Generalized Gauss map and geometry of strings (Q803563)

The tangent planes to a Euclidean string world sheet conformally immersed in \({\mathbb{R}}^ n\) define a map from the Riemann surface defined by the world sheet into the Grassmannian \(G_{2,n}\) (which may be realized as a quadric \(Q_{n-2}\) in \({\mathbb{C}}P^{n-1})\). This so-called Gauss map, together with its integrability conditions, defines a constrained Grassmannian \(\sigma\)-model, if the string actions of Nambu and Goto or that of \textit{A. M. Polyakov} [Nucl. Phys. B 268, 406-412 (1986)] involving the extrinsic curvature of the world sheet are chosen. The authors argue that their formulation is more amenable to a systematic perturbation study than the one involving the string coordinates. One- loop effects of the Polyakov action are evaluated in \(n=3\) and a geometrical interpretation of the instantons of the Grassmannian \(\sigma\)-model is given in \(n=4\).