Modular-type functions attached to mirror quintic Calabi-Yau varieties

TL;DR

This paper constructs a new algebra of modular-type functions linked to the periods of mirror quintic Calabi-Yau varieties, extending automorphic function theory beyond Hermitian symmetric domains.

Contribution

It introduces the first example of automorphic-type functions associated with non-Hermitian symmetric period domains of Calabi-Yau varieties.

Findings

Generated seven functions satisfying functional and differential equations.

Established parallels with classical Eisenstein series and Ramanujan equations.

Reformulated Griffiths' problem on automorphic functions for moduli spaces.

Abstract

In this article we study a differential algebra of modular-type functions attached to the periods of a one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of quintic threefolds. Such an algebra is generated by seven functions satisfying functional and differential equations in parallel to the modular functional equations of classical Eisenstein series and the Ramanujan differential equation. Our result is the first example of automorphic-type functions attached to varieties whose period domain is not Hermitian symmetric. It is a reformulation and realization of a problem of Griffiths around seventies on the existence of automorphic functions for the moduli of polarized Hodge structures.