Rank two quadratic pairs and surface group representations

TL;DR

This paper studies the moduli spaces of rank 2 quadratic pairs on compact Riemann surfaces, proving their connectedness under certain conditions and applying these results to classify components of specific surface group representations.

Contribution

It establishes the connectedness of moduli spaces of quadratic pairs of rank 2 and determines the connected components of the SO(2,3) character variety.

Findings

Moduli spaces of quadratic pairs of rank 2 are connected under certain topological constraints.

Connected components of the SO(2,3) character variety are explicitly determined.

Provides new insights into surface group representations via quadratic pairs.

Abstract

Let $X$ be a compact Riemann surface. A quadratic pair on $X$ consists of a holomorphic vector bundle with a quadratic form which takes values in fixed line bundle. We show that the moduli spaces of quadratic pairs of rank 2 are connected under some constraints on their topological invariants. As an application of our results we determine the connected components of the $\mathrm{SO}_0(2,3)$-character variety of $X$.