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

DOI:
https://doi.org/10.1090/S0002-9947-1991-1062870-5

MathSciNet review:
1062870

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 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 , 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".

**[AB]**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**, https://doi.org/10.1016/0370-2693(88)91292-0**[AJS]**Guy Arsenault, Michel Jacques, and Yvan Saint-Aubin,*Collapse and exponentiation of infinite symmetry algebras of Euclidean projective and Grassmannian 𝜎 models*, J. Math. Phys.**29**(1988), no. 6, 1465–1471. MR**944464**, https://doi.org/10.1063/1.527941**[At]**M. F. Atiyah,*Instantons in two and four dimensions*, Comm. Math. Phys.**93**(1984), no. 4, 437–451. MR**763752****[EL]**J. Eells and L. Lemaire,*A report on harmonic maps*, Bull. London Math. Soc.**10**(1978), no. 1, 1–68. MR**495450**, https://doi.org/10.1112/blms/10.1.1**[EW]**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**, https://doi.org/10.1112/jlms/s2-28.1.161**[Gul]**M. A. Guest,*Geometry of maps between generalized flag manifolds*, J. Differential Geom.**25**(1987), no. 2, 223–247. MR**880184****[Gu2]**M. A. Guest,*Harmonic two-spheres in complex projective space and some open problems*, Exposition. Math.**10**(1992), no. 1, 61–87. MR**1149882****[JS]**Michel Jacques and Yvan Saint-Aubin,*Infinite-dimensional Lie algebras acting on the solution space of various 𝜎 models*, J. Math. Phys.**28**(1987), no. 10, 2463–2479. MR**908018**, https://doi.org/10.1063/1.527736**[Po]**K. Pohlmeyer,*Integrable Hamiltonian systems and interactions through quadratic constraints*, Comm. Math. Phys.**46**(1976), no. 3, 207–221. MR**0408535****[PS]**Andrew Pressley and Graeme Segal,*Loop groups*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR**900587****[Ro]**Y. L. Rodin,*The Riemann boundary value problem on Riemann surfaces*, Reidel, Dordrecht, 1988.**[Se]**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****[Uh]**Karen Uhlenbeck,*Harmonic maps into Lie groups: classical solutions of the chiral model*, J. Differential Geom.**30**(1989), no. 1, 1–50. MR**1001271****[Va]**Giorgio Valli,*On the energy spectrum of harmonic 2-spheres in unitary groups*, Topology**27**(1988), no. 2, 129–136. MR**948176**, https://doi.org/10.1016/0040-9383(88)90032-8**[Wi]**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**, https://doi.org/10.1098/rsta.1985.0047**[Wo]**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**, https://doi.org/10.1112/plms/s3-58.3.608**[ZM]**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****[ZS]**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**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58E20,
22E67

Retrieve articles in all journals with MSC: 58E20, 22E67

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1991-1062870-5

Article copyright:
© Copyright 1991
American Mathematical Society