On Gromov--Witten invariants of $\mathbb{P}^1$-orbifolds and topological difference equations
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918086727172096 |
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| author | Huang, Zhengfei Yang, Di |
| author_facet | Huang, Zhengfei Yang, Di |
| contents | Let $(m_1, m_2)$ be a pair of positive integers. Denote by $\mathbb{P}^1$ the complex projective line, and by $\mathbb{P}^1_{m_1,m_2}$ the orbifold complex projective line obtained from $\mathbb{P}^1$ by adding $\mathbb{Z}_{m_1}$ and $\mathbb{Z}_{m_2}$ orbifold points. In this paper we introduce a matrix linear difference equation, prove existence and uniqueness of its formal Puiseux-series solutions, and use them to give conjectural formulas for $k$-point ($k\ge2$) functions of Gromov--Witten invariants of $\mathbb{P}^1_{m_1,m_2}$. Explicit expressions of the unique solutions are also obtained. We carry out concrete computations of the first few invariants by using the conjectural formulas. For the case when one of $m_1, m_2$ equals 1, we prove validity of the conjectural formulas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16375 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Gromov--Witten invariants of $\mathbb{P}^1$-orbifolds and topological difference equations Huang, Zhengfei Yang, Di Mathematical Physics Algebraic Geometry Exactly Solvable and Integrable Systems Let $(m_1, m_2)$ be a pair of positive integers. Denote by $\mathbb{P}^1$ the complex projective line, and by $\mathbb{P}^1_{m_1,m_2}$ the orbifold complex projective line obtained from $\mathbb{P}^1$ by adding $\mathbb{Z}_{m_1}$ and $\mathbb{Z}_{m_2}$ orbifold points. In this paper we introduce a matrix linear difference equation, prove existence and uniqueness of its formal Puiseux-series solutions, and use them to give conjectural formulas for $k$-point ($k\ge2$) functions of Gromov--Witten invariants of $\mathbb{P}^1_{m_1,m_2}$. Explicit expressions of the unique solutions are also obtained. We carry out concrete computations of the first few invariants by using the conjectural formulas. For the case when one of $m_1, m_2$ equals 1, we prove validity of the conjectural formulas. |
| title | On Gromov--Witten invariants of $\mathbb{P}^1$-orbifolds and topological difference equations |
| topic | Mathematical Physics Algebraic Geometry Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2504.16375 |