Unitary monodromies of rank two Fuchsian systems with $(n+1)$ singularities

TL;DR

This paper characterizes when rank two Fuchsian systems' monodromies are conjugate to subgroups of SU(p,q), providing conditions, constructions, and classifications for their unitarity on the Riemann sphere.

Contribution

It offers a necessary and sufficient condition for the unitarity of monodromies and characterizes the moduli space of such monodromies as an affine algebraic set.

Findings

Provides a criterion for monodromy unitarity in SU(p,q)

Constructs monodromies within the moduli space as algebraic sets

Classifies signatures of unitary monodromies

Abstract

We study the unitarity of monodromies of rank two Fuchsian systems of SL type with $(n+1)$ regular singularities on the Riemann sphere, namely, we give a sufficient and necessary condition for the monodromy group to be conjugate to a subgroup of a special unitary group $\mathrm{SU}(p,q)$. When $n\ge 3$, the moduli space of irreducible monodromies can be realized as an affine algebraic set in $\mathbb{C}^m$ for some $m \in \mathbb{N}$. In this paper, we give a characterization and construction of unitary monodromies in terms of this affine algebraic set. The signatures of unitary monodromies are also classified.