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Open/closed correspondence for the projective line

Zhengyu Zong

TL;DR

The work proves an open/closed Gromov-Witten correspondence for the projective line by equating $S^1$-equivariant disk invariants of $(\mathbb{P}^1,L)$ with genus-zero closed invariants of a carefully constructed toric surface $X$. The authors formulate both open and closed theories via virtual localization and graph-sum expressions, then construct a precise graph-level matching that yields a new, non-Calabi–Yau instance of open/closed duality. Descendant insertions are included on both sides, and a non-equivariant limit recovers the stationary open invariants, highlighting robustness beyond noncompact Calabi–Yau geometries. The result broadens the scope of open/closed correspondences, providing a concrete framework for compact targets and suggesting pathways to generalizations in broader settings. Overall, the paper delivers a detailed localization-based bridge between open disk GW theory on $(\mathbb{P}^1,L)$ and closed GW theory on a toric surface $X$, enriching the understanding of how boundary data encodes closed-string information.

Abstract

We establish a correspondence between the disk invariants of the complex projective line $\bP^1$ with boundary condition specified by an $S^1$-invariant Lagrangian sub-manifold $L$ and the genus-zero closed Gromov-Witten invariants of a toric surface $X$.

Open/closed correspondence for the projective line

TL;DR

The work proves an open/closed Gromov-Witten correspondence for the projective line by equating -equivariant disk invariants of with genus-zero closed invariants of a carefully constructed toric surface . The authors formulate both open and closed theories via virtual localization and graph-sum expressions, then construct a precise graph-level matching that yields a new, non-Calabi–Yau instance of open/closed duality. Descendant insertions are included on both sides, and a non-equivariant limit recovers the stationary open invariants, highlighting robustness beyond noncompact Calabi–Yau geometries. The result broadens the scope of open/closed correspondences, providing a concrete framework for compact targets and suggesting pathways to generalizations in broader settings. Overall, the paper delivers a detailed localization-based bridge between open disk GW theory on and closed GW theory on a toric surface , enriching the understanding of how boundary data encodes closed-string information.

Abstract

We establish a correspondence between the disk invariants of the complex projective line with boundary condition specified by an -invariant Lagrangian sub-manifold and the genus-zero closed Gromov-Witten invariants of a toric surface .
Paper Structure (16 sections, 6 theorems, 54 equations)

This paper contains 16 sections, 6 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Historical background and motivation
  3. Open/closed correspondence for Calabi-Yau 3-folds
  4. Open/closed correspondence for the projective line
  5. Statement of the main result
  6. Overview of the paper
  7. Geometric setup
  8. Equivariant cohomology of $\mathbb{P}^1$
  9. Geometry and equivariant cohomology of $X$
  10. Open Gromov-Witten theory of $\mathbb{P}^1$
  11. Open Gromov-Witten invariants
  12. Virtual localization formula
  13. Gromov-Witten theory of $X$
  14. Equivariant Gromov-Witten invariants
  15. Graph sum formula
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\beta=(d_1,d_2)\in \mathbb{Z}_{\geq 0}^2,d_1\neq d_2$. Then for $\alpha_1,\cdots,\alpha_n\in\{1,2\},a_1,\cdots,a_n\in\mathbb{Z}_{\geq 0}$, we have Here the sum is taken over $\{((\mu^1_i,d^1_i,A^1_i,b_i)_{1\leq i\leq l},(\mu^2_j,d^2_j,A^2_j,c_j)_{1\leq j\leq m},)\mid l,m\geq 0, l+m\geq 1,\mu^1_1\neq \mu^2_1 \textrm{ when } l=m=1, \mu^1_i,\mu^2_j>0, d^1_i,d^2_j,b_i,c_j\geq 0, \sum_{i=1}^{l}(d

Theorems & Definitions (11)

  • Theorem 1.1: =Theorem \ref{['thm:correspondence']}
  • Definition 3.1
  • Proposition 3.2
  • Definition 4.1
  • Proposition 4.2
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Theorem 5.3
  • ...and 1 more