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

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 $G_2,F_4,E_8$.