{
  "id": "1007.4252",
  "version": "v1",
  "published": "2010-07-24T07:23:27.000Z",
  "updated": "2010-07-24T07:23:27.000Z",
  "title": "On the non-Abelian monopoles on the background of spaces with constant curvature",
  "authors": [
    "V. Red'kov"
  ],
  "comment": "82 pages. Invited talk to BGL-7: International Conference on Non-Euclidean Geometry and its applications. 5 -- 9 July 2010, Kluj-Napoca, Romania",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-07-24T07:23:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "G.1"
    ],
    "keywords": [
      "constant curvature",
      "non-abelian monopoles",
      "background",
      "flat minkowski space",
      "covariant tetrad formalism"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 82,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 863208,
      "adsabs": "2010arXiv1007.4252R"
    }
  }
}