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ANALYTIC PROPERTIES OF MIRROR MAPS

Part of: Series expansions Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  22 November 2012

C. KRATTENTHALER  and
T. RIVOAL
C. KRATTENTHALER
Affiliation:
Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Vienna, Austria (email: Christian.Krattenthaler@univie.ac.at)
T. RIVOAL*
Affiliation:
Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d’Hères cedex, France
*
✉For correspondence; e-mail: rivoal@math.univ-lyon1.fr
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Abstract

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We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin. This family includes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated ‘quintic’ example of Candelas, de la Ossa, Green and Parkes. In a previous paper, we proved that all coefficients in the Taylor expansions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in their Taylor expansions at 0 are positive. Furthermore, we provide several results about the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address their analytic continuation, points of singularity, and radius of covergence. We present several very precise conjectures on the radius of covergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0.

Keywords

mirror mapscanonical coordinatesanalytic continuationsingular expansiongeneralised hypergeometric functionsmodular forms

MSC classification

Secondary: 30B10: Power series (including lacunary series) 14J32: Calabi-Yau manifolds 30B40: Analytic continuation 33C20: Generalized hypergeometric series, $_pF_q$
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 2 , April 2012 , pp. 195 - 235
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

The research of the first author was partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S9607-N13, the latter in the framework of the National Research Network ‘Analytic Combinatorics and Probabilistic Number Theory’. The research of the second author was partially supported by the project Q-DIFF, ANR-10-JCJC-0105, of the French ‘Agence Nationale de la Recherche’.

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