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

TL;DR

This paper reexamines BPS monopole solutions in SO(3) models on spaces with constant curvature, revealing only three geometrically linked solutions and analyzing fermion interactions, with solutions in curved spaces and symmetry properties.

Contribution

It identifies three unique BPS monopole solutions in curved spaces and explores fermion behavior, linking solutions to geometry and uncovering symmetry invariances.

Findings

Only three solutions correspond to specific geometries.

The trivial solution embeds Abelian monopoles into non-Abelian models.

Additional symmetry invariance in Lobachevsky and Riemann models.

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.