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Mirror symmetry and open/closed correspondence for the projective line

Jinghao Yu, Zhengyu Zong

Abstract

We study the open/closed correspondence for the projective line via mirror symmetry. More explicitly, we establish a correspondence between the generating function of disk Gromov-Witten invariants of the complex projective line $\mathbb{P}^1$ with boundary condition specified by an $S^1$-invariant Lagrangian sub-manifold $L$ and the asymptotic expansion of the $I$-function of a toric surface $\mathcal{S}$.

Mirror symmetry and open/closed correspondence for the projective line

Abstract

We study the open/closed correspondence for the projective line via mirror symmetry. More explicitly, we establish a correspondence between the generating function of disk Gromov-Witten invariants of the complex projective line with boundary condition specified by an -invariant Lagrangian sub-manifold and the asymptotic expansion of the -function of a toric surface .

Paper Structure

This paper contains 30 sections, 6 theorems, 98 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Historical background and motivation
  3. Open/closed correspondence for Calabi-Yau 3-folds
  4. Mirror symmetry and open/closed correspondence for the projective line
  5. Construction of toric Calabi-Yau manifold/orbifolds in open/closed correspondence
  6. Relation to open Gromov-Witten invariants via coherent boundary conditions
  7. Statement of the main result
  8. Overview of the paper
  9. Geometric setup
  10. Equivariant cohomology of $\mathbb{P}^1$
  11. The geometry of toric surface $\mathcal{S}$
  12. Comparison of geometry
  13. Gromov-Witten theory of $\mathbb{P}^1$
  14. Equivariant Gromov-Witten invariants of $\mathbb{P}^1$
  15. $S^1$-fixed locus and decorated graphs
  16. ...and 15 more sections

Key Result

Theorem 1.1

Under the relation $\log q_0 = t^0$, $q_1 = -\sqrt{q}X^{-1}$ and $q_2 = -\sqrt{q}X$, we have where the $I$-function is in the asymptotic expansion as $\mathsf{v} \rightarrow \infty$, and the exceptional term is $\text{Exc} := -\sqrt{q}X^{-1} + \sqrt{q}X- \frac{(t^0)^2}{2\mathsf{v}} - q\mathsf{v}^{-1}$.

Figures (3)

  • Figure 1: Interrelations among the mentioned topics
  • Figure 2: open/closed correspondence and mirror symmetry
  • Figure 3: The fan of $\Sigma$ and 1-skeleton of $\mathcal{S}$

Theorems & Definitions (11)

  • Theorem 1.1: =Theorem \ref{['thm:main-thm-i']}
  • Remark 2.1
  • Definition 3.1: Decorated graphs
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Theorem 4.1
  • Remark 4.2
  • Theorem 5.1
  • proof
  • ...and 1 more