Measuring Small Distances in N=2 Sigma Models

TL;DR

This paper investigates how string theory constrains the size of geometric features in Calabi-Yau moduli space, suggesting a minimal length scale in quantum geometry, with some lengths capable of shrinking to zero.

Contribution

It extends the notion of geometric length to conformal field theory using algebraic methods and mirror symmetry, revealing quantum restrictions on K"ahler forms in Calabi-Yau spaces.

Findings

String theory enforces minimal length scales in Calabi-Yau moduli space.

Some geometric lengths can shrink to zero, allowing topology change.

Comparison of stringy geometry with classical general relativity.

Abstract

We analyze global aspects of the moduli space of K\"ahler forms for $N$=(2,2) conformal $\sigma$-models. Using algebraic methods and mirror symmetry we study extensions of the mathematical notion of length (as specified by a K\"ahler structure) to conformal field theory and calculate the way in which lengths change as the moduli fields are varied along distinguished paths in the moduli space. We find strong evidence supporting the notion that, in the robust setting of quantum Calabi-Yau moduli space, string theory restricts the set of possible K\"ahler forms by enforcing ``minimal length'' scales, provided that topology change is properly taken into account. Some lengths, however, may shrink to zero. We also compare stringy geometry to classical general relativity in this context.