### 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) |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

**Existence of multiple vortices in supersymmetric gauge field theory.** / Chen, Shouxin; Yang, Yisong.

Research output: Contribution to journal › Article

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 468, no. 2148, pp. 3923-3946. https://doi.org/10.1098/rspa.2012.0159

}

TY - JOUR

T1 - Existence of multiple vortices in supersymmetric gauge field theory

AU - Chen, Shouxin

AU - Yang, Yisong

PY - 2012/12/8

Y1 - 2012/12/8

N2 - 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 2v2/8πN/ 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.

AB - 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 2v2/8πN/ 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.

KW - Calculus of variations

KW - Non-Abelian gauge theory

KW - Nonlinear elliptic equations

KW - Vortices

UR - http://www.scopus.com/inward/record.url?scp=84870337811&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84870337811&partnerID=8YFLogxK

U2 - 10.1098/rspa.2012.0159

DO - 10.1098/rspa.2012.0159

M3 - Article

VL - 468

SP - 3923

EP - 3946

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0080-4630

IS - 2148

ER -