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Degree two Gopakumar-Vafa invariants of local curves

Ben Davison, Naoki Koseki

TL;DR

The paper develops a detailed framework to compute degree-two Gopakumar–Vafa invariants for local curves Tot_C(N) with a rank-two bundle N and canonical determinant, using a two-part strategy tied to Hitchin fibrations and nearby hyperplanes. For genus-two bases, the authors carry out an explicit stratification of the Hitchin base, perform a degeneracy-locus calculation, and determine fibrewise GV contributions via HOMFLY polynomials, culminating in exact n_{g,2[C]} values and confirming GV/GW in this case. They also present a higher-genus analysis, deriving vanishing and leading nonzero terms for n_{g,2[C]}(Tot_C(L)) and establishing GV/GW correspondences in broader genus regimes with a perverse-sheaf framework. The results hinge on a concrete realization of moduli spaces as critical loci, explicit cohomology computations for M_L(2,1), and a careful tally of spectral-curve degeneracies, including generalized de Jonquières divisors. Overall, the work advances the GV/GW program for local curves by providing both exact genus-two invariants and structured higher-genus evidence via Hitchin-theoretic and degeneracy-locus methods.

Abstract

We investigate the Gopakumar-Vafa (GV) theory of local curves, namely, the total spaces of rank two vector bundles with canonical determinant on smooth projective curves. Under a certain genericity condition on the rank two bundles, we propose a general mechanism to compute the degree two GV invariants of local curves. In particular, we determine all the degree two GV invariants when the base curve has genus two. Combined with previous work by Bryan and Pandharipande, we obtain the GV/GW correspondence in this case. When the base curve has genus greater than two, we calculate GV invariants for some extremal genera, providing evidence for the GV/GW conjecture for curves of higher genus.

Degree two Gopakumar-Vafa invariants of local curves

TL;DR

The paper develops a detailed framework to compute degree-two Gopakumar–Vafa invariants for local curves Tot_C(N) with a rank-two bundle N and canonical determinant, using a two-part strategy tied to Hitchin fibrations and nearby hyperplanes. For genus-two bases, the authors carry out an explicit stratification of the Hitchin base, perform a degeneracy-locus calculation, and determine fibrewise GV contributions via HOMFLY polynomials, culminating in exact n_{g,2[C]} values and confirming GV/GW in this case. They also present a higher-genus analysis, deriving vanishing and leading nonzero terms for n_{g,2[C]}(Tot_C(L)) and establishing GV/GW correspondences in broader genus regimes with a perverse-sheaf framework. The results hinge on a concrete realization of moduli spaces as critical loci, explicit cohomology computations for M_L(2,1), and a careful tally of spectral-curve degeneracies, including generalized de Jonquières divisors. Overall, the work advances the GV/GW program for local curves by providing both exact genus-two invariants and structured higher-genus evidence via Hitchin-theoretic and degeneracy-locus methods.

Abstract

We investigate the Gopakumar-Vafa (GV) theory of local curves, namely, the total spaces of rank two vector bundles with canonical determinant on smooth projective curves. Under a certain genericity condition on the rank two bundles, we propose a general mechanism to compute the degree two GV invariants of local curves. In particular, we determine all the degree two GV invariants when the base curve has genus two. Combined with previous work by Bryan and Pandharipande, we obtain the GV/GW correspondence in this case. When the base curve has genus greater than two, we calculate GV invariants for some extremal genera, providing evidence for the GV/GW conjecture for curves of higher genus.
Paper Structure (28 sections, 25 theorems, 123 equations)

This paper contains 28 sections, 25 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Motivation and results
  3. GV invariants of a local curve and strategy of the proofs
  4. Relation to previous work
  5. Plan of the paper
  6. Preliminaries
  7. Twisted Higgs bundles
  8. Rigidity of vector bundles
  9. Hitchin fibrations and Gopakumar--Vafa invariants
  10. Genus two
  11. Stratification of the Hitchin base
  12. $\lambda=(6)$
  13. $\lambda=(5, 1)$
  14. $\lambda=(4, 2)$
  15. $\lambda=(4, 1, 1)$
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $C$ be a smooth projective curve of genus two, $L \in \mathop{\mathrm{Pic}}\nolimits^3(C)$ a generic line bundle. Let $[N] \in \mathbb{P} \mathop{\mathrm{Ext}}\nolimits^1(L, L^{-1} \otimes \omega_C)$ be a generic rank two vector bundle on $C$, and let $X=\mathop{\mathrm{Tot}}\nolimits_C(N)$. The In particular, Conjecture conj:GVGW holds in this case.

Theorems & Definitions (57)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • ...and 47 more