Deformation theory of periodic monopoles (with singularities)

We show that for generic choices of parameters the moduli spaces of periodic monopoles (with singularities), i.e. monopoles on $\mathbb{R}^{2} \times \mathbb{S}^{1}$ possibly singular at a finite collection of points, are either empty or smooth hyperk\"ahler manifolds. Furthermore, we prove an index theorem and therefore compute the dimension of the moduli spaces.