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,C), SL(2,C), and PSL(2,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 Z2-coefficients, we obtain in a unified manner the number of components of the character varieties for the real groups G = GL(2,R), SL(2,R), PGL(2,R), PSL(2,R), as well as the number of components of the Sp(4,R) and SO₀(2, 3)-character varieties with maximal Toledo invariant. We also use our results for GL(2,R) to compute the monodromy of the SO(2, 2) Hitchin map and determine the components of the SO(2, 2) character variety.