Modelling hyper-Kähler structures on moduli of parabolic Higgs bundles over the Riemann sphere

The non-abelian Hodge correspondence is a deep analytic result lying behind the existence of natural hyper-Kähler structures on moduli of parabolic Higgs bundles on compact Riemann surfaces. These structures arise in an infinite series of families, in part as a consequence of their dependence on choices of stability parameters. By the very nature of their construction, their characterisation beyond existence is a nontrivial problem, and although the wall-crossing phenomena associated with such dependence is well understood as a problem in birational geometry, the analogous differential-geometric problem for the hyper-Kähler structure is still outstanding.

In this talk I will present an overview of recent work on the construction of geometric models that could yield a more explicit dependence of the hyper-Kähler structure on stability parameters in the case of genus 0. In particular I will discuss results obtained in collaboration with Lynn Heller and Sebastian Heller, on how a suitable renormalised limit of the Hitchin metrics converge to the hyperpolygon-space metrics as the stability parameters are scaled down to 0.