Quadratic differentials and the monodromy of the Hitchin fibrations

Abstract

The purpose of this dissertation is to understand the monodromy action on the Hitchin fibration for the group $SO(2,2)$. We denote by $\mathcal{M}_{SO(2,2)}$ the moduli space of $SO(2,2)$-Higgs bundles; $\mathcal{A}_{SO_4}$ the Hitchin base for $SO_4$. Since $SO(2,2)$ is the split real form of $SO_4(\mathbb{C})$, the generic fibers of the Hitchin map $h:\mathcal{M}_{SO(2,2)}\rightarrow\mathcal{A}_{SO_4}$ correspond to the $2$-torsion points of the $SO_4(\mathbb{C})$-Hitchin fibers. By using the braid group as a tool, the present research is to study the fundamental group of the regular Hitchin bases involving quadratic differentials. For example, the Hitchin fibration for $SL_2$ is determined by a single quadratic differential. Therefore, a natural step further is to examine the Hitchin fibration for $SO(2, 2)$. This Hitchin base involves two quadratic differentials. We consider the subset $\mathcal{A}_{SO_4}^{reg}\subset\mathcal{A}_{SO_4}$ over which the fibers are isomorphic to abelian varieties. In the present study, the mixed braid group is introduced to study the fundamental group of the regular $SO(2, 2)$-Hitchin base $\mathcal{A}_{SO_4}^{reg}$, which has a rich structure already.

In addition, by using the data from $\pi_1(\mathcal{A}_{SO_4}^{reg})$, one can study the monodromy action on the Hitchin fibration for the group $SO(2,2)$. The study is able to develop the monodromy representation for $SO(2, 2)$-Hitchin fibration directly without using the method involving low-rank exceptional isogeny $\mathcal{M}_{SL_2(\mathbb{R})\times SL_2(\mathbb{R})}\rightarrow \mathcal{M}_{SO(2,2)}$.

A direct application of this monodromy representation is to count the number of orbits under the monodromy actions. Then the number of components of the $SO(2, 2)$-character variety is calculable. The orbit count based on this study is smaller than Baraglia and Schaposnik’s result because this study takes an extra type of generators into consideration.