Abelianisation of λ-Connections
In this talk, I will describe some interesting geometry on the moduli space of λ-connections. First, I will explain what I mean by a λ-connection and show how this object can be abelianised to a λ-connection of rank one by extending the spectral correspondence for Higgs bundles to flat bundles. I will then describe how this abelianisation correspondence can be used to define interesting geometric structures on the moduli space of SL(2,C)-λ-connections: namely, a complex hyperKähler structure as well as what looks like a completely integrable system but with fibres being abelian character varieties rather than abelian varieties. Based on joint work in progress with Tom Bridgeland and Menelaos Zikidis.