A Superstring Theory in Four Curved Space-time Dimensions *

Neveu-Schwarz-Ramond type heterotic and type-II superstrings in four dimensional curved space-time are constructed as exact N = 1 superconformal theories. The tachyon is eliminated with a GSO projection. The theory is based on the N=1 superconformal gauged WZW model for the anti-de Sitter coset SO(3, 2)/SO(3, 1) with integer central extension k = 5. The model has dynamical duality properties in its space-time metric that are similar to the large-small (R → 1/R) duality of tori. To first order in a 1/k expansion we give expressions for the metric, the dilaton, the Ricci tensor and their dual generalizations. The curvature scalar has several singularities at various locations in the 4-dimensional manifold. This provides a new singular solution to Einstein's equations in the presence of matter in four dimensions. A non-trivial path integral measure which we conjectured in previous work for gauged WZW models is verified.