Actions of loop groups on harmonic maps

Bergvelt, M. J.; Guest, M. A.

doi:10.1090/S0002-9947-1991-1062870-5

Actions of loop groups on harmonic maps
HTML articles powered by AMS MathViewer

by M. J. Bergvelt and M. A. Guest PDF

Trans. Amer. Math. Soc. 326 (1991), 861-886 Request permission

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".

References

J. Avan and M. Bellon, Groups of dressing transformations for integrable models in dimension two, Phys. Lett. B 213 (1988), no. 4, 459–465. MR 967732, DOI 10.1016/0370-2693(88)91292-0
Guy Arsenault, Michel Jacques, and Yvan Saint-Aubin, Collapse and exponentiation of infinite symmetry algebras of Euclidean projective and Grassmannian $\sigma$ models, J. Math. Phys. 29 (1988), no. 6, 1465–1471. MR 944464, DOI 10.1063/1.527941
M. F. Atiyah, Instantons in two and four dimensions, Comm. Math. Phys. 93 (1984), no. 4, 437–451. MR 763752
J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, DOI 10.1112/blms/10.1.1
S. Erdem and J. C. Wood, On the construction of harmonic maps into a Grassmannian, J. London Math. Soc. (2) 28 (1983), no. 1, 161–174. MR 703474, DOI 10.1112/jlms/s2-28.1.161
M. A. Guest, Geometry of maps between generalized flag manifolds, J. Differential Geom. 25 (1987), no. 2, 223–247. MR 880184
M. A. Guest, Harmonic two-spheres in complex projective space and some open problems, Exposition. Math. 10 (1992), no. 1, 61–87. MR 1149882
Michel Jacques and Yvan Saint-Aubin, Infinite-dimensional Lie algebras acting on the solution space of various $\sigma$ models, J. Math. Phys. 28 (1987), no. 10, 2463–2479. MR 908018, DOI 10.1063/1.527736
K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys. 46 (1976), no. 3, 207–221. MR 408535
Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR 900587

The Riemann boundary value problem on Riemann surfaces

Graeme Segal, Loop groups and harmonic maps, Advances in homotopy theory (Cortona, 1988) London Math. Soc. Lecture Note Ser., vol. 139, Cambridge Univ. Press, Cambridge, 1989, pp. 153–164. MR 1055875, DOI 10.1017/CBO9780511662614.015
Karen Uhlenbeck, Harmonic maps into Lie groups: classical solutions of the chiral model, J. Differential Geom. 30 (1989), no. 1, 1–50. MR 1001271
Giorgio Valli, On the energy spectrum of harmonic $2$-spheres in unitary groups, Topology 27 (1988), no. 2, 129–136. MR 948176, DOI 10.1016/0040-9383(88)90032-8
G. Wilson, Infinite-dimensional Lie groups and algebraic geometry in soliton theory, Philos. Trans. Roy. Soc. London Ser. A 315 (1985), no. 1533, 393–404. New developments in the theory and application of solitons. MR 836741, DOI 10.1098/rsta.1985.0047
John C. Wood, Explicit construction and parametrization of harmonic two-spheres in the unitary group, Proc. London Math. Soc. (3) 58 (1989), no. 3, 608–624. MR 988105, DOI 10.1112/plms/s3-58.3.608
V. E. Zakharov and A. V. Mikhaĭlov, Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method; Russian transl., Soviet Phys. JETP 74 (1978), no. 6, 1953–1973. MR 524247
V. E. Zaharov and A. 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