Nonlinearity and Geometry - seminar

This talk is devoted to the study of an invariant formulation of completely integrable CP^N-1 Euclidean sigma models in two dimensions, defined on the Riemann sphere, having finite actions. Surfaces connected with the CP^N-1 models, invariant recurrence relations linking the successive projection operators and immersion functions of the surfaces are discussed in detail. We show that the immersion functions of 2D-surfaces associated with the CP^N-1 model are contained in 2D-spheres in the su(N) algebra. Making use of the fact that the immersion functions of the surfaces satisfy the same Euler-Lagrange equations as the original projector variables, we derive surfaces induced by surfaces and prove that the stacked surfaces coincide with each other, which demonstrates the idempotency of the recurrent procedure. We also demonstrate that the CP^N-1 model equations admit larger classes of solutions than the ones corresponding to rank-1 Hermitian projectors. This fact allows us to generalize the Weierstrass formula for the immersion of 2D-surfaces in the su(N) algebra and show that in general these surfaces cannot be conformally parametrized. Finally, we consider the connection between the structure of the projective formalism and the possibility of spin representations of the su(N) algebra in quantum mechanics.