Picard-Fuchs Uniformization: Modularity of the Mirror Map and Mirror-Moonshine

Details Picard-Fuchs Uniformization: Modularity of the Mirror Map and Mirror-Moonshine Picard-Fuchs Uniformization: Modularity of the Mirror Map and Mirror-Moonshine

1998-12-31

arxiv.org/abs/math/9812162v1

Mathematics

Algebraic Geometry

Submitted to the Proceedings of NATO-ASI and CRM Summer School on the Arithmetic and Geometry of Algebraic Cycles

Scientific paper

Motivated by a conjecture of Lian and Yau concerning the mirror map in string theory, we determine when the mirror map q-series of certain elliptic curve and K3 surface families are Hauptmoduln (genus zero modular functions). Our geometric criterion for modularity characterizes orbifold uniformization properties of their Picard-Fuchs equations, effectively demystifying the mirror-moonshine phenomenon. A longer, more comprehensive treatment of these results will appear shortly.

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