SU(2) Calorons and Magnetic Monopoles

TL;DR

This paper addresses the problem of understanding calorons on $R^3\times S^1$ with SU(2) broken by a nontrivial Wilson loop. It uses the Nahm construction to produce an explicit field configuration for a single caloron and demonstrates that it is a bound state of two fundamental monopoles of opposite magnetic charge, providing exact expressions and exploring several limits, including massless monopoles and zero temperature. The authors derive the full moduli-space metric, showing the relative moduli space is Taub-NUT with a $Z_2$ orbifold and that the total space is $\mathbb{R}^3\times (\mathbb{R}\times {\cal M}_0)/\mathbb{Z}$, with a flat limit at zero temperature. These results deepen the constituent-monopole picture of calorons, connect finite-temperature Yang-Mills configurations to periodic instantons, and suggest avenues for understanding nonperturbative phenomena such as confinement and chiral symmetry in gauge theories.

Abstract

We investigate the self-dual Yang-Mills gauge configurations on $R^3\times S^1$ when the gauge symmetry SU(2) is broken to U(1) by the Wilson loop. We construct the explicit field configuration for a single instanton by the Nahm method and show that an instanton is composed of two self-dual monopoles of opposite magnetic charge. We normalize the moduli space metric of an instanton and study various limits of the field configuration and its moduli space metric.