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St. Petersburg Mathematical Journal

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Geometry and analysis in nonlinear sigma models


Authors: D. Auckly, L. Kapitanski and J. M. Speight
Translated by: the authors
Original publication: Algebra i Analiz, tom 18 (2006), nomer 1.
Journal: St. Petersburg Math. J. 18 (2007), 1-19
MSC (2000): Primary 81T13
DOI: https://doi.org/10.1090/S1061-0022-06-00940-X
Published electronically: November 27, 2006
MathSciNet review: 2225211
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Abstract | References | Similar Articles | Additional Information

Abstract: The configuration space of a nonlinear sigma model is the space of maps from one manifold to another. This paper reviews the authors' work on nonlinear sigma models with target a homogeneous space. It begins with a description of the components, fundamental group, and cohomology of such configuration spaces, together with the physical interpretations of these results. The topological arguments given generalize to Sobolev maps. The advantages of representing homogeneous-space-valued maps by flat connections are described, with applications to the homotopy theory of Sobolev maps, and minimization problems for the Skyrme and Faddeev functionals. The paper concludes with some speculation about the possibility of using these techniques to define new invariants of manifolds.


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Additional Information

D. Auckly
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506

L. Kapitanski
Affiliation: Department of Mathematics, University of Miami, Coral Gabels, Florida 33124

J. M. Speight
Affiliation: School of Mathematics, University of Leeds, Leeds LS2 9JT, England

DOI: https://doi.org/10.1090/S1061-0022-06-00940-X
Keywords: Skyrme model, Faddeev model, Yang--Mills equations
Received by editor(s): June 16, 2005
Published electronically: November 27, 2006
Additional Notes: The first author was partially supported by NSF grant DMS-0204651. The second author was partially supported by NSF grant DMS-0436403. The third author was partially supported by EPSRC grant GR/R66982/01.
Article copyright: © Copyright 2006 American Mathematical Society

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