Stable pairs and Gopakumar-Vafa type invariants on holomorphic symplectic 4-folds

TL;DR

This paper introduces a sheaf-theoretic interpretation of Gopakumar-Vafa type invariants for holomorphic symplectic 4-folds, extending the analogy from Calabi-Yau 3-folds and utilizing moduli spaces of stable pairs.

Contribution

It provides a sheaf-theoretic framework for Gopakumar-Vafa type invariants on holomorphic symplectic 4-folds, building on reduced Gromov-Witten theory.

Findings

Sheaf-theoretic interpretation of invariants

Connection to moduli spaces of stable pairs

Extension of invariants to holomorphic symplectic 4-folds

Abstract

As an analogy to Gopakumar-Vafa conjecture on Calabi-Yau 3-folds, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold $X$ using Gromov-Witten theory. When $X$ is holomorphic symplectic, Gromov-Witten invariants vanish and one can consider the corresponding reduced theory. In a companion work, we propose a definition of Gopakumar-Vafa type invariants for such a reduced theory. In this paper, we give them a sheaf theoretic interpretation via moduli spaces of stable pairs.