Monopoles in Even Dimensions

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. An n-dimensional monopole over the sphere 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 more accurate descriptions of 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 [2]. Second, we replace the Chern class with the Euler class to calculate the charge of the SO(2n)-monopoles studied in [18].