Higgs bundles for the non-compact dual of the special orthogonal group

Abstract

Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group G. In this paper we examine the case G=SO*(2n). We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli space is connected in this maximal Toledo case. The Morse theory also allows us to show connectedness when the Toledo invariant is zero. The correspondence between Higgs bundles and surface group representations thus allows us to count the connected components with zero and maximal Toledo invariant in the moduli space of representations of the fundamental group of the surface in SO*(2n).

43 pages; v2: minor corrections and improvements, appendix significantly shortened (the removed material will appear elsewhere); v3: final version with many minor corrections and improvements, to appear in Geometriae Dedicata

References in corpus (5)

• Topological Invariants of Anosov Representations
• Nonabelianization of Higgs bundles
• Spectral data for U(m,m)-Higgs bundles
• Rank two quadratic pairs and surface group representations
• Cohomology of U(2,1) representation varieties of surface groups

Cited by in corpus (6)

• Higgs bundles for real groups and the Hitchin-Kostant-Rallis section
• Higgs bundles, the Toledo invariant and the Cayley correspondence
• Birationality of moduli spaces of twisted $\mathrm{U}(p,q)$-Higgs bundles
• Cyclic Higgs bundles and the Toledo invariant
• Maximal Higgs bundles for adjoint forms via Cayley correspondence
• Generalized theta functions, strange duality, and odd orthogonal bundles on curves