Considerations of One-Modulus Calabi-Yau Compactifications: Picard-Fuchs Equations, K\"ahler Potentials and Mirror Maps

TL;DR

This paper analyzes one-modulus Calabi-Yau compactifications by solving Picard-Fuchs equations to understand K"ahler potentials and mirror maps, using classical geometry techniques to access quantum information.

Contribution

It provides an exact solution for the quantum theory of one-modulus Calabi-Yau compactifications using Picard-Fuchs equations and mirror symmetry, emphasizing classical geometric methods.

Findings

Exact solutions for quantum moduli dependence

Explicit form of Picard-Fuchs equations

Insights into mirror symmetry and global properties

Abstract

We consider Calabi-Yau compactifications with one K\"ahler modulus. Following the method of Candelas et al. we use the mirror hypothesis to solve the quantum theory exactly in dependence of this modulus by performing the calculation for the corresponding complex structure deformation on the mirror manifold. Here the information is accessible by techniques of classical geometry. It is encoded in the Picard-Fuchs differential equation which has to be supplemented by requirements on the global properties of its solutions.