Explicit Non-Abelian Monopoles and Instantons in SU(N) Pure Yang-Mills Theory

TL;DR

This paper constructs explicit non-Abelian monopole and instanton solutions in SU(N) Yang-Mills theory on specific curved spaces, revealing new static and chain-like configurations with potential physical interpretations.

Contribution

It introduces a method to obtain non-Abelian monopole and instanton solutions in pure Yang-Mills theory on curved spaces by reducing to a non-Abelian 4 kink model, a novel approach for such configurations.

Findings

Existence of static SU(N) monopoles on R^2 d7 S^2 and R^1 d7 S^1 d7 S^2.

Construction of chain solutions of monopole-antimonopole pairs on R^1 d7 S^1 d7 S^2.

Identification of instanton configurations in 2+1 dimensions and Euclidean spaces.

Abstract

It is well known that there are no static non-Abelian monopole solutions in pure Yang-Mills theory on Minkowski space R^{3,1}. We show that such solutions exist in SU(N) gauge theory on the spaces R^2\times S^2 and R^1\times S^1\times S^2 with Minkowski signature (-+++). In the temporal gauge they are solutions of pure Yang-Mills theory on T^1\times S^2, where T^1 is R^1 or S^1. Namely, imposing SO(3)-invariance and some reality conditions, we consistently reduce the Yang-Mills model on the above spaces to a non-Abelian analog of the \phi^4 kink model whose static solutions give SU(N) monopole (-antimonopole) configurations on the space R^{1,1}\times S^2 via the above-mentioned correspondence. These solutions can also be considered as instanton configurations of Yang-Mills theory in 2+1 dimensions. The kink model on R^1\times S^1 admits also periodic sphaleron-type solutions describing chains of n kink-antikink pairs spaced around the circle S^1 with arbitrary n>0. They correspond to chains of n static monopole-antimonopole pairs on the space R^1\times S^1\times S^2 which can also be interpreted as instanton configurations in 2+1 dimensional pure Yang-Mills theory at finite temperature (thermal time circle). We also describe similar solutions in Euclidean SU(N) gauge theory on S^1\times S^3 interpreted as chains of n instanton-antiinstanton pairs.