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

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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 languageEnglish (US)
Pages (from-to)579-609
Number of pages31
JournalCommunications in Mathematical Physics
Volume249
Issue number3
DOIs
StatePublished - Aug 2004

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Gauge Field Theories
Riemann Surface
Pole
poles
Zero
Imply
strings
Existence Theory
Cosmic Strings
Sigma Models
Gravitation
Einstein Equations
Gauge Field
Einstein equations
Nonexistence
equivalence
Equations of Motion
Torus
equations of motion
topology

ASJC Scopus subject areas

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

Cite this

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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.",
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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.

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