arXiv2001
We formulate models of complex scalar fields in the space-time that has a two-dimensional sphere as extra dimensions. The Dirac-Wu-Yang monopole is set in two-sphere S^2 as a background gauge field. The nontrivial topology of the monopole induces topological defects, i.e. vortices. When the radius of S^2 is larger than a critical radius, the scalar field develops a vacuum expectation value and creates vortices in S^2. Then the vortices break the rotational symmetry of S^2. We exactly evaluate the critical radius as r_q = \sqrt{|q|}/μ, where q is the monopole number and μis the imaginary mass of the scalar. We show that the vortices repel each other. We analyze the vacua of the 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 diametrical points on S^2; when q=3/2, three vortices sit at the vertices of the largest triangle on S^2. The symmetry of the model G = U(1) x SU(2) x CP is broken to H_{1/2} = U(1)', H_{1} = U(1)'' x CP, H_{3/2} = D_{3h}, respectively. Here D_{3h} is the symmetry group of a regular triangle. We extend our analysis to the 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 the exact 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) x SU(2) x SU(2j+1)_f x CP to H_{multiplet} = SU(2)' x CP'. Our results caution that a careful analysis of dynamics of the topological defects is required for construction of a reliable model that possesses such a defect structure.