MEASURING SMALL DISTANCES IN N = 2 SIGMA-MODELS

We analyze global aspects of the moduli space of Kahler forms for N = (2,2) conformal sigma-models. Using algebraic methods and mirror symme try we study extensions of the mathematical notion of length (as speci fied by a Kahler structure) to conformal field theory and calculate th e way in which lengths change as the moduli fields are varied along di stinguished paths in the moduli space. We find strong evidence support ing the notion that, in the robust setting of quantum Calabi-Yau modul i space, string theory restricts the set of possible Kahler forms by e nforcing ''minimal length'' scales, provided that topology change is p roperly taken into account. Some lengths, however, may shrink to zero. We also compare stringy geometry to classical general relativity in t his context.