 3 invariants on Y . Examples are computed for both compact and non-compact Fano 3-folds to support our conjecture.Although K S is non-compact, the moduli spaces used to define GW/ DT 3 invariants are compact, so Katz's conjecture still makes sense on K S . By Proposition 2.5, it is easy to show: Corollary 2.7. In the setting of Proposition 2.5, Conjecture 1.2 is true for Y = S × P 1 with β ∈ H 2 (S, Z) ⊆ H 2 (Y, Z) if and only if Katz's conjecture holds for K S with β ∈ H 2 (S, Z).Combining with the previous checks of Katz's conjecture, we obtain:Theorem 2.8. Let S be a toric del-Pezzo surface and Y = S × P 1 be the product. Then Conjecture 1.2 is true for β ∈ H 2 (S, Z) ⊆ H 2 (Y, Z).Proof. By Corollary 2.7, we are reduced to prove the Katz's conjecture for K S . This is done by combining DT/PT/GW correspondence and geometric vanishing in [5, Corollary A.7.].Remark 2.9. When Y = O P 1 (−1) × P 1 , Conjecture 1.2 is true and can be reduced to the Aspinwall-Morrison multiple cover formula GW 0,d (X) = d|n 1 d 3 , DT 3 (X)(d) = 1, if d = 1 ; DT 3 (X)(d) = 0, if d > 1, for X = O P 1 (−1, −1).

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Genus zero Gopakumar–Vafa type invariants for Calabi–Yau 4-folds

Cited by 44 publications

References 66 publications

Counting conics on sextic $4$-folds

Counting conics on sextic $4$-folds

Determinantal Calabi-Yau varieties in Grassmannians and the Givental I-functions

Genus zero Gopakumar–Vafa type invariants for Calabi–Yau 4-folds II: Fano 3-folds

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