Gauged Harmonic Maps, Born-Infeld Electromagnetism, and Magnetic Vortices

Research output: Contribution to journalArticle

Abstract

We study maps from a 2-surface into the standard 2-sphere coupled with Born-Infeld geometric electromagnetism through an Abelian gauge field. Such a formalism extends the classical harmonic map model, known as the σ-model, governing the spin vector orientation in a ferromagnet and allows us to obtain the coexistence of vortices and antivortices characterized by opposite, self-excited, magnetic flux lines. We show that the Born-Infeld free parameter may be used to achieve arbitrarily high local concentration of magnetic flux lines and that the total minimum energy is an additive function of these quantized flux lines realized as the numbers of vortices and antivortices. In the case where the underlying surface, or the domain, is compact, we obtain a necessary and sufficient condition for the existence of a unique solution representing a prescribed distribution of vortices and antivortices. In the case where the domain is the full plane, we prove the existence of a unique solution representing an arbitrary distribution of vortices and antivortices. Furthermore, we also consider the Einstein gravitation induced by these vortices, known as cosmic strings, and establish the existence of a solution representing a prescribed distribution of cosmic strings and cosmic antistrings under a necessary and sufficient condition that makes the underlying surface a complete surface with respect to the induced gravitational metric.

Original languageEnglish (US)
Pages (from-to)1631-1665
Number of pages35
JournalCommunications on Pure and Applied Mathematics
Volume56
Issue number11
DOIs
StatePublished - Nov 2003

Fingerprint

Electromagnetism
Harmonic Maps
Vortex
Vortex flow
Cosmic Strings
Magnetic flux
Unique Solution
Line
Necessary Conditions
Additive Function
Ferromagnet
Sufficient Conditions
Gravitation
Gauge Field
Coexistence
Gages
Albert Einstein
Fluxes
Metric
Arbitrary

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Gauged Harmonic Maps, Born-Infeld Electromagnetism, and Magnetic Vortices. / Lin, Fanghua; Yang, Yisong.

In: Communications on Pure and Applied Mathematics, Vol. 56, No. 11, 11.2003, p. 1631-1665.

Research output: Contribution to journalArticle

@article{6454b2d7fc7347e0a6efaeb13f6cf2e8,
title = "Gauged Harmonic Maps, Born-Infeld Electromagnetism, and Magnetic Vortices",
abstract = "We study maps from a 2-surface into the standard 2-sphere coupled with Born-Infeld geometric electromagnetism through an Abelian gauge field. Such a formalism extends the classical harmonic map model, known as the σ-model, governing the spin vector orientation in a ferromagnet and allows us to obtain the coexistence of vortices and antivortices characterized by opposite, self-excited, magnetic flux lines. We show that the Born-Infeld free parameter may be used to achieve arbitrarily high local concentration of magnetic flux lines and that the total minimum energy is an additive function of these quantized flux lines realized as the numbers of vortices and antivortices. In the case where the underlying surface, or the domain, is compact, we obtain a necessary and sufficient condition for the existence of a unique solution representing a prescribed distribution of vortices and antivortices. In the case where the domain is the full plane, we prove the existence of a unique solution representing an arbitrary distribution of vortices and antivortices. Furthermore, we also consider the Einstein gravitation induced by these vortices, known as cosmic strings, and establish the existence of a solution representing a prescribed distribution of cosmic strings and cosmic antistrings under a necessary and sufficient condition that makes the underlying surface a complete surface with respect to the induced gravitational metric.",
author = "Fanghua Lin and Yisong Yang",
year = "2003",
month = "11",
doi = "10.1002/cpa.10106",
language = "English (US)",
volume = "56",
pages = "1631--1665",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
publisher = "Wiley-Liss Inc.",
number = "11",

}

TY - JOUR

T1 - Gauged Harmonic Maps, Born-Infeld Electromagnetism, and Magnetic Vortices

AU - Lin, Fanghua

AU - Yang, Yisong

PY - 2003/11

Y1 - 2003/11

N2 - We study maps from a 2-surface into the standard 2-sphere coupled with Born-Infeld geometric electromagnetism through an Abelian gauge field. Such a formalism extends the classical harmonic map model, known as the σ-model, governing the spin vector orientation in a ferromagnet and allows us to obtain the coexistence of vortices and antivortices characterized by opposite, self-excited, magnetic flux lines. We show that the Born-Infeld free parameter may be used to achieve arbitrarily high local concentration of magnetic flux lines and that the total minimum energy is an additive function of these quantized flux lines realized as the numbers of vortices and antivortices. In the case where the underlying surface, or the domain, is compact, we obtain a necessary and sufficient condition for the existence of a unique solution representing a prescribed distribution of vortices and antivortices. In the case where the domain is the full plane, we prove the existence of a unique solution representing an arbitrary distribution of vortices and antivortices. Furthermore, we also consider the Einstein gravitation induced by these vortices, known as cosmic strings, and establish the existence of a solution representing a prescribed distribution of cosmic strings and cosmic antistrings under a necessary and sufficient condition that makes the underlying surface a complete surface with respect to the induced gravitational metric.

AB - We study maps from a 2-surface into the standard 2-sphere coupled with Born-Infeld geometric electromagnetism through an Abelian gauge field. Such a formalism extends the classical harmonic map model, known as the σ-model, governing the spin vector orientation in a ferromagnet and allows us to obtain the coexistence of vortices and antivortices characterized by opposite, self-excited, magnetic flux lines. We show that the Born-Infeld free parameter may be used to achieve arbitrarily high local concentration of magnetic flux lines and that the total minimum energy is an additive function of these quantized flux lines realized as the numbers of vortices and antivortices. In the case where the underlying surface, or the domain, is compact, we obtain a necessary and sufficient condition for the existence of a unique solution representing a prescribed distribution of vortices and antivortices. In the case where the domain is the full plane, we prove the existence of a unique solution representing an arbitrary distribution of vortices and antivortices. Furthermore, we also consider the Einstein gravitation induced by these vortices, known as cosmic strings, and establish the existence of a solution representing a prescribed distribution of cosmic strings and cosmic antistrings under a necessary and sufficient condition that makes the underlying surface a complete surface with respect to the induced gravitational metric.

UR - http://www.scopus.com/inward/record.url?scp=2442549733&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2442549733&partnerID=8YFLogxK

U2 - 10.1002/cpa.10106

DO - 10.1002/cpa.10106

M3 - Article

AN - SCOPUS:2442549733

VL - 56

SP - 1631

EP - 1665

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 11

ER -