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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Actions of loop groups on harmonic maps

Authors: M. J. Bergvelt and M. A. Guest
Journal: Trans. Amer. Math. Soc. 326 (1991), 861-886
MSC: Primary 58E20; Secondary 22E67
MathSciNet review: 1062870
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Abstract: We describe a general framework in which subgroups of the loop group $ \Lambda G{l_n}\mathbb{C}$ act on the space of harmonic maps from $ {S^2}$ to $ G{l_n}\mathbb{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 $ \Lambda G{l_n}\mathbb{C}$, 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".

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  • [AB] J. Avan and G. Bellon, Groups of dressing transformations for integrable models in dimension two, Phys. Lett. B 213 (1988), 459-465. MR 967732 (90e:58056)
  • [AJS] G. Arsenault, M. Jacques, and Y. Saint-Aubin, Collapse and exponentiation of infinite symmetry algebras of Euclidean projective and Grassmannian sigma models, J. Math. Phys. 29 (1988), 1465-1471. MR 944464 (89k:58298)
  • [At] M. F. Atiyah, Instantons in two and four dimensions, Comm. Math. Phys. 93 (1984), 437-451. MR 763752 (86m:32042)
  • [EL] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. MR 495450 (82b:58033)
  • [EW] S. Erdem and J. C. Wood, On the construction of harmonic maps into a Grassmannian, J. London Math. Soc. (2) 28 (1983), 161-174. MR 703474 (85j:58044)
  • [Gul] M. A. Guest, Geometry of maps between generalized flag manifolds, J. Differential Geom. 25 (1987), 223-247. MR 880184 (88f:58033)
  • [Gu2] -, Harmonic two-spheres in complex projective space and some open problems, Exposition. Math. (to appear). MR 1149882 (93c:58053)
  • [JS] M. Jacques and Y. Saint-Aubin, Infinite dimensional Lie algebras acting on the solution space of various sigma models, J. Math. Phys. 28 (1987), 2463-2479. MR 908018 (89c:58027)
  • [Po] K. Pohlmeyer, Integrable Hamiltonian systems and interactions through constraints, Comm. Math. Phys. 46 (1976), 207-221. MR 0408535 (53:12299)
  • [PS] A. N. Pressley and G. B. Segal, Loop groups, Oxford Univ. Press, 1986. MR 900587 (88i:22049)
  • [Ro] Y. L. Rodin, The Riemann boundary value problem on Riemann surfaces, Reidel, Dordrecht, 1988.
  • [Se] G. B. Segal, Loop groups and harmonic maps, Advances in Homotopy Theory (S. M. Salamon, B. Steer, and W. A. Sutherland, eds.), LMS Lecture Notes 139, Cambridge Univ. Press, 1989, pp. 153-164. MR 1055875 (91m:58043)
  • [Uh] K. Uhlenbeck, Harmonic maps into Lie groups (Classical solutions of the chiral model), J. Differential Geom. 30 (1989), 1-50. MR 1001271 (90g:58028)
  • [Va] G. Valli, On the energy spectrum of harmonic $ 2$-spheres in unitary groups, Topology 27 (1988), 129-136. MR 948176 (90f:58042)
  • [Wi] G. Wilson, Infinite dimensional Lie groups and algebraic geometry in soliton theory, Philos. Trans. Roy. Soc. London Ser. A 315 (1985), 393-404. MR 836741 (87i:58089)
  • [Wo] J. C. Wood, Explicit construction and parametrization of harmonic two-spheres in the unitary group, Proc. London Math. Soc. 58 (1989), 608-624. MR 988105 (90k:58055)
  • [ZM] V. E. Zakharov and A. V. Mikhailov, Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method, Soviet Phys. JETP 47 (1978), 1017-1027. MR 524247 (80c:81115)
  • [ZS] V. E. Zakharov and A. B. Shabat, Integration of non-linear equations of mathematical physics by the inverse scattering method. II, Functional Anal. Appl. 13 (1979), 13-22. MR 545363 (82m:35137)

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