Gluing constructions for Higgs bundles over a complex connected sum

Georgios Kydonakis (UIUC)

For a compact Riemann surface of genus $g\ge 2$, the components of the moduli space of $\text{Sp}(4,\mathbb{R})$-Higgs bundles, or equivalently the $\text{Sp}(4,\mathbb{R})$-character variety, are partially labeled by an integer $d$ known as the Toledo invariant. The subspace for which this integer attains a maximum has been shown to have $3\cdot {{2}^{2g}}+2g-4$ many components. A gluing construction between parabolic Higgs bundles over a connected sum of Riemann surfaces provides model Higgs bundles in a subfamily of particular significance. This construction is formulated in terms of solutions to the Hitchin equations, using the linearization of a relevant elliptic operator.