The Zinger deformation of differential equations with maximal unipotent monodromy

Abstract

In this thesis, in the first part we study the Zinger deformation for the holomorphic solution of a differential equation with maximal unipotent monodromy. It is basically a generalization of the example made by Zinger, who used it to compute the reduced genus one Gromov-Witten invariants for the hypersurfaces.<br /> We define also a differential operator and we show that the action of the Zinger deformation is periodic under this operator. Then we give some structral properties of this deformation at zero. In the case of Calabi-Yau equation we give a connection of this deformation and the Yukawa coupling.<br /> Then we study the asymptotic expansion of the Zinger deformation at infinity and we give a perturbative expansion for it and furthermore we prove a conjecture made by Zagier about the logarithmic derivative of this perturbative expansion. We show that the $s$th term of this logarithmic derivative, up to a simple factor is a polynomial of two variable, namely $n, X$.<br /> Studying of these polynomials for the Zinger example is the aim of the second part. For these polynomials we give explicite formulas for the first and the second top coefficients (with respect to $n$) and in general for fixed $\ell$ we give a recursive formula to compute the $\ell$th top coefficient of $P_s(n,X)$ where $s$ varies. We show that these coefficients under a map belong to the image of elementary functions.