A family of flat connections on the projective space having dihedral monodromy and algebraic Garnier solutions

TL;DR

This paper generalizes a construction of flat connections with dihedral monodromy on projective spaces, linking them to algebraic solutions of the Garnier system, expanding understanding of monodromy representations and integrable systems.

Contribution

It introduces an explicit n-parameter family of flat connections on projective spaces, extending previous work and connecting these to algebraic Garnier solutions.

Findings

Constructed explicit n-parameter family of flat connections.

Established relation between connections and Garnier system.

Generalized previous dihedral monodromy constructions.

Abstract

A. Girand has constructed an explicit two-parameter family of flat connections over the complex projective plane $\mathbb{P}^2$. These connections have dihedral monodromy and their polar locus is a prescribed quintic composed of a conic and three tangent lines. In this paper, we give a generalization of this construction. That is, we construct an explicit $n$-parameter family of flat connections over the complex projective space $\mathbb{P}^n$. Moreover, we discuss the relation between these connections and the Garnier system.