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On one-leg orbifold topological vertex in refined Gromov-Witten theory

Jinghao Yu, Zhengyu Zong

Abstract

We define the one-leg orbifold topological vertex in refined Gromov-Witten theory \cite{BS24}. There are two cases where the leg is effective or gerby. The main result of this paper is the computation of the effective case. In the smooth case, this result matches the one-leg refined vertex in \cite{IKV09}. As an application, we compute the refined Gromov-Witten invariants of the local football.

On one-leg orbifold topological vertex in refined Gromov-Witten theory

Abstract

We define the one-leg orbifold topological vertex in refined Gromov-Witten theory \cite{BS24}. There are two cases where the leg is effective or gerby. The main result of this paper is the computation of the effective case. In the smooth case, this result matches the one-leg refined vertex in \cite{IKV09}. As an application, we compute the refined Gromov-Witten invariants of the local football.
Paper Structure (39 sections, 6 theorems, 129 equations, 5 figures)

This paper contains 39 sections, 6 theorems, 129 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Historical background
  3. Topological vertex and its orbifold generalization
  4. Refined topological vertex
  5. One-leg orbifold refined topological vertex
  6. The effective case
  7. The gerby case
  8. Statement of the main result
  9. Overview of the paper
  10. Mariño-Vafa formula in orbifold Gromov-Witten theory
  11. One-leg orbifold topological vertex in refined Gromov-Witten theory
  12. Moduli spaces and $\mathbb{C}^*$-actions
  13. The branch morphism and double Hurwitz numbers
  14. The obstruction bundle
  15. Refined relative Gromov-Witten invariants
  16. ...and 24 more sections

Key Result

Theorem 1.1

Let $t=e^{-\epsilon_1 u}, q_l=\xi_a^{-1}e^{-\sum_{i=1}^{a-1}\frac{w_a^{-2il}}{a}(w_a^i-w_a^{-i})x_i},l=1, \cdots,a-1$, where $\xi_a=e^{\frac{2\pi i}{a}},w_a=e^{\frac{\pi i}{a}}$. Then in the framing condition $\tau = \epsilon_1$, we have where $\tilde{t}_{\bullet} = (\tilde{T},\tilde{T}q_{a-1},\dots, \tilde{T}q_1\cdots q_{a-1})$ and $\tilde{T} = (1,t,t^2,t^3,\dots)$. In the smooth case (when $a=

Figures (5)

  • Figure 1: The weight diagram of $Z$
  • Figure 2: The weight diagram of $\mathcal{Y}$
  • Figure 3: The weight diagram of $Z$
  • Figure 4: refined topological vertex
  • Figure 5: refined vertex attached at $p_\infty$

Theorems & Definitions (9)

  • Theorem 1.1: = Theorem \ref{['thm:refine-vertex']}
  • Theorem 2.1: Zong15
  • Theorem 2.2: Generalized Mariño-Vafa formula Zong15
  • Proposition 3.1: Liu-Liu-Zhou2
  • Proposition 3.2: Burnside formula Dij
  • Theorem 5.1
  • Remark 5.2
  • Remark 6.1
  • Definition 6.1