This material has been published in
J. Austral. Math. Soc. 92 (2012), 195-235,
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Christian Krattenthaler and
Analytic properties of mirror maps
We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced
[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 P4(C).
In [Duke Math. J. 151 (2010),
175-218], 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 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.
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