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The Gopakumar-Vafa finiteness conjecture

Aleksander Doan, Eleny-Nicoleta Ionel, Thomas Walpuski

TL;DR

This work proves the finiteness part of the Gopakumar--Vafa conjecture for Calabi–Yau classes on closed symplectic 6-manifolds by replacing Gromov compactness of maps with compactness for J-holomorphic cycles and employing geometric measure theory. It introduces an upgraded cluster formalism based on the cycle space, establishes compactness and openness results for embedded cycles, and analyzes wall-crossing via a cluster isotopy theorem to show that cluster contributions remain finite and computable. The paper then shows that Gromov--Witten invariants in Calabi--Yau classes admit a representation in terms of integer BPS-type coefficients with finite genus support, mirroring the structure predicted by GV. It also sketches the Fano-class analogue and Castelnuovo-type bounds, highlighting the broader applicability of the cycle-based approach to enumerative symplectic geometry.

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.

The Gopakumar-Vafa finiteness conjecture

TL;DR

This work proves the finiteness part of the Gopakumar--Vafa conjecture for Calabi–Yau classes on closed symplectic 6-manifolds by replacing Gromov compactness of maps with compactness for J-holomorphic cycles and employing geometric measure theory. It introduces an upgraded cluster formalism based on the cycle space, establishes compactness and openness results for embedded cycles, and analyzes wall-crossing via a cluster isotopy theorem to show that cluster contributions remain finite and computable. The paper then shows that Gromov--Witten invariants in Calabi--Yau classes admit a representation in terms of integer BPS-type coefficients with finite genus support, mirroring the structure predicted by GV. It also sketches the Fano-class analogue and Castelnuovo-type bounds, highlighting the broader applicability of the cycle-based approach to enumerative symplectic geometry.

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.

Paper Structure

This paper contains 16 sections, 45 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. The space of pseudo-holomorphic cycles
  4. Definitions and results
  5. The monotonicity formula
  6. Allard's regularity theorem
  7. Convergence of submanifolds
  8. Convergence of embedded pseudo-holomorphic curves
  9. The proof of the Gopakumar--Vafa conjecture
  10. Gromov--Witten invariants of symplectic $6$--manifolds
  11. Cluster formalism
  12. Proof of the cluster isotopy theorem
  13. Contributions of super-rigid curves
  14. Conclusion of the proof of the Gopakumar--Vafa conjecture
  15. The Gopakumar–Vafa conjecture for Fano classes
  16. ...and 1 more sections

Key Result

theorem 1.7

Conj_GopakumarVafa holds.

Theorems & Definitions (119)

  • Conjecture 1.2: The Gopakumar--Vafa conjecture Gopakumar1998Gopakumar1998a[Conjecture 1.2]Bryan2001
  • remark 1.3
  • remark 1.4
  • remark 1.5
  • theorem 1.7
  • example 2.2
  • Definition 2.3
  • theorem 2.4: Blaschke1956 Blaschke1956[Theorem 7.3.8]Burago2001
  • remark 2.5
  • Definition 2.6
  • ...and 109 more