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

Abstract

In analogy with the Gopakumar-Vafa conjecture on CY 3-folds, Klemm and Pandharipande defined GV type invariants on Calabi-Yau 4-folds using Gromov-Witten theory and conjectured their integrality. In this paper, we propose a sheaf-theoretic interpretation of their genus zero invariants using Donaldson-Thomas theory on CY 4-folds. More specifically, we conjecture genus zero GV type invariants are $\mathrm{D}{\mathrm{T}}_4$ invariants for one-dimensional stable sheaves on CY 4-folds. Some examples are computed for both compact and non-compact CY 4-folds to support our conjectures. We also propose an equivariant version of the conjectures for local curves and verify them in certain cases.

30 pages

References in corpus (1)

• Enumerative geometry of Calabi-Yau 4-folds

Cited by in corpus (25)

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• Stable maps to Looijenga pairs
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• Gopakumar-Vafa type invariants on Calabi-Yau 4-folds via descendent insertions
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• Determinantal Calabi-Yau varieties in Grassmannians and the Givental $I$-functions
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