Higgs bundles and surface group representations in the real symplectic group

Abstract

In this paper we study the moduli space of representations of a surface group (i.e., the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n,R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor-Wood type inequality. Our main result is a count of the number of connected components of the moduli space of maximal representations, i.e. representations with maximal Toledo invariant. Our approach uses the non-abelian Hodge theory correspondence proved in a companion paper arXiv:0909.4487 [math.DG] to identify the space of representations with the moduli space of polystable Sp(2n,R)-Higgs bundles. A key step is provided by the discovery of new discrete invariants of maximal representations. These new invariants arise from an identification, in the maximal case, of the moduli space of Sp(2n,R)-Higgs bundles with a moduli space of twisted Higgs bundles for the group GL(n,R).

55 pages; v2: main results are unchanged but paper has been completely reorganized and presentation significantly improved; v3, v4: various corrections and improvements; Part 1 of v2 has been split off as the independent paper arXiv:0909.4487v1 [math.DG]

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• Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces
• Surface group representations in PU(p,q) and Higgs bundles
• The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations
• Homotopy groups of moduli spaces of representations
• Representations of surface groups in the general linear group
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Cited by in corpus (21)

• The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations
• Deformations of maximal representations in Sp(4,R)
• Asymptotics of certain families of Higgs bundles in the Hitchin component
• Higgs bundles, the Toledo invariant and the Cayley correspondence
• The geometry of maximal components of the PSp(4,R) character variety
• Higgs bundles for the non-compact dual of the special orthogonal group
• Positivity and representations of surface groups
• A general Cayley correspondence and higher Teichmüller spaces
• Birationality of moduli spaces of twisted $\mathrm{U}(p,q)$-Higgs bundles
• Higgs bundles and the real symplectic group
• A Milnor-Wood inequality for complex hyperbolic lattices in quaternionic space
• Noncommutative coordinates for symplectic representations
• Stability of Quadric bundles
• Cyclic Higgs bundles and the Toledo invariant
• Maximal Higgs bundles for adjoint forms via Cayley correspondence
• Metha-Ramanathan for ε and k-semistable Decorated Sheaves
• Morse Theory, Higgs fields and Yang-Mills-Higgs functionals
• Signature, Toledo invariant and surface group representations in the real symplectic group
• Boundary of the Gothen components
• Connected components of representation spaces of non-orientable surfaces
• A properness result for degenerate Quadratic and Symplectic Bundles on a smooth projective curve