On Gromov--Witten invariants of ℙ¹-orbifolds and topological difference equations

2025/04/23 by Z Huang, Di Yang, Huang, Zhengfei +1
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2504.16375

Abstract

Let (m₁, m₂) be a pair of positive integers. Denote by ℙ¹ the complex projective line, and by ℙ¹_m1,m2 the orbifold complex projective line obtained from ℙ¹ by adding ℤ_m1 and ℤ_m2 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≥2) functions of Gromov--Witten invariants of ℙ¹_m1,m2. 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 m₁, m₂ equals 1, we prove validity of the conjectural formulas.

Citations

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Related

• Lectures on Gromov-Witten invariants of orbifolds
• On Frobenius Manifolds from Gromov--Witten Theory of Orbifold Projective Lines with r orbifold points
• Algebraic reduced genus one Gromov-Witten invariants for complete intersections in projective spaces, Part 2
• A Uniqueness Theorem for Frobenius Manifolds and Gromov--Witten Theory for Orbifold Projective Lines
• Difference equation for the Gromov-Witten potential of the resolved conifold