On the non-Abelian monopoles on the background of spaces with constant curvature

Abstract

Procedure of constructing the BPS solutions in SO(3) model on the background of 4D-space-time with the spatial part as a model of constant curvature: Euclid, Riemann, Lobachevsky, is reexamined. It is shown that among possible solutions$W^{k}_{\alpha}(x)$ there exist just three ones which in a one-to-one correspondence can be associated with respective geometries, the known non-singular BPS-solution in the flat Minkowski space can be understood as a result of somewhat artificial combining the Minkowski space model with a possibility naturally linked up with the Loba\-chevsky geometry. A special solution $W^{k}_{(triv)\alpha} (x)$ in three spaces is described, which can be understood as result of embedding the Abelian monopole potential into the non-Abelian model. The problem of Dirac fermion doublet in the~external BPS-monopole potential in these curved spaces is examined on the base of generally covariant tetrad formalism by Tetrode-Weyl-Fock-Ivanenko. In the frame of spherical coordinates, and (Schr\"{o}dinger's) tetrad basis, and special unitary basis in isotopic space, an analog of Schwin\-ger's one in Abelian case, there arises a Schr\"{o}dinger's structure for extended operator ${\bf J} = {\bf l} + {\bf S} + {\bf T}$. Correspondingly, instead of monopole harmonics, the language of conventional Wigner $D$-functions is used, radial equations are founds in all three models, and solved in the case of trivial W^{k}_{(triv)\alpha} (x)$ in Lobachevsky and Riemann models. In the particular case $W^{k}_{(triv)\alpha} (x)$, the~doublet-monopole Hamiltonian is invariant under additional one-para\-metric group. This symmetry results in a freedom in choosing a~discrete operator $\hat{N}_{A}$ entering the complete set of quantum variables.