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We formulate models of complex scalar fields in the space-time that has atwo-dimensional sphere as extra dimensions. The Dirac-Wu-Yang monopole is setin two-sphere S^2 as a background gauge field. The nontrivial topology of themonopole induces topological defects, i.e. vortices. When the radius of S^2 islarger than a critical radius, the scalar field develops a vacuum expectationvalue and creates vortices in S^2. Then the vortices break the rotationalsymmetry of S^2. We exactly evaluate the critical radius as r_q =\sqrt{|q|}/\mu, where q is the monopole number and \mu is the imaginary mass ofthe scalar. We show that the vortices repel each other. We analyze the vacua ofthe models with one scalar field in each case of q=1/2, 1, 3/2 and find that:when q=1/2, a single vortex exists; when q=1, two vortices sit at diametricalpoints on S^2; when q=3/2, three vortices sit at the vertices of the largesttriangle on S^2. The symmetry of the model G = U(1) x SU(2) x CP is broken toH_{1/2} = U(1)', H_{1} = U(1)'' x CP, H_{3/2} = D_{3h}, respectively. HereD_{3h} is the symmetry group of a regular triangle. We extend our analysis tothe doublet scalar fields and show that the symmetry is broken from G_{doublet}= U(1) x SU(2) x SU(2)_f x P to H_{doublet} = SU(2)' x P. Finally we obtain theexact vacuum of the model with the multiplet (q_1, q_2,..., q_{2j+1}) = (j,j,..., j) and show that the symmetry is broken from G_{multiplet} = U(1) xSU(2) x SU(2j+1)_f x CP to H_{multiplet} = SU(2)' x CP'. Our results cautionthat a careful analysis of dynamics of the topological defects is required forconstruction of a reliable model that possesses such a defect structure.Comment: 31 pages, 1 eps figure, LaTeX2e; the title changed, references added,  corrections mad

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

Spontaneous breaking of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mo>,</mml:mo><mml:mi> </mml:mi><mml:mi>P</mml:mi><mml:mo>,</mml:mo></mml:math>and rotational symmetries by topological defects in two extra dimensions

Spontaneous breaking of theC, P,and rotational symmetries by topological defects in two extra dimensions