Monodromy of rank 2 twisted Hitchin systems and real character varieties

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 L -twisted G G -Higgs bundles for the groups G = G L ( 2 , C ) G = GL(2,\mathbb {C}) , S L ( 2 , C ) SL(2,\mathbb {C}) , and P S L ( 2 , C ) 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 L -twisted Higgs bundles of rank 2 2 and degree deg ⁡ ( L ) + 1 \deg (L)+1 . By counting orbits of the monodromy action with Z 2 \mathbb {Z}_2 -coefficients, we obtain in a unified manner the number of components of the character varieties for the real groups G = G L ( 2 , R ) G = GL(2,\mathbb {R}) , S L ( 2 , R ) SL(2,\mathbb {R}) ,