This talk will present a new approach to the geometry of moduli spaces of hyperbolic monopoles.  It is well-known that the L^2 metric on the moduli space of hyperbolic monopoles, defined using a Coulomb gauge fixing condition, diverges. Recently we have shown that a supersymmetry-inspired gauge-fixing condition cures this divergence, resulting in a pluricomplex geometry that generalises the hyperkaehler geometry of euclidean monopole moduli spaces.  We will compare this with metrics introduced by Nash and Bielawski—Schwachhofer, and present explicit calculations of both metrics for charge 2 monopoles.

I will start the talk by discussing renormlisation group in perturbative algebraic quantum field theory (pAQFT) and its non-perturbative incarnation acting on the Buchholz-Fredenhagen dynamical C*-algebra. I will also explain how pAQFT can be used to derive functional renormlisation group (FRG) equations that generalize Wetterich equations to globally hyperbolic Lorentzian manifolds and arbitrary states (beyond the usual FRG in the vacuum).

Gravitational instantons are complete 4-dimensional hyperkähler manifolds with square-integrable curvature tensor. I will address the question whether all gravitational instantons (of type ALG) can be obtained as Hitchin moduli spaces. In particular, I will explain how to compute the (hyperkähler) Torelli map for (weakly) parabolic Higgs bundles on the 4-punctured sphere. This is based on recent joint work with Fredrickson, Mazzeo and Swoboda.