Rational curves on hypersurfaces [after A. Givental]

TL;DR

This paper reviews Givental's formalism linking hypergeometric series to quantum differential equations for hypersurfaces, providing an algebro-geometric proof of rational curve counts on the Calabi-Yau quintic 3-fold.

Contribution

It offers an algebro-geometric proof of the Mirror prediction for rational curves on the Calabi-Yau quintic, using localization on the moduli space of stable maps.

Findings

Proof of the Mirror prediction for the quintic 3-fold

Connection between hypergeometric series and quantum differential equations

Localization formula on moduli space of stable maps

Abstract

This article accompanies my June 1998 seminaire Bourbaki talk on Givental's work. After a quick review of descendent integrals in Gromov-Witten theory, I discuss Givental's formalism relating hypergeometric series to solutions of quantum differential equations arising from hypersurfaces in projective space. A particular case of this relationship is a proof of the Mirror prediction for the numbers of rational curves on the Calabi-Yau quintic 3-fold. The approach taken here is entirely algebro-geometric and relies upon a localization formula on the moduli space of stable genus 0 maps to projective space. A different proof of the quintic Mirror prediction may be found in the work of Lian, Liu, and Yau.