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 alsoallows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by O. Guichard and A. Wienhard.
Highlights
Let Σ be a closed connected and oriented surface of genus g ≥ 2 and G be a connected semisimple Lie group
Fixing a complex structure J on the surface Σ transforms this into a Riemann surface X = (Σ, J) and opens the way for holomorphic techniques using the theory of Higgs bundles
It is important that a gluing construction of parabolic Higgs bundles over the complex connected sum X# of two distinct compact Riemann surfaces X1 and X2 with a divisor of s-many distinct points on each, is formulated so that the gluing of stable parabolic pairs is providing a polystable Higgs bundle over X#
Summary
Let Σ be a closed connected and oriented surface of genus g ≥ 2 and G be a connected semisimple Lie group. It is important that a gluing construction of parabolic Higgs bundles over the complex connected sum X# of two distinct compact Riemann surfaces X1 and X2 with a divisor of s-many distinct points on each, is formulated so that the gluing of stable parabolic pairs is providing a polystable Higgs bundle over X#. Let (E1, Φ1) → X1 and (E2, Φ2) → X2 be parabolic stable Sp(4,R)-Higgs bundles with corresponding solutions to the Hitchin equations (A1, Φ1) and (A2, Φ2) Assume that these solutions agree with model solutions A1m,poid , Φ1m,poid and A2m,qojd , Φ2m,qojd near the points pi ∈. 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 Mmax, which are fully distinguished by the calculation of the degree of a line bundle. These invariants live naturally in different cohomology groups
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