Abstract
In this chapter some aspects of two-dimensional nonlinear sigma models are discussed. The solutions to the field equations are (pseudo-)harmonic maps of two-dimensional Minkowski space into some homogeneous Riemannian manifolds.
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Bibliography
R. Abraham and J. Marsden, Foundations of Mechanics, Addison-Wesley, 1978.
J. Avan and M. Talon, Rational and trigonometric constant non-antisymmetric r-matrices, Phys. Lett. B 241 (1990), 77–82.
O. Babelon and C.-M. Viallet, Hamiltonian structures and Lax equations, Phys. Lett. 237 B (1990), 411–416.
M. Bordemann, Generalized Lax pairs, the modified classical Yang-Baxter-equation, and affine geometry of Lie groups, Comm Math. Phys. 135 (1990), 201–216.
M. Bordemann, Nondegenerate bilinear forms on nonassociative algebras, Universität Freiburg preprint THEP 92 /3, April 1992.
M. Bordemann, M Forger and H. Römer, Homogeneous Kähler manifolds: Paving the way to new supersymmetric sigma models, Comm Math. Phys. 102 (1986), 605–647.
M. Bordemann, M. Forger, J. Laartz and U. Schäper, The Lie-Poisson structure of integrable classical non-linear sigma models, Comm. Math. Phys. 152 (1993), 167–190.
F.E. Burstall, D. Ferus, F. Pedit and U. Pinkall, Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras, Ann. of Math. 138 (1993), 173–212.
H.J. de Vega, H. Eichenherr and J.-M. Maillet, Classical and quantum algebras of non-local charges in sigma models, Comm Math. Phys. 92 (1984), 507–524.
H. Eichenherr, SU(N)-invariante nichtlineare sigma-modelle, Universität Heidelberg, Fakultät für Physik und Astronomie, Dissertation SS 78–2 (1978); published as: SU(N)-invariant nonlinear a-models, Nuclear Physics B 146 (1978), 215–223.
H. Eichenherr and M. Forger, On the dual symmetry of the nonlinear sigma models, Nucl. Phys. B 155 (1979), 381–393.
H. Eichenherr and M. Forger, More about nonlinear sigma models on symmetric spaces,Nucl. Phys. B 164 (1980), 528–535 and B 282 (1987), 745–746 (erratum).
H. Eichenherr and M. Forger, Higher local conservation laws for nonlinear sigma models on symmetric spaces, Comm Math. Phys. 82 (1981), 227–255.
L.D. Faddeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer-Verlag, Berlin 1987.
M. Forger, Nonlinear sigma models on symmetric spaces. In: Nonlinear Partial Differential Operators and Quantization Procedures, Proceedings, Clausthal, Germany 1981, ed. S.I. Andersson and H.D. Doebner, Lect. Notes in Math. 1037, Springer-Verlag, Berlin 1983.
M. Forger, Differential Geometric Methods in Nonlinear a-models and Gauge Theories, Dissertation, FU Berlin, Fachbereich Physik, 1979.
M. Forger, J. Laartz and U. Schäper, Current algebra of classical non-linear sigma models, Comm. Math. Phys. 146 (1992), 397–402.
L. Freidel and J.M. Maillet, Quadratic algebras and integrable systems,preprint LPTHE-24/91; On classical and quantum integrable field theories associated to KacMoody current algebras,preprint LPTHE-25/91.
M. Gell-Mann and M. Lévy, The axial vector current in beta decay, Nuovo Cimento 16 (1960), 705–726.
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York 1978.
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience, New York, 1963 ( Vol.I) and 1969 ( Vol.II).
B. Kostant, On differential geometry and homogeneous spaces II, Proc. Nat. Acad. Sci. 42 (1956), 354–357.
M. Löscher and K. Pohlmeyer, Scattering of massless lumps and nonlocal charges in the two-dimensional nonlinear a-model, Nuclear Physics B 137 (1978), 46–54.
J.-M. Maillet, Kac-Moody algebra and extended Yang-Baxter relations in the 0(N) non-linear sigma model, Phys. Lett. 162 B (1985), 137–142.
J.-M. Maillet, Hamiltonian structures for integrable classical theories from graded Kac-Moody algebras,Phys. Lett. 167 B (1986), 401–405.
J.-M. Maillet, New integrable canonical structures in two-dimensional models, Nucl. Phys. B 269 (1986), 54–76.
M. Malias, The principal chiral model as an integrable system,this volume.
L. Michel, Minima of Higgs-Landau polynomials, CERN preprint Ref.Th.2716-CERN (1979).
K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys. 46 (1976), 207–221.
M.A. Semenov-Tyan-Shanskii, What is a classical r-matrix, Funk. Anal. i Prilozh. 17 (1983), 17–33; English transl.: Funct. Anal. Appl. 17 (1983), 259–272.
K. Uhlenbeck, Harmonic maps into Lie groups (Classical solutions of the chiral model), J. Diff. Geom. 30 (1989), 1–50.
J.C. Wood, Harmonic maps into symmetric spaces and integrable systems,this volume.
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, Zh. Eksp. Teor. Fiz. 74 (1978), 1953–1973); Engl transi.: Soy. Phys. JETP 47 (6) (1978), 1017–1027.
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Bordemann, M., Forger, M., Laartz, J., Schäper, U. (1994). 2-dimensional nonlinear sigma models: zero curvature and Poisson structure. In: Fordy, A.P., Wood, J.C. (eds) Harmonic Maps and Integrable Systems. Aspects of Mathematics, vol E 23. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-14092-4_8
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DOI: https://doi.org/10.1007/978-3-663-14092-4_8
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
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