The Gopakumar-Vafa formula for symplectic manifolds

TL;DR

This paper proves the Gopakumar-Vafa formula, relating Gromov-Witten invariants to BPS numbers, for symplectic Calabi-Yau 6-manifolds, extending the conjecture beyond complex algebraic settings.

Contribution

It establishes the Gopakumar-Vafa formula within symplectic geometry for a broad class of manifolds, including all symplectic Calabi-Yau 6-manifolds.

Findings

Gopakumar-Vafa formula holds for symplectic Calabi-Yau 6-manifolds

Extension to all symplectic 6-manifolds and genus zero invariants

Validation of the conjecture using symplectic Gromov-Witten theory

Abstract

The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa formula holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.