Transparent connections over negatively curved surfaces

TL;DR

This paper classifies transparent unitary connections on negatively curved surfaces, linking them to solutions of specific PDEs and establishing a correspondence with meromorphic functions, revealing new geometric and analytical insights.

Contribution

It provides a classification of transparent connections on negatively curved surfaces and relates them to solutions of natural PDEs and meromorphic functions.

Findings

Classification of transparent connections modulo gauge

Existence of a transparent SU(2)-connection for each meromorphic function

Local uniqueness of the trivial connection

Abstract

Let $(M,g)$ be a closed oriented negatively curved surface. A unitary connection on a Hermitian vector bundle over $M$ is said to be transparent if its parallel transport along the closed geodesics of $g$ is the identity. We study the space of such connections modulo gauge and we prove a classification result in terms of the solutions of certain PDE that arises naturally in the problem. We also show a local uniqueness result for the trivial connection and that there is a transparent SU(2)-connection associated to each meromorphic function on $M$.