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].