Deformation theory of periodic monopoles (with singularities)

TL;DR

This paper studies the structure of moduli spaces of periodic monopoles with singularities, proving they are either empty or smooth hyperkähler manifolds, and computes their dimensions via an index theorem.

Contribution

It establishes the smoothness and hyperkähler structure of moduli spaces for generic parameters and provides an index theorem to determine their dimensions.

Findings

Moduli spaces are either empty or smooth hyperkähler manifolds.

An index theorem for these moduli spaces is proved.

The dimension of the moduli spaces is explicitly computed.

Abstract

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.