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Filtrations, Factorizations and Explicit Formulae for Harmonic Maps

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Abstract

We use filtrations of the Grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a Riemann surface, and so all harmonic maps from the 2-sphere, to the unitary group for a general class of factorizations by unitons. We show how these specialize to give explicit formulae for such harmonic maps to each of the classical compact Lie groups and their inner symmetric spaces—the nonlinear σ-model of particle physics. Our methods also give an explicit Iwasawa decomposition of the algebraic loop group.

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Authors and Affiliations

  1. Department of Mathematics & Computer Science, and CP3-Origins, Centre of Excellence for Particle Physics Phenomenology, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark

    Martin Svensson

  2. Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    John C. Wood

Authors
  1. Martin Svensson
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  2. John C. Wood
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Correspondence to Martin Svensson.

Additional information

Communicated by A. Connes

The first author was supported by the Danish Council for Independent Research and the Danish National Research Foundation.

The second author thanks the Department of Mathematics and Computer Science of the University of Southern Denmark, Odense, for support and hospitality during the preparation of this work.

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Svensson, M., Wood, J.C. Filtrations, Factorizations and Explicit Formulae for Harmonic Maps. Commun. Math. Phys. 310, 99–134 (2012). https://doi.org/10.1007/s00220-011-1398-3

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  • DOI: https://doi.org/10.1007/s00220-011-1398-3

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