### Abstract

This is a survey article concerning some recent mathematical results for the static selfdual cosmic-string solutions in the Abelian Higgs model and in the Weinberg-Salam standard model unifying electromagnetic and weak interactions, both coupled with gravity through the Einstein equations. For the Abelian Higgs strings there is a nearly complete picture. If the Riemann surface M on which the strings reside is compact, it can be shown that M must be S^{2} up to topological equivalence and there are only countably many values of the Higgs vacuum states for strings to exist. When M is noncompact and conformally a plane there are exact obstructions to the finiteness of energies and geodesic completeness of solutions. For the Weinberg-Salam strings, much is to be achieved. It Can be shown in this case that self-dual strings generated from W and Higgs condensation lead to an explicit formula for a positive cosmological constant and the gravitational metric is always noncomplete. This feature leads to the properties that the metric decays sufficiently rapidly at infinity and there exist non-Abelian electroweak strings of finite energies. It is established that for any integer N there are always suitable ranges of the electroweak parameters to allow the existence of W- and Higgs-condensed N-vortex solutions of finite energies.

Original language | English (US) |
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Pages (from-to) | 203-227 |

Number of pages | 25 |

Journal | International Journal of Modern Physics A |

Volume | 11 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1996 |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics
- Nuclear and High Energy Physics
- Astronomy and Astrophysics