20th February 2019: Hyperholomorphic line bundles and energy functionals

Florian Beck (Hamburg University)

Title: Hyperholomorphic line bundles and energy functionals

Abstract: 

For any Hyperkähler manifold M with an appropriate circle action, Haydys introduced a hyperholomorphic line bundle on M. This line bundle induces a holomorphic line bundle L on the twistor space Z of M. The latter captures the whole sphere of Kähler structures on M and fibers over the complex projective line. Hitchin constructed the line bundle L purely in terms of the geometry of Z. This twistorial approach gives further insights, for example it shows the existence of meromorphic connections on L with certain properties. 

In a joint project with Sebastian Heller and Markus Röser, we examine these meromorphic connections in more detail and show how they induce a natural functional on the space of sections of Z. In the particular case when M is the moduli space of self-duality equations on a Riemann surface (of rank 2), we prove that this functional is a natural extension of the energy of Higgs fields. We apply this generalized energy functional to study the space of sections of Z and comment on the relationship to the non-abelian Hodge correspondence. 

Villa Battelle, Wednesday 20th February, 14:00