Model Higgs Bundles in Exceptional Components of the Sp(4,R)-Character Variety

Abstract

We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the Sp(4,R)-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the 2g − 3 exceptional components of the maximal Sp(4,R)-Higgs bundle moduli space, which correspond to components solely consisted of Zariski dense representations. This also allows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by O. Guichard and A. Wienhard.for s-many possible pairs of points (p i , q j ). Then there is a polystable Sp(4,R)-Higgs bundle (E # , Φ # ) → X # , constructed over the connected sum of Riemann surfaces X # = X 1 #X 2 of genus g 1 + g 2 + s − 1, which agrees with the initial data over X # \X 1 and X # \X 2 .By analogy with the terminology introduced by O. Guichard and A. Wienhard in their construction of hybrid representations, we call the polystable Higgs bundles corresponding to such exact solutions hybrid. The construction can have a wider applicability for identifying smooth points in moduli of Higgs bundles. As one application, we build Higgs bundles corresponding to Zariski dense representations into Sp(4,R). For this purpose, we look at how the Higgs bundle topological invariants behave under the complex connected sum operation. We first show the following: Proposition 1.2. Let X # = X 1 #X 2 be the complex connected sum of two closed Riemann surfaces X 1 and X 2 with divisors D 1 and D 2 of s-many distinct points on each surface, and let V 1 , V 2 be parabolic principal H C -bundles over X 1 and X 2 respectively. For a parabolic subgroup P ⊂ H C , a holomorphic reduction σ of the structure group of E from H C to P and an antidominant character χ of P , the following identity holds:Note that an analogous additivity property for the Toledo invariant was established by M. Burger, A. Iozzi and A. Wienhard in [8] from the point of view of fundamental group representations. It implies in particular that the connected sum of maximal parabolic G-Higgs bundles is again a maximal (non-parabolic) G-Higgs bundle.We find model Higgs bundles in all exceptional components of the maximal Sp(4,R)-Higgs bundle moduli space; these models are described by hybrid Higgs bundles. In the case when G = Sp(4,R), considering all possible decompositions of a surface Σ along a simple, closed, separating geodesic is sufficient in order to obtain representations in the desired components of M max , which are fully distinguished by the calculation of the degree of a line bundle. This degree equals the Euler class for a hybrid representation as defined by O. Guichard and A. Wienhard, although these invariants live naturally in different cohomology groups.

From hyperbolic Dehn filling to surgeries in representation varieties

“…Examples of models in the case of the group Sp (4, R) were obtained in [55] (see also [54]), while for G = SO (p, p + 1) we will demonstrate some examples in §6.…”

“…for a reductive Lie group G. It is important that copies of a maximal compact subgroup of SL(2,R) are mapped via φ into copies of a maximal compact subgroup of G and that the norm of the infinitesimal deformation φ * on the complexified Lie algebra g C satisfies a Lipschitz condition. Assuming that this is indeed the case for an embedding φ (examples can be found in [55] and will be demonstrated in §6), one gets by extension via the embedding φ a G C -pair satisfying the G-Hitchin equations up to an error, which we have good control of. For i = 1, 2, let X i be a closed Riemann surface of genus g i and let D 1 = {p 1 , .…”

“…Proposition 5.11 (Proposition 8.1 in [55]). Let X # = X 1 #X 2 be the complex connected sum of two closed Riemann surfaces X 1 and X 2 with divisors D 1 and D 2 of s-many distinct points on each surface, and let V 1 , V 2 be parabolic vector bundles over X 1 and X 2 respectively.…”

“…The argument providing the existence of such a gauge is translated into a Banach fixed point theorem argument and involves the study of the linearization of a relevant elliptic operator. We briefly describe these steps in the sequel; for complete proofs we refer to [54] and [55]. 5.4.1.…”

“…When the Lie group is G = Sp (4, R), hybrid Higgs bundles in the exceptional connected components of the maximal G = Sp (4, R)-Higgs bundles identified by P. Gothen in [36] were obtained in [55]. We next provide such examples in the case of the group G = SO (p, p + 1), which involves an extra parameter compared to the Sp (4, R)-case; note, however, that a maximality property is not apparent in this case (apart from when p = 2).…”