Zinger Functions and Yukawa Couplings

Abstract

For the domain $\mathcal{P}=1+x\mathbb{Q}(w)_0[[x]]$, where $\mathbb{Q}(w)_0=\mathbb{Q}(w)\cap\mathbb{Q}[[w]]$, and a function $f(w,x)\in\mathcal{P}$, we consider the Zinger operator

$$\mathbf{M} f(w,x)=\biggl(1+\frac xw\frac{\partial}{\partial x}\biggr)\frac{f(w,x)}{f(0,x)}$$

and define $I_p(x)=\mathbf{M}^p(f(w,x))\mid_{w=0}$. In this article, we study a class of periodic functions under the iterations of $\mathbf{M}$ and show that $I_p$ have interesting properties. A typical element of this class is constructed from the holomorphic solution of a differential equation with maximal unipotent monodromy. For this solution we define a kind of deformation (Zinger deformation) as a member of $\mathcal{P}$. This deformation is a natural generalization of what Zinger did for the hypergeometric function

$$\mathcal{F}(x)=\sum_{d=0}^\infty\biggl(\frac{(nd)!}{(d!)^n}\biggr)x^d.$$

Finally for a family of Calabi–Yau manifolds, we consider the associated Picard–Fuchs equation. Then under the mirror symmetry hypothesis, we show that the Yukawa couplings can be interpreted as these new functions $I_p$.