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Unramified Gromov-Witten and Gopakumar-Vafa invariants

Denis Nesterov

TL;DR

The paper develops an epsilon-parameterized unramified Gromov–Witten theory using Fulton–MacPherson degenerations and Zhou’s entanglement framework to establish a wall-crossing formula that relates standard GW and unramified GW invariants. By analyzing these wall-crossings in threefolds, the authors prove Pandharipande’s conjecture that unramified GW invariants equal Gopakumar–Vafa invariants for Fano and primitive Calabi–Yau classes, providing an algebro-geometric construction of GV invariants in these cases. The approach hinges on master spaces, relative/log variants, and a detailed localization analysis that connects GW-type data to Hodge integrals on moduli spaces of curves, yielding divisor and dilaton equations in the relative setting. The results extend the dictionary between GW theory and GV theory beyond one-dimensional targets, offering new tools to compute and interpret GV invariants in higher dimensions for key classes. Overall, the work bridges stable map moduli, log geometry, and enumerative invariants to realize GV counts via unramified and relative GW data on threefolds.

Abstract

Kim, Kresch and Oh defined unramified Gromov-Witten invariants. For a threefold, Pandharipande conjectured that they are equal to Gopakumar-Vafa invariants (BPS invariants) in the case of Fano classes and primitive Calabi-Yau classes. We prove the conjecture using a wall-crossing technique. This provides an algebro-geometric construction of Gopakumar-Vafa invariants in these cases.

Unramified Gromov-Witten and Gopakumar-Vafa invariants

TL;DR

The paper develops an epsilon-parameterized unramified Gromov–Witten theory using Fulton–MacPherson degenerations and Zhou’s entanglement framework to establish a wall-crossing formula that relates standard GW and unramified GW invariants. By analyzing these wall-crossings in threefolds, the authors prove Pandharipande’s conjecture that unramified GW invariants equal Gopakumar–Vafa invariants for Fano and primitive Calabi–Yau classes, providing an algebro-geometric construction of GV invariants in these cases. The approach hinges on master spaces, relative/log variants, and a detailed localization analysis that connects GW-type data to Hodge integrals on moduli spaces of curves, yielding divisor and dilaton equations in the relative setting. The results extend the dictionary between GW theory and GV theory beyond one-dimensional targets, offering new tools to compute and interpret GV invariants in higher dimensions for key classes. Overall, the work bridges stable map moduli, log geometry, and enumerative invariants to realize GV counts via unramified and relative GW data on threefolds.

Abstract

Kim, Kresch and Oh defined unramified Gromov-Witten invariants. For a threefold, Pandharipande conjectured that they are equal to Gopakumar-Vafa invariants (BPS invariants) in the case of Fano classes and primitive Calabi-Yau classes. We prove the conjecture using a wall-crossing technique. This provides an algebro-geometric construction of Gopakumar-Vafa invariants in these cases.
Paper Structure (41 sections, 36 theorems, 322 equations, 3 figures)

This paper contains 41 sections, 36 theorems, 322 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Gopakumar--Vafa invariants
  4. Integrality and finiteness
  5. Ciocan-Fontanine--Kim--Zhou formula
  6. Organisation of the paper
  7. Acknowledgments
  8. Conventions
  9. Moduli spaces of unramified maps
  10. Fulton--MacPherson degenerations
  11. Blow-ups and contractions
  12. Unramified maps
  13. Moduli spaces of $\epsilon$-unramified maps
  14. Weight functions
  15. Definition of $\epsilon$-unramified maps
  16. ...and 26 more sections

Key Result

Theorem 1

Let $X$ be a smooth projective threefold. There is a wall-crossing formula relating standard Gromov--Witten (GW) invariants and unramified Gromov--Witten (uGW) invariants of $X$ in a class $\beta$,

Figures (3)

  • Figure 1: Tree
  • Figure 2: Tree with markings
  • Figure 3: Depiction of $n$-unstable components

Theorems & Definitions (69)

  • Theorem : Theorem \ref{['pand']}
  • Corollary
  • Definition 2.1
  • Definition 2.2: KKO
  • Definition 2.3
  • Theorem 2.4: KKO
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 59 more