Soliton surfaces associated with sigma models: differential and algebraic aspects

Abstract

In this paper, we consider both differential and algebraic properties of surfaces associated with sigma models. It is shown that surfaces defined by the generalized Weierstrass formula for immersion for solutions of the sigma model with finite action, defined in the Riemann sphere, are themselves solutions of the Euler–Lagrange equations for sigma models. On the other hand, we show that the Euler–Lagrange equations for surfaces immersed in the Lie algebra with conformal coordinates, that are extremals of the area functional, subject to a fixed polynomial identity, are exactly the Euler–Lagrange equations for sigma models. In addition to these differential constraints, the algebraic constraints, in the form of eigenvalues of the immersion functions, are systematically treated. The spectrum of the immersion functions, for different dimensions of the model, as well as its symmetry properties and its transformation under the action of the ladder operators are discussed. Another approach to the dynamics is given, i.e. description in terms of the unitary matrix which diagonalizes both the immersion functions and the projectors constituting the model.