Zinger Functions and Yukawa Couplings
Kh. M. Shokri Shahid Beheshti University
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
$I_p$.
Keywords:
Zinger functions, Yukawa couplings, maximal unipotent monodromy, Calabi–Yau equations, mirror symmetry.
UDC:
517.95 Received: 21.07.2021
Revised: 06.03.2022
DOI:
10.4213/mzm13232
Fulltext:
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