### Abstract

In this paper, we study the existence of prescribed cosmic strings and antistrings realized as the static solutions with prescribed zeros and poles of the Einstein equations coupled with the classical sigma model and an Abelian gauge field over a compact Riemann surface. We show that the equations of motion are equivalent to a system of self-dual equations and the presence of string defects are necessary and sufficient for gravitation which implies the equivalence of topology and geometry in the model. More precisely, we prove that the absence of a solution with zeros and poles implies that the underlying Riemann surface S must be a flat 2-torus and that the existence of a solution with zeros and poles implies that S must be a 2-sphere. Furthermore, we develop an existence theory for solutions with prescribed zeros and poles. We also obtain some nonexistence results.

Original language | English (US) |
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Pages (from-to) | 579-609 |

Number of pages | 31 |

Journal | Communications in Mathematical Physics |

Volume | 249 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2004 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

**Prescribing zeros and poles on a compact Riemann surface for a gravitationally coupled Abelian gauge field theory.** / Yang, Y.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Prescribing zeros and poles on a compact Riemann surface for a gravitationally coupled Abelian gauge field theory

AU - Yang, Y.

PY - 2004/8

Y1 - 2004/8

N2 - In this paper, we study the existence of prescribed cosmic strings and antistrings realized as the static solutions with prescribed zeros and poles of the Einstein equations coupled with the classical sigma model and an Abelian gauge field over a compact Riemann surface. We show that the equations of motion are equivalent to a system of self-dual equations and the presence of string defects are necessary and sufficient for gravitation which implies the equivalence of topology and geometry in the model. More precisely, we prove that the absence of a solution with zeros and poles implies that the underlying Riemann surface S must be a flat 2-torus and that the existence of a solution with zeros and poles implies that S must be a 2-sphere. Furthermore, we develop an existence theory for solutions with prescribed zeros and poles. We also obtain some nonexistence results.

AB - In this paper, we study the existence of prescribed cosmic strings and antistrings realized as the static solutions with prescribed zeros and poles of the Einstein equations coupled with the classical sigma model and an Abelian gauge field over a compact Riemann surface. We show that the equations of motion are equivalent to a system of self-dual equations and the presence of string defects are necessary and sufficient for gravitation which implies the equivalence of topology and geometry in the model. More precisely, we prove that the absence of a solution with zeros and poles implies that the underlying Riemann surface S must be a flat 2-torus and that the existence of a solution with zeros and poles implies that S must be a 2-sphere. Furthermore, we develop an existence theory for solutions with prescribed zeros and poles. We also obtain some nonexistence results.

UR - http://www.scopus.com/inward/record.url?scp=4544310404&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4544310404&partnerID=8YFLogxK

U2 - 10.1007/s00220-004-1119-2

DO - 10.1007/s00220-004-1119-2

M3 - Article

AN - SCOPUS:4544310404

VL - 249

SP - 579

EP - 609

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -