TOPOLOGICAL PROPERTIES OF STRING VACUA

We address the study of geometrical and topological properties of string vacua. In particular, the distance in the space of couplings of two-dimensional quantum field theories is studied, as specified by the Zamolodchikov metric. We show that for world-sheet supersymmetric theories, the Witten index (target space Euler number) cannot be changed while moving a finite distance, and illustrate this for N = 1 and N = 2 minimal series as well as for Calabi-Yau manifolds. We also comment on the properties of such theories when they are coupled to two-dimensional gravity.