Mirror Symmetry for Plane Cubics Revisited
Jie Zhou
TL;DR
This paper explores the arithmetic properties of mirror symmetry for plane cubic curves, utilizing Picard-Fuchs equations to connect geometric invariants with arithmetic aspects and extending these ideas to Calabi-Yau varieties.
Contribution
It revisits mirror symmetry for plane cubics, highlighting the role of Picard-Fuchs equations in understanding arithmetic properties and Gromov-Witten invariants.
Findings
Picard-Fuchs equations reveal arithmetic properties of mirror symmetry.
Connections between Gromov-Witten invariants and arithmetic geometry.
Insights into Weil-Petersson geometry on moduli spaces.
Abstract
In this expository note we discuss some arithmetic aspects of the mirror symmetry for plane cubic curves. We also explain how the Picard-Fuchs equation can be used to reveal part of these arithmetic properties. The application of Picard-Fuchs equations in studying the genus zero Gromov-Witten invariants of more general Calabi-Yau varieties and the Weil-Petersson geometry on their moduli spaces will also be discussed.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Advanced Combinatorial Mathematics