Monodromy of rank 2 twisted Hitchin systems and real character varieties

TL;DR

This paper introduces a new method to compute monodromy in rank 2 twisted Hitchin systems, determining monodromy, Chern classes, and component counts for various real and complex character varieties.

Contribution

It presents a novel approach for monodromy computation in rank 2 twisted Hitchin systems and applies it to classify components of related character varieties.

Findings

Complete monodromy determination for specific groups

Counting components of real and maximal Toledo invariant character varieties

Calculation of monodromy for the $SO(2,2)$ Hitchin map

Abstract

We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of $L$-twisted $G$-Higgs bundles, for the groups $G = GL(2,\mathbb{C})$, $SL(2,\mathbb{C})$ and $PSL(2,\mathbb{C})$. We also determine the twisted Chern class of the regular locus, which obstructs the existence of a section of the moduli space of $L$-twisted Higgs bundles of rank $2$ and degree $deg(L)+1$. By counting orbits of the monodromy action with $\mathbb{Z}_2$-coefficients, we obtain in a unified manner the number of components of the character varieties for the real groups $G = GL(2,\mathbb{R})$, $SL(2,\mathbb{R})$, $PGL(2,\mathbb{R})$, $PSL(2,\mathbb{R})$, as well as the number of components of the $Sp(4,\mathbb{R})$-character variety with maximal Toledo invariant. We also use our results for $GL(2,\mathbb{R})$ to compute the monodromy of the $SO(2,2)$ Hitchin map and determine the components of the $SO(2,2)$ character variety.