Analytic properties of mirror maps

TL;DR

This paper studies the analytic and algebraic properties of mirror maps, proving positivity of their Taylor coefficients, exploring their singularities, and conjecturing about their convergence behavior.

Contribution

It establishes the positivity of Taylor coefficients for a broad family of mirror maps and analyzes their complex analytic properties, including singularities and convergence.

Findings

All Taylor coefficients at 0 are positive.

Mirror maps exhibit specific singularity structures.

Conjectures on the radius of convergence and coefficient sign patterns.

Abstract

We consider a multi-parameter family of canonical coordinates and mirror maps o\ riginally introduced by Zudilin [Math. Notes 71 (2002), 604-616]. This family includes 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 $\mathbb P^4(\mathbb C)$. In [Duke Math. J. 151 (2010), 175-218], we proved that all coeffi\ cients 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 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 convergence 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.