## Abstract

We study the semilocal version of Popov's vortex equations on ^{S2}. Though they are not integrable, we construct two families of exact solutions which are expressed in terms of rational functions on ^{S2}. One family is a trivial embedding of Liouville-type solutions of the Popov equations obtained by Manton, where the vortex number is an even integer. The other family of solutions is constructed through a field redefinition which relates the semilocal Popov equation to the original Popov equation but with the ratio of radii 3/2, which is not integrable. These solutions have vortex number N = 3n - 2 where n is a positive integer, and hence N = 1 solutions belong to this family. In particular, we show that the N = 1 solution with reflection symmetry is the well-known C^{P1} lump configuration with unit size where the scalars lie on ^{S3} with radius 3/2. It generates the uniform magnetic field of a Dirac monopole with unit magnetic charge on ^{S2}.

Original language | English |
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Pages (from-to) | 253-257 |

Number of pages | 5 |

Journal | Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics |

Volume | 733 |

DOIs | |

State | Published - 2 Jun 2014 |

### Bibliographical note

Funding Information:This work was supported by the Mid-career Researcher Program grant No. 2012-045385/2013-056327 and the WCU grant No. R32-10130 through NRF of Korea funded by the Korean government (MEST), and the Research fund No. 1-2008-2935-001-2 by Ewha Womans University .