A note on the integrality of mirror maps

TL;DR

This paper presents examples of A-hypergeometric systems where mirror maps exhibit integrality, with solutions involving elementary congruences ensuring the exponential of a function has integral coefficients.

Contribution

It introduces a class of A-hypergeometric systems with solutions demonstrating integrality of mirror maps, expanding understanding of their algebraic properties.

Findings

Solutions have the form 1 and log-linear plus G functions.

The exponential of G has integral coefficients.

Elementary congruences suffice for the proof.

Abstract

We give a class of examples of $A$-hypergeometric systems that display integrality of mirror maps. Specifically, these systems have solutions $F(\lambda_1,\dots,\lambda_N) = 1$ and $\log\lambda^l + G(\lambda_1,\dots,\lambda_N)$ (for certain $l\in{\mathbb Z}^N$) such that $\exp G(\lambda)$ has integral coefficients. The proof requires only some elementary congruences.