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Spontaneous breaking of the rotational symmetry induced by monopoles in extra dimensions. (English) Zbl 0971.81043
Summary: We propose a field theoretical model that exhibits spontaneous breaking of the rotational symmetry. The model has a two-dimensional sphere as extra dimensions of the space-time and consists of a complex scalar field and a background gauge field. The Dirac monopole, which is invariant under the rotations of the sphere, is taken as the background field. We show that when the radius of the sphere is larger than a certain critical radius, the vacuum expectation value of the scalar field develops vortices, which pin down the rotational symmetry to lower symmetries. We evaluate the critical radius and calculate configurations of the vortices by the lowest approximation. The original model has a $$U(1)\times SU(2)$$ symmetry and it is broken to $$U(1), U(1), D_3$$ for each case of the monopole number $$q=1/2,1,3/2$$, respectively, where $$D_3$$ is the symmetry group of a regular triangle. Moreover, we show that the vortex configurations are stable against higher corrections of the perturbative approximation.

##### MSC:
 81R40 Symmetry breaking in quantum theory 83E15 Kaluza-Klein and other higher-dimensional theories 81T13 Yang-Mills and other gauge theories in quantum field theory
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