All genus open mirror symmetry for the projective line

TL;DR

The article establishes all-genus open mirror symmetry for the pair $(\mathbb{P}^1,\mathbb{R}\mathbb{P}^1)$ by equating the A-model open Gromov-Witten generating functions with the B-model Chekhov-Eynard-Orantin invariants on the mirror curve. It develops a comprehensive equivariant framework for both closed and open sectors, including S-, J-, and R-matrix formalisms, and derives graph-sum representations for descendant invariants. Central results show that disk and annulus invariants satisfy $F_{0,1} = -W_{0,1}$ and $F_{0,2} = -W_{0,2}$, with all higher-genus/descendant cases governed by a parallel A-B correspondence through mirror maps and SYZ duals. The work further connects the A-model fixed-locus computations to a geometric counting perspective via coherent boundary conditions, providing a recursive and computable algorithm for higher-genus open GW invariants and their geometric interpretations.

Abstract

We prove that the Chekhov-Eynard-Orantin recursion on the mirror curve of $\mathbb{P}^1$ encodes all genus equivariant open Gromov-Witten invariants of $(\mathbb{P}^1, \mathbb{R}\mathbb{P}^1)$. This result can be viewed as an all genus equivariant open mirror symmetry for $(\mathbb{P}^1, \mathbb{R}\mathbb{P}^1)$.

Figures (3)

• Figure 1: The left figure is an example of the graph $\vec{\Gamma}$$\in$$G_{g,n}(\mathbb{P}^1,\mathbb{R}\mathbb{P}^1|\beta',\vec{\mu})$. The right figure is an example of stable map corresponding to $\vec{\Gamma}$.
• Figure 2: SYZ $T$-dual of $\mathcal{E}$ in $\mathbb{C}$
• Figure 3: boundary contribution graph

Theorems & Definitions (44)

• Theorem 1.1: = Theorem \ref{['thm:stableMS']}
• Theorem 1.2: = Theorem \ref{['thm:SYZ-open-mirror']}
• Theorem 1.3: = Theorem \ref{['thm:geo-MS']}
• Remark 2.1
• Theorem 2.2
• Theorem 2.3: Giv98b
• Definition 3.1: Decorated graphs (a)
• Proposition 3.2
• Definition 3.3: Decorated graphs (b)
• Proposition 3.4