Monodromy of rank 2 twisted Hitchin systems and real character varieties

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 Tate-Shafarevich class of the abelian torsor defined by 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 $Sp(4,\mathbb {R})$ and $SO_0(2,3)$-character varieties 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.