Surfaces immersed in su(N+1) Lie algebras obtained from the CP^N sigma models

Abstract

We study some geometrical aspects of two dimensional orientable surfaces arrising from the study of CP^N sigma models. To this aim we employ an identification of R^(N(N+2)) with the Lie algebra su(N+1) by means of which we construct a generalized Weierstrass formula for immersion of such surfaces. The structural elements of the surface like its moving frame, the Gauss-Weingarten and the Gauss-Codazzi-Ricci equations are expressed in terms of the solution of the CP^N model defining it. Further, the first and second fundamental forms, the Gaussian curvature, the mean curvature vector, the Willmore functional and the topological charge of surfaces are expressed in terms of this solution. We present detailed implementation of these results for surfaces immersed in su(2) and su(3) Lie algebras.

32 pages, 1 figure; changes: major revision of presentation, clarifications added

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• Constant mean curvature surfaces via integrable dynamical system
• On the Spinor Representation of Surfaces in Euclidean 3-Space
• Representations of the Weyl group and Wigner functions for SU(3)
• Generalized Weierstrass-Enneper inducing, conformal immersions, and gravity
• Surfaces immersed in su(N+1) Lie algebras obtained from the CP^N sigma models
• Links between different analytic descriptions of constant mean curvature surfaces
• Description of surfaces associated with $C{P}^{N-1}$ sigma models on Minkowski space

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