### Abstract

Two sharp existence and uniqueness theorems are presented for solutions of multiple vortices arising in a six-dimensional brane-world supersymmetric gauge field theory under the general gauge symmetry group G =U(1) × SU(N) and with N Higgs scalar fields in the fundamental representation of G. Specifically, when the space of extra dimension is compact so that vortices are hosted in a 2-torus of volume /ω/, the existence of a unique multiple vortex solution representing n1, . . . , nN , respectively, prescribed vortices arising in the N species of the Higgs fields is established under the explicitly stated necessary and sufficient condition ni <g^{2}v^{2}/8πN/ ω/ + 1/N(1 -1/N) [g/e]^{2})n, i =1, . . . ,N, where e and g are the U(1) electromagnetic and SU(N) chromatic coupling constants, v measures the energy scale of broken symmetry and n =σ ^{N}_{i=1} ni is the total vortex number; when the space of extra dimension is the full plane, the existence and uniqueness of an arbitrarily prescribed n-vortex solution of finite energy is always ensured. These vortices are governed by a system of nonlinear elliptic equations, which may be reformulated to allow a variational structure. Proofs of existence are then developed using the methods of calculus of variations.

Original language | English (US) |
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Pages (from-to) | 3923-3946 |

Number of pages | 24 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 468 |

Issue number | 2148 |

DOIs | |

State | Published - Dec 8 2012 |

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### Keywords

- Calculus of variations
- Non-Abelian gauge theory
- Nonlinear elliptic equations
- Vortices

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)
- Physics and Astronomy(all)