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We establish a gluing construction for Higgs bundles over a connected sum ofRiemann surfaces in terms of solutions to the$\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Hitchin equations using thelinearization of a relevant elliptic operator. The construction can be used toprovide model Higgs bundles in all the $2g-3$ exceptional components of themaximal $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Higgs bundle moduli space,which correspond to components solely consisted of Zariski denserepresentations. This also allows a comparison between the invariants formaximal Higgs bundles and the topological invariants for Anosov representationsconstructed by O. Guichard and A. Wienhard.

Model Higgs bundles in exceptional components of the $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-character variety