Irreducible flat SL(2,R)-connections on the trivial holomorphic bundle

TL;DR

This paper constructs a specific type of irreducible flat SL(2,R)-connection on a trivial bundle over a compact Riemann surface, addressing a previously open question in the field.

Contribution

It provides the first explicit construction of an irreducible holomorphic connection with SL(2,R)-monodromy on the trivial bundle, solving an open problem.

Findings

Constructed an irreducible holomorphic SL(2,R)-connection on the trivial bundle.

Confirmed the existence of such connections on compact Riemann surfaces.

Answered a question posed by Calsamiglia, Deroin, Heu, and Loray.

Abstract

We construct an irreducible holomorphic connection with SL(2,R)-monodromy on the trivial holomorphic vector bundle of rank two over a compact Riemann surface. This answers a question of Calsamiglia, Deroin, Heu and Loray in \cite{CDHL}.