The Gopakumar-Vafa finiteness conjecture

TL;DR

This paper proves the finiteness aspect of the Gopakumar-Vafa conjecture, establishing that BPS invariants of symplectic 6-manifolds are finite in number and vanish beyond certain classes, using geometric measure theory techniques.

Contribution

It provides the first proof of the finiteness part of the Gopakumar-Vafa conjecture, extending previous results on integrality by employing a modified cluster formalism.

Findings

Finiteness of BPS invariants in symplectic 6-manifolds established

Method involves a modification of Ionel and Parker's cluster formalism

Utilizes geometric measure theory to prove the conjecture

Abstract

The Gopakumar-Vafa conjecture predicts that the BPS invariants of a symplectic 6-manifold, defined in terms of the Gromov-Witten invariants, are integers and all but finitely many vanish in every homology class. The integrality part of this conjecture was proved earlier by Ionel and Parker. This article proves the finiteness part. The proof relies on a modification of Ionel and Parker's cluster formalism using results from geometric measure theory.