On the integrality of the Taylor coefficients of mirror maps

TL;DR

This paper proves the integrality of Taylor coefficients of mirror maps derived from hypergeometric solutions, confirming a conjecture and refining previous results in Calabi-Yau geometry.

Contribution

It establishes the integrality of mirror map coefficients and determines the maximal integer root preserving integrality, advancing understanding in mirror symmetry and hypergeometric functions.

Findings

Proves Taylor coefficients of certain mirror maps are integers.

Determines the largest integer root for which coefficients remain integral.

Confirms Zudilin's integrality conjecture for mirror maps.

Abstract

We show that the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$ are integers, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at $z=0$. We also address the question of finding the largest integer $u$ such that the Taylor coefficients of $(z ^{-1}{\bf q}(z))^{1/u}$ are still integers. As consequences, we are able to prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi-Yau complete intersections in weighted projective spaces, which improve and refine previous results by Lian and Yau, and by Zudilin. In particular, we prove the general "integrality" conjecture of Zudilin about these mirror maps.