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

We formulate models of complex scalar fields in a space-time that has a two-dimensional sphere as extra dimensions. The two-sphere S2 is assumed to have the Dirac-Wu-Yang monopole as a background gauge field. The nontrivial topology of the monopole induces topological defects, i.e., vortices, in the scalar field. When the radius of S2 is larger than a critical radius, the scalar field develops a vacuum expectation value and creates vortices in S2. Then the vortices break the rotational symmetry of S2. We exactly evaluate the critical radius as rq=√(|q|)/μ, where q is the monopole number and μ is 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 S2, and when q=3/2, three vortices sit at the vertices of the largest triangle on S2. The symmetry of the model G=U(1)xSU(2)xCP, is broken to H1/2=U(1)', H1=U(1)''xCP, H3/2=D3h, respectively. Here D3h 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 Gdoublet=U(1)xSU(2)xSU(2)fxP to Hdoublet=SU(2)'xP. Finally, we obtain the exact vacuum solution of the model with the multiplet (q1,q2,...,q2j+1)=(j,j,...,j) and show that the symmetry is broken from Gmultiplet=U(1)xSU(2)xSU(2j+1)fxCP to Hmultiplet=SU(2)'xCP'. Our results caution that a careful analysis of the dynamics of the topological defects is required for construction of a reliable model that possesses such a defect structure