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 R2 x S1. 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 hyperkähler 4–manifolds with decaying curvature) of
type ALG. Recently, Hein constructed ALG metrics by solving a complex Monge-
Ampère 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 hyperkähler 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.