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

Abstract

In analogy with the Gopakumar-Vafa (GV) conjecture on Calabi-Yau (CY) 3-folds, 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 $\mathrm{D}{\mathrm{T}}_4$ 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 $\mathrm{D}{\mathrm{T}}_3$ invariants on $Y$. Examples are computed for both compact and non-compact Fano 3-folds to support our conjecture.

15 pages, typos fixed, to appear in Commun. Contemp. Math

References in corpus (4)

• Enumerative geometry of Calabi-Yau 4-folds
• Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds
• Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds
• Gromov-Witten invariants of varieties with holomorphic 2-forms

Cited by in corpus (10)

• Curve counting and DT/PT correspondence for Calabi-Yau 4-folds
• Curve counting via stable objects in derived categories of Calabi-Yau 4-folds
• Stable pair invariants of local Calabi-Yau 4-folds
• Gopakumar-Vafa type invariants on Calabi-Yau 4-folds via descendent insertions
• Canonical vertex formalism in DT theory of toric Calabi-Yau 4-folds
• Counting perverse coherent systems on Calabi-Yau 4-folds
• Enumerative invariants and wall-crossing formulae in abelian categories
• Cosection Localization and Vanishing for Virtual Fundamental Classes of D-Manifolds
• Open/Closed BPS Correspondence and Integrality
• Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local Calabi-Yau 4-folds over V_5 and V_22