Dirac-Yang monopoles and their regular counterparts

The Dirac-Yang monopoles are singular Yang–Mills field configurations in all Euclidean dimensions. The regular counterpart of the Dirac monopole in D = 3 is the t Hooft-Polyakov monopole, the former being simply a gauge transform of the asymptotic fields of the latter. Here, regular counterparts of Dirac-Yang monopoles in all dimensions, are described. In the first part of this talk the hierarchy of Dirac– Yang (DY) monopoles will be defined, in the second part the motivation to study these in a topoical context will be briefly presented, and in the last part, two classes of regular counterparts to the DY hierarchy will be presented. 1 The Dirac–Yang hierarchy in D ≥ 3 The Dirac [1] monopole can be constructed by gauge transforming the asymptotic ’t HooftPolyakov monopole [2] in D = 3, which can be taken to be spherically symmetric , such that the SO(3) isovector Higgs field is gauged to a (trivial) constant, and the SU(2) ∼ SO(3) gauge group of the Yang-Mills (YM) connection breaks down to U(1) ∼ SO(2), the resulting Abelian connection developing a line singularity on the positive or negative (x3 =)z-axis. In exactly the same way, the Yang [3] monopole can be constructed by gauge transforming the asymptotic D = 5 dimensional ’monopole’ [4] such that the such that the SO(5) isovector Higgs field is gauged to a (trivial) constant, and the SO(5) gauge group of the YM connection breaks down to SO(4), the resulting non Abelian connection developing a line singularity on the positive or negative x5-axis. In fact, the residual non Abelian connection can take its values in one or other chiral representations of SU(2), as formulated by Yang [3], but this is a low dimensional accident which does not apply to the higher diemnsional analogues to be defined below, all of which are SO(D−1) connctions. It is not infact necessary to restrict to spherically symmetric fields only. By choosing to start with the asymptotic axially symmetric fields characterised with vorticity n, the gauge tranformed connection is just n times the usual Dirac monopole field.