On Gromov--Witten invariants of ${\mathbb{P}}^1$-orbifolds and topological difference equations

Let $(m_1,m_2)$ be a pair of positive integers. Denote by ${\mathbb{P}}^1$ the complex projective line, and by ${\mathbb{P}}_{m_1,m_2}^1$ 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 ($k\ge 2$) functions of Gromov–Witten invariants of ${\mathbb{P}}_{m_1,m_2}^1$. 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.