### Abstract

We describe a general framework in which subgroups of the loop group AGl_{n}C act on the space of harmonic maps from S^{2} to Gl_{n}C. This represents a simplification of the action considered by Zakharov-Mikhailov-Shabat [ZM, ZS] in that we take the contour for the Riemann-Hilbert problem to be a union of circles; however, it reduces the basic ingredient to the well-known Birkhoff decomposition of AGl_{n}nC, and this facilitates a rigorous treatment. We give various concrete examples of the action, and use these to investigate a suggestion of Uhlenbeck [Uh] that a limiting version of such an action (“completion”) gives rise to her fundamental process of “adding a uniton”. It turns out that this does not occur, because completion preserves the energy of harmonic maps. However, in the special case of harmonic maps from S^{2} to complex projective space, we describe a modification of this completion procedure which does indeed reproduce “adding a uniton”.

Original language | English |
---|---|

Pages (from-to) | 1-26 |

Number of pages | 26 |

Journal | Transactions of the American Mathematical Society |

Volume | 326 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1991 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*326*(2), 1-26. https://doi.org/10.1090/S0002-9947-1991-1062870-5

**Actions of loop groups on harmonic maps.** / Bergvel, M. J.; Guest, Martin.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 326, no. 2, pp. 1-26. https://doi.org/10.1090/S0002-9947-1991-1062870-5

}

TY - JOUR

T1 - Actions of loop groups on harmonic maps

AU - Bergvel, M. J.

AU - Guest, Martin

PY - 1991

Y1 - 1991

N2 - We describe a general framework in which subgroups of the loop group AGlnC act on the space of harmonic maps from S2 to GlnC. This represents a simplification of the action considered by Zakharov-Mikhailov-Shabat [ZM, ZS] in that we take the contour for the Riemann-Hilbert problem to be a union of circles; however, it reduces the basic ingredient to the well-known Birkhoff decomposition of AGlnnC, and this facilitates a rigorous treatment. We give various concrete examples of the action, and use these to investigate a suggestion of Uhlenbeck [Uh] that a limiting version of such an action (“completion”) gives rise to her fundamental process of “adding a uniton”. It turns out that this does not occur, because completion preserves the energy of harmonic maps. However, in the special case of harmonic maps from S2 to complex projective space, we describe a modification of this completion procedure which does indeed reproduce “adding a uniton”.

AB - We describe a general framework in which subgroups of the loop group AGlnC act on the space of harmonic maps from S2 to GlnC. This represents a simplification of the action considered by Zakharov-Mikhailov-Shabat [ZM, ZS] in that we take the contour for the Riemann-Hilbert problem to be a union of circles; however, it reduces the basic ingredient to the well-known Birkhoff decomposition of AGlnnC, and this facilitates a rigorous treatment. We give various concrete examples of the action, and use these to investigate a suggestion of Uhlenbeck [Uh] that a limiting version of such an action (“completion”) gives rise to her fundamental process of “adding a uniton”. It turns out that this does not occur, because completion preserves the energy of harmonic maps. However, in the special case of harmonic maps from S2 to complex projective space, we describe a modification of this completion procedure which does indeed reproduce “adding a uniton”.

UR - http://www.scopus.com/inward/record.url?scp=84966255655&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966255655&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1991-1062870-5

DO - 10.1090/S0002-9947-1991-1062870-5

M3 - Article

AN - SCOPUS:84966255655

VL - 326

SP - 1

EP - 26

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -