Minimum Distances in Non-Trivial String Target Spaces

TL;DR

This paper investigates the concept of minimum distances in complex string target spaces, especially those with singularities, using mirror symmetry to analyze how distances can be zero or infinite in these geometries.

Contribution

It extends the understanding of minimum distances beyond simple geometries to complex Calabi-Yau spaces with singularities, revealing new phenomena related to zero and infinite distances.

Findings

Zero distances can occur in complex geometries.

Zero distances often require infinite distances elsewhere.

In some cases, zero distances occur without infinite distances.

Abstract

The idea of minimum distance, familiar from R <-> 1/R duality when the string target space is a circle, is analyzed for less trivial geometries. The particular geometry studied is that of a blown-up quotient singularity within a Calabi-Yau space and mirror symmetry is used to perform the analysis. It is found that zero distances can appear but that in many cases this requires other distances within the same target space to be infinite. In other cases zero distances can occur without compensating infinite distances.