The Deligne-Hitchin moduli space of a Riemann surface is the complex analytic reincarnation of the twistor space of the hyper-kaehler moduli space of solutions of Hitchin’s selfduality equations. It is constructed by gluing the moduli spaces of lambda-connections on the Riemann surface with that of the complex conjugate Riemann surface. The twistor lines correspond to solutions of the self-duality equations. In this talk I will show that the Deligne-Hitchin moduli space contains various other complex lines which (in the rank 2 case) give rise to equivariant harmonic maps into the 3-sphere, anti-deSitter and deSitter 3-space. The talk is based on joint work with Lynn Heller, and on joint work with Indranil Biswas and Markus Roeser.
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