TL;DR
This paper explains the genus-zero mirror theorems for the quintic threefold using localization techniques, emphasizing elementary recursions over complex machinery, aimed at graduate students.
Contribution
It provides an accessible exposition of mirror theorems in Gromov-Witten and Fan-Jarvis-Ruan-Witten theories, highlighting elementary localization recursions.
Findings
Proofs of genus-zero mirror theorems for the quintic threefold
Emphasis on elementary localization recursions
Clarification of the mirror theorems' formulas
Abstract
These notes were born out of a five-hour lecture series for graduate students at the May 2018 Snowbird workshop Crossing the Walls in Enumerative Geometry. After a short primer on equivariant cohomology and localization, we provide proofs of the genus-zero mirror theorems for the quintic threefold, first in Fan-Jarvis-Ruan-Witten theory and then in Gromov-Witten theory. We make no claim to originality, except in exposition, where special emphasis is placed on peeling away the standard technical machinery and viewing the mirror theorems as closed-formula manifestations of elementary localization recursions.
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