Eisenstein type series for Calabi–Yau varieties

Abstract

In this article we introduce an ordinary differential equation associated to the one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of smooth quintic three folds. It is satisfied by seven functions written in the q-expansion form and the Yukawa coupling turns out to be rational in these functions. We prove that these functions are algebraically independent over the field of complex numbers, and hence, the algebra generated by such functions can be interpreted as the theory of quasi-modular forms attached to the one parameter family of Calabi-Yau varieties. Our result 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. It is a generalization of the Ramanujan differential equation satisfied by three Eisenstein series.

Algebraic Structure of tt * Equations for Calabi-Yau Sigma Models

“…These differential rings coincide with those of Ref. [13] and are expected to provide generalizations of classical quasi modular forms. When subgroups of SL(2, Z) appear in limits of the monodromy groups of the mirror CY threefolds [19,20], the differential rings coincide with the differential rings of quasi modular forms of Kaneko and Zagier [21].…”

Polynomial Structure of Topological String Partition Functions

“…It would be interesting to see whether these rings could help construct ring of modular objects (see for example [Mov11]), and how the global properties of the generators could help solve for the topological string partition functions from the holomorphic anomaly equations with boundary conditions for more general CY 3-fold families.…”

Арифметические Гипергеометрические Ряды

“…Отметим, что приведенное в [89] уравнение очень длинное, и лишь недавно Х. Мовасати [99] удалось построить очень элегантную систему нелинейных дифференциальных уравне-ний для семейства трехмерных квинтик (74) и отвечающего ему уравнения Пикара-Фукса (75), которое напоминает систему Рамануджана (77):…”

“…При доказательстве алгебраической независимости в [161] и [99] рассматри-ваемые функции связываются с фундаментальным решением линейного диф-ференциального уравнения (75) и используется структура монодромии послед-него. Вопрос о структуре монодромии рассматривался в ряде работ по этой тематике.…”