Periodic Monopoles With Singularities And N=2 Super-QCD

TL;DR

This paper explores the mathematical structure of monopole solutions with singularities and their relation to supersymmetric gauge theories, revealing new insights into the geometry of Coulomb branches and spectral curves.

Contribution

It establishes a novel correspondence between monopole solutions with singularities and Hitchin equations, and connects these to the quantum Coulomb branch of N=2 supersymmetric gauge theories.

Findings

Nahm transform links monopole solutions to Hitchin equations.

Moduli spaces have unique hyperkahler metrics.

Spectral curves match Seiberg-Witten curves for these theories.

Abstract

We study solutions of the Bogomolny equation on R^2\times S^1$ with prescribed singularities. We show that 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 hyperkahler 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 R^3\times S^1. 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 R^2\times S^1. In particular, this allows us to rederive Witten's results without recourse to the M-theory fivebrane.