A Lagrangian definition of a large family of (0, 2) supersymmetric conformal field theories may be made by an appropriate gauge invariant combination of a gauged Wess-Zumino-Witten model, right-moving supersymmetry fermions, and left-moving current algebra fermions. Throughout this paper, use is made of the interplay between field theoretic and algebraic techniques (together with supersymmetry) which is facilitated by such a definition. These heterotic coset models are thus studied in some detail, with particular attention paid to the (0, 2) analogue of the N =2 minimal models, which coincide with the 'monopole' theory of Giddings, Polchinski and Strominger. A family of modular invariant partition functions for these (0, 2) minimal models is presented. Some examples of N =1 supersymmetric four dimensional string theories with gauge groups E 6 × G and SO(10)× G are presented, using these minimal models as building blocks. The factor G represents various enhanced symmetry groups made up of products of SU (2) and U (1).

The large N limit of a one-dimensional infinite chain of random matrices is investigated.It is found that in addition to the expected Kosterlitz-Thouless phase transition this model exhibits an infinite series of phase transitions at special values of the lattice spacing ǫ pq = sin(πp/2q). An unusual property of these transitions is that they are totally invisible in the double scaling limit. A method which allows us to explore the transition regions analytically and to determine certain critical exponents is developed. It is argued that phase transitions of this kind can be induced by the interaction of two-dimensional vortices with curvature defects of a fluctuating random lattice.

We propose a contraction of the de Sitter quantum group leading to the quantum Poincaré group in any dimensions. The method relies on the coaction of the de Sitter quantum group on a non-commutative space, and the deformation parameter q is sent to one. The bicrossproduct structure of the quantum Poincaré group is exhibited and shown to be dual to the one of the κ-Poincaré Hopf algebra, at least in two dimensions.