Picard-Fuchs Uniformization: Modularity of the Mirror Map and Mirror-Moonshine
Charles F. Doran
TL;DR
This paper investigates when mirror maps associated with elliptic curves and K3 surfaces are modular functions, linking geometric properties of Picard-Fuchs equations to the mirror-moonshine phenomenon.
Contribution
It provides a geometric criterion for the modularity of mirror maps, connecting orbifold uniformization to the mirror-moonshine phenomenon.
Findings
Identifies conditions for mirror maps to be Hauptmoduln
Characterizes orbifold uniformization via Picard-Fuchs equations
Demystifies the mirror-moonshine phenomenon
Abstract
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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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Black Holes and Theoretical Physics · Nonlinear Waves and Solitons