Nahm Transform For Periodic Monopoles And N=2 Super Yang-Mills Theory

TL;DR

This paper explores the properties of periodic monopoles on R^2×S^1, establishing a Nahm transform correspondence with Hitchin equations, and demonstrates their relevance to solving N=2 super Yang-Mills theory on a circle.

Contribution

It introduces a novel class of infinite-energy solutions called periodic monopoles and links their moduli spaces to hyperkähler manifolds relevant in quantum gauge and string theories.

Findings

Periodic monopoles correspond to Hitchin solutions on a cylinder.

Moduli spaces form a new class of hyperkähler manifolds.

Exact solutions to N=2 super Yang-Mills theory are derived from monopole moduli spaces.

Abstract

We study Bogomolny equations on $R^2\times S^1$. 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 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 hyperk\"ahler 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 ${\cal N}=2$ super Yang-Mills theory with gauge group $SU(k)$ compactified on a circle of arbitrary radius.