Periodic monopoles with singularities and N = 2 super-QCD

We study solutions of the Bogomolny equation on ℝ² × S¹ with prescribed singularities. We show that the Nahm transform establishes a one-to-one correspondence between such solutions and solutions of the Hitchin equations on a punctured cylinder with the eigenvalues of the Higgs field growing at infinity in a particular manner. The moduli spaces of solutions have natural hyperkähler metrics of a novel kind. We show that these metrics describe the quantum Coulomb branch of certain N = 2 d = 4 supersymmetric gauge theories on ℝ³ × S¹. The Coulomb branches of the corresponding uncompactified theories have been previously determined by E. Witten using the M-theory fivebrane. We show that the Seiberg-Witten curves of these theories are identical to the spectral curves associated to solutions of the Bogomolny equation on ℝ² × S¹. In particular, this allows us to rederive Witten's results without recourse to the M-theory fivebrane.