Nahm transform for periodic monopoles and N = 2 super Yang-Mills theory

We study Bogomolny equations on ℝ² × double-struck S sign¹1. Although they do not admit nontrivial finite-energy solutions, we show that there are interesting infinite-energy solutions with Higgs field growing logarithmically at infinity. We call these solutions periodic monopoles. Using the Nahm transform, we show that periodic monopoles are in one-to-one correspondence with solutions of Hitchin equations on a cylinder with Higgs field growing exponentially at infinity. The moduli spaces of periodic monopoles belong to a novel class of hyperkähler manifolds and have applications to quantum gauge theory and string theory. For example, we show that the moduli space of k periodic monopoles provides the exact solution of N = 2 super Yang-Mills theory with gauge group SU(k) compactified on a circle of arbitrary radius.