Higgs bundles, the Toledo invariant and the Cayley correspondence

Abstract

We define the Toledo invariant of a G-Higgs bundle on a Riemann surface, where G is a real semisimple group of Hermitian type, and we prove a Milnor-Wood type bound for this invariant when the bundle is semistable. We prove rigidity results when the Toledo invariant is maximal, establishing in particular a Cayley correspondence when the symmetric space defined by G is of tube type. This gives a new proof of the Milnor-Wood inequality of Burger-Iozzi-Wienhard for representations of the fundamental group of a Riemann surface into G. Compared to previous results using Higgs bundles, it uses general theory and avoids any case by case study.

To appear in Journal of Topology

Cited by in corpus (9)

• Involutions and higher order automorphisms of Higgs bundle moduli spaces
• Higgs bundles for real groups and the Hitchin-Kostant-Rallis section
• SO(p,q)-Higgs bundles and higher Teichmüller components
• Exotic components of $\mathrm{SO}(p,q)$ surface group representations, and their Higgs bundle avatars
• Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces
• Positivity and representations of surface groups
• Homotopy Type of Moduli Spaces of G-Higgs Bundles and Reducibility of the Nilpotent Cone
• Higgs bundles, abelian gerbes and cameral data
• Cyclic Higgs bundles and the Toledo invariant