Instantons and Magnetic Monopoles on $R^3\times S^1$ with Arbitrary Simple Gauge Groups

TL;DR

This paper explores the structure of instantons and magnetic monopoles in Yang-Mills theories on $R^3 imes S^1$ with various gauge groups, revealing their composition from fundamental monopoles and introducing new monopole solutions.

Contribution

It demonstrates that instantons are composed of fundamental magnetic monopoles linked to the extended Dynkin diagram and identifies new monopole solutions in exceptional gauge groups.

Findings

Instantons are made of fundamental magnetic monopoles corresponding to roots.

Number of monopoles per instanton equals the dual Coxeter number.

New $S^1$-dependent monopole solutions found for $G_2$, $F_4$, and $E_8$.

Abstract

We investigate Yang-Mills theories with arbitrary gauge group on $R^3\times S^1$, whose symmetry is spontaneously broken by the Wilson loop. We show that instantons are made of fundamental magnetic monopoles, each of which has a corresponding root in the extended Dynkin diagram. The number of constituent magnetic monopoles for a single instanton is the dual Coxeter number of the gauge group, which also accounts for the number of instanton zero modes. In addition, we show that there exists a novel type of the $S^1$ coordinate dependent magnetic monopole solutions in $G_2,F_4,E_8$.