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

TL;DR

This paper proposes a sheaf-theoretic interpretation of genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds, connecting Gromov-Witten and Donaldson-Thomas theories and supporting the conjecture with computed examples.

Contribution

It introduces a conjectural sheaf-theoretic framework relating GV invariants to DT4 invariants on CY 4-folds, extending the understanding of enumerative invariants.

Findings

Computed examples for compact and non-compact CY 4-folds support the conjecture.

Proposed an equivariant version of the conjecture for local curves.

Verified the conjecture in specific cases.

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 $\mathrm{DT_{4}}$ 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.