Monopoles in arbitrary dimension

TL;DR

This paper provides a comprehensive algebraic and geometric analysis of monopole configurations in pure Yang-Mills theories across arbitrary dimensions, correcting previous misconceptions and offering coordinate-free charge calculations.

Contribution

It introduces a unified geometric framework for monopoles in any dimension, proves the uniqueness of canonical invariant connections, and corrects earlier results on monopole charges.

Findings

Principal bundles over symmetric spaces admit unique K-invariant connections.

Yang monopoles are invariant under Spin(5), not SO(5).

Correct characteristic class for monopole charge calculation is identified.

Abstract

A self-contained study of monopole configurations of pure Yang-Mills theories and a discussion of their charges is carried out in the language of principal bundles. A n-dimensional monopole over the sphere S^n is a particular type of principal connection on a principal bundle over a symmetric space K/H which is K-invariant, where K=SO(n+1) and H=SO(n). It is shown that principal bundles over symmetric spaces admit a unique K-invariant principal connection called canonical, which also satisfy Yang-Mills equations. The geometrical framework enables us to describe their associated field strengths in purely algebraic terms and compute the charge of relevant (Yang-type) monopoles avoiding the use of coordinates. Besides, two corrections on known results are performed in this paper. First, it is proven that the Yang monopole should be considered a connection invariant by Spin(5) instead of by SO(5), as Yang did in his original article J. Math. Phys. 19(1), pp. 320-328 (1978). Second, unlike the way suggested in Class. Quantum Grav. 23, pp. 4873-4885 (2006), we give the correct characteristic class to be used to calculate the charge of the monopoles studied by Gibbons and Townsend.