The Gopakumar–Vafa finiteness conjecture

From To appear in forthcoming issues by Aleksander Doan, Eleny-Nicoleta Ionel, Thomas Walpuski

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.

Keywords
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Mathematical Subject Classification

Primary 2020: Secondary:

Milestones
Received: 2021-03-17
Revised: 2025-10-07
Accepted: 2025-11-25

Authors

Aleksander Doan

Department of Mathematics, University College London, London, UK

Eleny-Nicoleta Ionel

Department of Mathematics, Stanford, University, Stanford, CA, USA

Thomas Walpuski

Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany