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On Gromov--Witten invariants of $\mathbb{P}^1$

Boris Dubrovin, Di Yang

TL;DR

The paper proposes a conjectural explicit generating series (Main Conjecture) for Gromov–Witten invariants of $\mathbb{P}^1$ across all degrees and genera, focusing on the stationary sector and encoding invariants in a matrix-resolvent formalism. An algorithm is developed that computes these invariants recursively via matrix series $R^{\bf b}_{K}(\lambda;\epsilon)$ built from a base resolvent $\mathcal{R}(\lambda;\epsilon)$, yielding two- and multi-point correlator expressions in terms of traces of products of $R$-matrices. The authors verify the conjecture in key cases (notably degree 1, and Hurwitz-number cases for $b=1$), provide new data for analogues of polygon numbers, and derive large-genus asymptotics, highlighting integrality patterns and connections to classical enumerative problems. They also discuss the Toda conjecture as a foundational framework, contrasting weak and strong forms and noting partial confirmations and the potential for broader integration with topological recursion and the Dubrovin–Zhang program. Overall, the work proposes a unifying, computable description of $\mathbb{P}^1$ GW invariants with implications for integrable hierarchies and enumerative geometry.

Abstract

We propose a conjectural explicit formula of generating series of a new type for Gromov--Witten invariants of $\mathbb{P}^1$ of all degrees in full genera.

On Gromov--Witten invariants of $\mathbb{P}^1$

TL;DR

The paper proposes a conjectural explicit generating series (Main Conjecture) for Gromov–Witten invariants of across all degrees and genera, focusing on the stationary sector and encoding invariants in a matrix-resolvent formalism. An algorithm is developed that computes these invariants recursively via matrix series built from a base resolvent , yielding two- and multi-point correlator expressions in terms of traces of products of -matrices. The authors verify the conjecture in key cases (notably degree 1, and Hurwitz-number cases for ), provide new data for analogues of polygon numbers, and derive large-genus asymptotics, highlighting integrality patterns and connections to classical enumerative problems. They also discuss the Toda conjecture as a foundational framework, contrasting weak and strong forms and noting partial confirmations and the potential for broader integration with topological recursion and the Dubrovin–Zhang program. Overall, the work proposes a unifying, computable description of GW invariants with implications for integrable hierarchies and enumerative geometry.

Abstract

We propose a conjectural explicit formula of generating series of a new type for Gromov--Witten invariants of of all degrees in full genera.

Paper Structure

This paper contains 12 sections, 7 theorems, 46 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Acknowledgements
  4. An algorithm for computing GW invariants of $\mathbb{P}^1$
  5. Examples
  6. Degree $1$ GW invariants of $\mathbb{P}^1$.
  7. Analogues of polygon numbers
  8. The case $b=0.$
  9. The case $b=1.$
  10. Further examples. Some large genus asymptotics.
  11. Concluding remarks
  12. Toda Conjecture.

Key Result

Lemma 2

In the particular case of $b_1=b_2=b_3=\dots=b$ we have Moreover, the following formulae hold true for $R^b_m(\lambda;\epsilon),\,m\geq 1$

Theorems & Definitions (13)

  • Definition 1
  • Conjecture 2
  • Remark 3
  • Remark 4
  • Example 5
  • Definition 1
  • Lemma 2
  • Proposition 3: *
  • Proposition 1: *
  • Proposition 2: *
  • ...and 3 more