Coset Models and Differential Geometry *

String propagation on a curved background defines an embedding problem of surfaces in differential geometry. Using this, we show that in a wide class of backgrounds the classical dynamics of the physical degrees of freedom of the string involves 2–dim σ–models corresponding to coset conformal field theories. Contribution to the proceedings of the Conference on Gauge Theories, Applied Supersymmetry and Quantum Gravity, Imperial College, London, 5-10 July 1996 and the e–proceedings of Summer 96 Theory Institute, Topics in Non-Abelian Duality, Argonne, IL, 27 June 12 July 1996. e–mail address: [email protected] Coset models have been used in string theory for the construction of classical vacua, either as internal theories in string compactification or as exact conformal field theories representing curved spacetimes. Our primary aim in this note, based on [1], is to reveal their usefulness in a different context by demonstrating that certain classical aspects of constraint systems are governed by 2–dim σ–models corresponding to some specific coset conformal field theories. In particular, we will examine string propagation on arbitrary curved backgrounds with Lorentzian signature which defines an embedding problem in differential geometry, as it was first shown for 4–dim Minkowski space by Lund and Regge [2]. Choosing, whenever possible, the temporal gauge one may solve the Virasoro constraints and hence be left with D − 2 coupled non–linear differential equations governing the dynamics of the physical degrees of freedom of the string. By exploring their integrability properties, and considering as our Lorentzian background D–dim Minkowski space or the product form R⊗KD−1, where KD−1 is any WZW model for a semi–simple compact group, we will establish connection with the coset model conformal field theories SO(D − 1)/SO(D − 2). This universal behavior irrespectively of the particular WZW model KD−1 is rather remarkable, and sheds new light into the differential geometry of embedding surfaces using concepts and field variables, which so far have been natural only in conformal field theory. Let us consider classical propagation of closed strings on a D–dim background that is the direct product of the real line R (contributing a minus in the signature matrix) and a general manifold (with Euclidean signature) KD−1. We will denote σ ± = 1 2 (τ ±σ), where τ and σ are the natural time and spatial variables on the world–sheet Σ. Then, the 2–dim σ–model action is given by