Periodic Monopoles from Spectral Curves

TL;DR

This paper studies periodic monopoles in SU(2) gauge theory using spectral curves, analyzing their large size behavior, symmetries, and extending methods to higher charge and rank systems for comparison with periodic Yang-Mills theories.

Contribution

It introduces a spectral curve approach to approximate periodic monopoles and explores symmetries to analyze their dynamics, extending to higher charge and rank cases.

Findings

Spectral curve effectively approximates monopole fields in large size limit.

Symmetries of Nahm transform facilitate analysis of effective dynamics.

Results enable comparison with other periodic Yang-Mills systems.

Abstract

We consider $\text{SU}(2)$ Bogomolny equations on $\mathbb{R}^2\times\hat{S}^1$ and use the spectral curve defined by the holonomy in the periodic direction to approximate the fields in the limit of large size to period ratio. Symmetries of the Nahm transform allow a study of the effective two dimensional dynamics, which is compared with known results on the full moduli space. The techniques are applied to systems of higher charge and higher rank gauge group, allowing a direct comparison to other periodic Yang-Mills systems.