Dirac Monopoles, from d = 2 to d = 5, Lecture I

Abstract

We review abelian (Dirac type) monopole solutions in an increasing number of dimensions. In doing so, we tie together three remarkable ideas, all of which date back to the twenties and thirties; the Dirac monopole (1931), the Hopf map (bundle) (1931) and the Kaluza-Klein dimensional reduction (compactification) scheme. Starting from Maxwell’s equations on the two sphere, we arrive via euclidean selfdual Einstein spaces, at the recently discovered Kaluza-Klein monopoles. These are regular time independent solutions to the five-dimensional theory of general relativity corresponding to monopoles in the Kaluza-Klein frame-work. Various properties of these solutions are briefly discussed.