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 hyperk?hler 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. Received: 9 March 2001 / Accepted: 15 January 2002 Published online: 20 January 2003