Classical nonlinear σ models on Grassmann manifolds of compact or noncompact type

Abstract

Grassmannian σ models are reexamined in the light of a new geometrical result. Namely, the Cartan immersion of a Riemannian symmetric space G/H into its isometry group G is not always a diffeomorphism onto the set Mσ introduced by Eichenherr and Forger [Nucl. Phys. B 164, 528 (1980)]. This is the case, in particular, for Grassmann manifolds of all types: their set Mσ is shown to be disconnected and its structure is completely analyzed. As a consequence, a new constraint must be introduced in the Bäcklund transformation (BT) method proposed by Harnad, Saint-Aubin, and Shnider [Commun. Math. Phys. 92, 329 (1984)]. It is shown, however, to always be satisfied for Grassmann manifolds of compact type (i.e., G compact), but the problem remains open in most other cases. On the other hand, the BT method is extended to the Euclidean regime.