×

zbMATH — the first resource for mathematics

Description of surfaces associated with \({\mathbb C} P^{N-1}\) sigma models on Minkowski space. (English) Zbl 1092.53044
The authors study two-dimensional smooth and orientable surfaces \(F\) immersed in the Lie-algebra su\((N)\) which are associated with the \({\mathbb C} P^{N-1}\) sigma model defined on two-dimensional Minkowski space. They determine the first and the second fundamental form to \(F\) and compute the Gauss-Weingarten and the Gauss-Codazzi-Ricci equations for \(F.\) Formulae for the Gaussian curvature, the mean curvature and the Willmore functional are computed, too. Then they construct a standard representation of a moving frame of \(F\) immersed in su\((N).\) To this end they use the adjoint representation of the group SU\((N).\) This delivers a normal representation harnessing an orthogonalizing process similar to the method of Gram-Schmidt. The case \(N=2\) leads to surfaces with constant negative Gaussian curvature. An example of this type closes the paper.

MSC:
53C40 Global submanifolds
53C43 Differential geometric aspects of harmonic maps
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] F. Helein, Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems, Lect. Math., Birkhäuser, Boston, 2001. · Zbl 1158.53301
[2] A. Enneper, Nachr. Königl. Gesell. Wissensch. Georg-Augustus-Univ. Göttingen 12 (1868) 258-277, 421-443.
[3] Enneper, A.; Enneper, A., Nachr. Königl. gesell. wissensch. georg-augustus-univ. Göttingen, Nachr. Königl. gesell. wissensch. georg-augustus-univ. Göttingen, 26, (1880)
[4] Dobriner, H., Acta math., 9, 73-104, (1886)
[5] Thomsen, G., Abh. math. sem. Hamburg, 3, 31-56, (1923)
[6] Melko, M.; Sterling, I., Integrable systems, harmonic maps and the classical theory of surfaces, () · Zbl 0814.58014
[7] Pinkall, U.; Sterling, I., Math. intelligencer, 9, 2, 38-43, (1987)
[8] Helein, F., Harmonic maps, conservation laws and moving frames, (2002), Cambridge University Press Cambridge · Zbl 1010.58010
[9] Helein, F., J. diff. geom., 50, 331-385, (1998)
[10] Bobenko, A., Surfaces in terms of 2 by 2 matrices, () · Zbl 0841.53003
[11] Fokas, A.S.; Gelfand, I.M.; Finkel, F.; Liu, Q.M., Selecta math. new series, 6, 347-375, (2000)
[12] Fokas, A.S.; Gelfand, I.M., Commun. math. phys., 177, 203-220, (1996)
[13] M.A. Guest, Harmonic Maps, Loop Groups and Integrable Systems, London Mathematical Society Student Texts 38, Cambridge University Press, Cambridge, 1997. · Zbl 0898.58010
[14] Ferapontov, E.V.; Grundland, A.M., J. nonlinear math. phys., 7, 14-21, (2000)
[15] Grundland, A.M.; Zakrzewski, W.J., J. math. phys., 44, 3370-3382, (2003)
[16] A.M. Grundland, A. Strasburger, W.J. Zakrzewski, Surfaces on \(S U(N)\) groups via \(C P^{N - 1}\) harmonic maps, J. Math. Phys. (submitted for publication). · Zbl 1097.53005
[17] Nelson, D.; Piran, T.; Weinberg, S., Statistical mechanics of membranes and surfaces, (1992), World Scientific Singapore
[18] Charvolin, J.; Joanny, J.F.; Zinn-Justin, J., Liquids at interfaces, (1989), Elsevier Amsterdam
[19] J. Polchinski, String Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1998. · Zbl 1006.81521
[20] M.B. Green, J.H. Schwarz E. Witten, Superstring Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1988.
[21] Konopelchenko, B.G.; Landolfi, G., Modern phys. lett., 12, 3161-3168, (1997)
[22] Konopelchenko, B.G.; Landolfi, G., Stud. appl. math., 104, 129-169, (2000)
[23] Ablowitz, M.; Chakravarty, S.; Halburd, R., J. math. phys., 44, 3147-3173, (2003)
[24] F. David, P. Ginsparg, Y. Zinn-Justin (Eds.), Fluctuating Geometries in Statistical Mechanics and Field Theory, Elsevier, Amsterdam, 1996.
[25] Seifert, U., Adv. phys., 46, 13-137, (1997)
[26] Ou-Yang, Z.; Lui, J.; Xie, Y., Geometric methods in elastic theory of membranes in liquid crystal phases, (1999), World Scientific Singapore
[27] Safram, S.A., Statistical thermodynamics of surfaces, interfaces and membranes, (1994), Addison-Wesley New York
[28] Zakrzewski, W.J., Low dimensional sigma models, (1989), Adam Hilger Bristol · Zbl 0787.53072
[29] K. Weierstrass, Fortsetzung der Untersuchung über die Minimalflächen, Mathematische Werke, vol. 3, Verlagsbuchhandlung, Hillesheim, 1866. · JFM 09.0283.04
[30] Blank, J.; Exner, P.; Havlíček, M., Hilbert space operators in quantum physics, (1994), AIP New York · Zbl 0873.46038
[31] A.M. Grundland, L. Šnobl, Surfaces in \(s u(N)\) algebra via \(\mathbb{C} P^{N - 1}\) sigma models on Minkowski space, in: Č. Burdík, O. Navrátil, S. Pošta (Eds.), Proceedings of International Conference Symmetry Methods in Physics, JINR, Dubna, 2004 (CD-ROM), ISBN 5-9530-0069-3.
[32] Kobayashi, S.; Nomizu, K., Foundation of differential geometry, (1963), John Wiley New York · Zbl 0119.37502
[33] Willmore, T.J., Riemannian geometry, (1993), Clarendon Oxford · Zbl 0797.53002
[34] Melko, M.; Sterling, I., Ann. global anal. geom., 11, 65-107, (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.