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I will discuss joint work with C. Ross aimed at describing the behaviour of Yang-Mills instantons on ALF spaces in the collapsed limit where these 4-dimensional geometries are close to a 3-dimensional collapsed limit space. In this limit, the instantons are well approximated by superpositions of simple localised building blocks constructed out of monopoles on R^3 (with singularities). I will discuss how the construction can be used to describe the asymptotics of the hyperkähler L^2-metric on the moduli space of instantons, providing examples of a class of higher dimensional non-compact hyperkähler spaces with an asymptotic geometry, sometimes referred to as QALF, that generalises to higher dimensions the geometry of 4-dimensional ALF spaces. If time permits, I will discuss joint work in progress with R. Bielawski on a conjectural definition of a large class of holomorphic symplectic spaces expected to carry QALF metrics from Hilbert schemes of hypetoric varieties.

GT Program Seminar: Instantons On ALF Spaces, Codimension-1 Collapse And QALF Metrics