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

TL;DR

The paper introduces periodic monopoles on ${\mathbb R}^2\times {\mathbb S}^1$ and proves a Nahm transform to Hitchin equations on a cylinder, establishing a one-to-one correspondence between periodic monopoles (charge $k$) and $U(k)$ Hitchin pairs with specified exponential asymptotics. It develops a detailed spectral-data framework, showing the monopole and Hitchin spectral curves coincide and that the Nahm and inverse Nahm transforms are mutually compatible via cohomological formalisms and spectral sequences. The authors connect these mathematical structures to ${\cal N}=2$ super Yang-Mills theory on a circle, predicting a hyperkähler moduli space of dimension $4(k-1)$ for the centered sector and providing a classical route to the quantum Coulomb branch through geometric analysis. The work also outlines existence arguments for periodic monopoles, explicit $k=1$ solutions, and a rigorous circle of ideas binding brane configurations, spectral geometry, and gauge theory, with future work on the full moduli-space metric. Overall, this framework offers a robust, dual description of the Coulomb branch and a rich geometric structure for periodic monopoles, with potential applications in string theory and nonperturbative gauge dynamics.

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 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 ${\cal N}=2$ super Yang-Mills theory with gauge group $SU(k)$ compactified on a circle of arbitrary radius.