English

40 Pages

Gain access to the library to view online

__
Learn more
__

Description

Niveau: Supérieur, Doctorat, Bac+8

ANALYTIC PROPERTIES OF MIRROR MAPS C. KRATTENTHALER† AND T. RIVOAL†† Abstract. We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin [Math. Notes 71 (2002), 604–616]. This family in- cludes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated example of Candelas, de la Ossa, Green and Parkes [Nucl. Phys. B359 (1991), 21–74] associated to the quintic hypersurface in P4(C). In [Duke Math. J. 151 (2010), 175–218], we proved that all coefficients in the Taylor expan- sions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in the Taylor expansions at 0 of these canonical coordinates are positive. Furthermore, we provide several results pertaining to the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address analytic continuation, points of singularity, and radius of conver- gence of these functions. We present several very precise conjectures on the radius of convergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0. 1. Introduction In the focus of this article there is a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin [39] (to be defined below).

ANALYTIC PROPERTIES OF MIRROR MAPS C. KRATTENTHALER† AND T. RIVOAL†† Abstract. We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin [Math. Notes 71 (2002), 604–616]. This family in- cludes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated example of Candelas, de la Ossa, Green and Parkes [Nucl. Phys. B359 (1991), 21–74] associated to the quintic hypersurface in P4(C). In [Duke Math. J. 151 (2010), 175–218], we proved that all coefficients in the Taylor expan- sions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in the Taylor expansions at 0 of these canonical coordinates are positive. Furthermore, we provide several results pertaining to the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address analytic continuation, points of singularity, and radius of conver- gence of these functions. We present several very precise conjectures on the radius of convergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0. 1. Introduction In the focus of this article there is a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin [39] (to be defined below).

- variable mirror
- generalised hypergeometric
- fuchsian differential
- mirror maps
- theorem provides precise
- multi-variable mirror
- nk ≥
- hypergeometric differential operator

Subjects

Informations

Published by | mijec |

Reads | 11 |

Language | English |

Report a problem