Explicit Non-Abelian Monopoles and Instantons in SU(N) Pure Yang-Mills Theory
Abstract
It is well known that there are no static non-Abelian monopole solutions in pure Yang-Mills theory on Minkowski space R3,1. We show that such solutions exist in SU(N) gauge theory on the spaces R2× S2 and R1× S1× S2 with Minkowski signature (-+++). In the temporal gauge they are solutions of pure Yang-Mills theory on T1× S2, where T1 is R1 or S1. 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 φ4 kink model whose static solutions give SU(N) monopole (-antimonopole) configurations on the space R1,1× S2 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 R1× S1 admits also periodic sphaleron-type solutions describing chains of n kink-antikink pairs spaced around the circle S1 with arbitrary n>0. They correspond to chains of n static monopole-antimonopole pairs on the space R1× S1× S2 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 S1× S3 interpreted as chains of n instanton-antiinstanton pairs.