Actions of loop groups on harmonic maps
Authors:
M. J. Bergvelt and M. A. Guest
Journal:
Trans. Amer. Math. Soc. 326 (1991), 861886
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 act on the space of harmonic maps from to . This represents a simplification of the action considered by ZakharovMikhailovShabat [ZM, ZS] in that we take the contour for the RiemannHilbert problem to be a union of circles; however, it reduces the basic ingredient to the wellknown Birkhoff decomposition of , 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 to complex projective space, we describe a modification of this completion procedure which does indeed reproduce "adding a uniton".
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V.
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B. Šabat, Integration of the nonlinear equations of
mathematical physics by the method of the inverse scattering problem.
II, Funktsional. Anal. i Prilozhen. 13 (1979),
no. 3, 13–22 (Russian). MR 545363
(82m:35137)
 [AB]
 J. Avan and G. Bellon, Groups of dressing transformations for integrable models in dimension two, Phys. Lett. B 213 (1988), 459465. MR 967732 (90e:58056)
 [AJS]
 G. Arsenault, M. Jacques, and Y. SaintAubin, Collapse and exponentiation of infinite symmetry algebras of Euclidean projective and Grassmannian sigma models, J. Math. Phys. 29 (1988), 14651471. MR 944464 (89k:58298)
 [At]
 M. F. Atiyah, Instantons in two and four dimensions, Comm. Math. Phys. 93 (1984), 437451. MR 763752 (86m:32042)
 [EL]
 J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 168. 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), 161174. MR 703474 (85j:58044)
 [Gul]
 M. A. Guest, Geometry of maps between generalized flag manifolds, J. Differential Geom. 25 (1987), 223247. MR 880184 (88f:58033)
 [Gu2]
 , Harmonic twospheres in complex projective space and some open problems, Exposition. Math. (to appear). MR 1149882 (93c:58053)
 [JS]
 M. Jacques and Y. SaintAubin, Infinite dimensional Lie algebras acting on the solution space of various sigma models, J. Math. Phys. 28 (1987), 24632479. MR 908018 (89c:58027)
 [Po]
 K. Pohlmeyer, Integrable Hamiltonian systems and interactions through constraints, Comm. Math. Phys. 46 (1976), 207221. 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. 153164. MR 1055875 (91m:58043)
 [Uh]
 K. Uhlenbeck, Harmonic maps into Lie groups (Classical solutions of the chiral model), J. Differential Geom. 30 (1989), 150. MR 1001271 (90g:58028)
 [Va]
 G. Valli, On the energy spectrum of harmonic spheres in unitary groups, Topology 27 (1988), 129136. 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), 393404. MR 836741 (87i:58089)
 [Wo]
 J. C. Wood, Explicit construction and parametrization of harmonic twospheres in the unitary group, Proc. London Math. Soc. 58 (1989), 608624. MR 988105 (90k:58055)
 [ZM]
 V. E. Zakharov and A. V. Mikhailov, Relativistically invariant twodimensional models of field theory which are integrable by means of the inverse scattering problem method, Soviet Phys. JETP 47 (1978), 10171027. MR 524247 (80c:81115)
 [ZS]
 V. E. Zakharov and A. B. Shabat, Integration of nonlinear equations of mathematical physics by the inverse scattering method. II, Functional Anal. Appl. 13 (1979), 1322. MR 545363 (82m:35137)
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DOI:
http://dx.doi.org/10.1090/S00029947199110628705
PII:
S 00029947(1991)10628705
Article copyright:
© Copyright 1991
American Mathematical Society
