Invariant connections and magnetic monopoles

Abstract

Given a Lie group K acting on a principal fiber bundle P(M, G), we study the K-invariance of connections in P and in bundles associated to P. The geometry of spontaneous symmetry breaking is discussed from this point of view. The results are applied to the Wu–Yang and ’t Hooft–Polyakov models of a magnetic monopole.

Classical Yang-Mills fields with non-compact symmetry and an arbitrary compact gauge group

“…This is the so-called canonical connection on H3 [ 131 and therefore the corresponding curvature F = Dw automatically satisfies the sourceless YM equation D*F=O, as shown by Nowakowski and Trautman [ 141.The interesting point here is that w is the only SO(3) invariant connection on H3. The same situation was found by Cant[3] for the Dirac monopole.For the sake of completeness we compute w and F explicitly. Given y = (yo, y) E H3, its image under the Cartan immersion X: H3+ S0,(3,1) is given byc(Y) =2(TY)(TY)T-T E soo(3, 1)(4.2)where 77 =diag(+l, -1, -1, -1).…”

Dirac’s magnetic monopole and the Kontsevich star product

“…This geometric description reveals the topological origin of Dirac's charge quantization condition [6,7], shows the role of the Hopf fibration [8] in the monopole context, and gives adequate tools to analyse the symmetry properties [9,10,11]. Although there is a vast body of literature on this subject and the charge-monopole system has been deeply studied from various angles (see review by Milton [12]), the geometric and functional analytic aspects of constructing the Weyl correspondence between symbols and operators in this topologically nontrivial case deserve more study, especially in connection with developments in the so-called magnetic Weyl calculus [13,14,15,16,17,18].…”