On moduli spaces of periodic monopoles and gravitational instantons

Abstract

The topic of this thesis is the study of moduli spaces of periodic monopoles (with singularities), i.e. (singular) solutions to the Bogomolny equation (the dimensional reduction of the anti-self-duality equation to 3 dimensions) on R×S. Using arguments from physics, Cherkis and Kapustin gave strong evidence that 4–dimensional moduli spaces of (singular) periodic monopoles yield examples of gravitational instantons (i.e. complete hyperkahler 4–manifolds with decaying curvature) of type ALG. Recently, Hein constructed ALG metrics by solving a complex MongeAmpere equation on the complement of a fibre in a rational elliptic surface. The thesis is the first step in a programme aimed to verify Cherkis and Kapustin’s predictions and understand them in relation to Hein’s construction. More precisely: (i) We construct moduli spaces of periodic monopoles (with singularities) and show that they are smooth hyperkahler manifolds for generic choices of parameters. (ii) For each admissible choice of charge and number of singularities (and under additional conditions on the parameters in certain cases), we show that moduli spaces of periodic monopoles (with singularities) are non-empty by gluing methods. After presenting these results, we will conclude the thesis with an outline of the other steps in the programme.