Number theory casting a look at the mirror

TL;DR

This paper explores mirror symmetry for quintic threefolds using hypergeometric functions, establishing the non-algebraic nature of the Yukawa coupling through differential equations and transcendence results.

Contribution

It introduces a new analytic approach to mirror symmetry, demonstrating the existence of high-order non-linear differential equations for the mirror map and proving the transcendence of the Yukawa coupling.

Findings

Mirror map generalizes modular maps.

Yukawa coupling does not satisfy low-order algebraic differential equations.

Established existence of high-order differential equations for mirror maps.

Abstract

In this work, we give a purely analytic introduction to the phenomenon of mirror symmetry for quintic threefolds via classical hypergeometric functions and differential equations for them. Starting with a modular map and recent transcendence results for its values, we regard a mirror map $z(q)$ as a concept generalizing the modular one. We give an alternative approach demonstrating the existence of non-linear differential equations for the mirror map, and exploit both an elegant construction of Klemm-Lian-Roan-Yau and the Ax theorem to prove that the Yukawa coupling $K(q)$ does not satisfy any algebraic differential equation of order less than 7 with coefficients from $\mathbb{C}(q)$.