Rational curves on hypersurfaces [after A. Givental]

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.

arXiv comment33 pages, Latex2e

References in corpus (5)

• Mirror Manifolds And Topological Field Theory
• Topological Field Theory and Rational Curves
• Mirror Manifolds in Higher Dimension
• Enumeration of $n$-fold tangent hyperplanes to a surface
• Quantum cohomology of complete intersections

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• An integral structure in quantum cohomology and mirror symmetry for toric orbifolds
• Computing Genus-Zero Twisted Gromov-Witten Invariants
• Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence
• Quantum D-modules and generalized mirror transformations
• Quantum Cohomology and Periods
• The Crepant Transformation Conjecture for Toric Complete Intersections
• Disk enumeration on the quintic 3-fold
• Real and integral structures in quantum cohomology I: toric orbifolds
• Psi-floor diagrams and a Caporaso-Harris type recursion
• Eisenstein type series for Calabi-Yau varieties
• On the work of Givental relative to mirror symmetry
• Covariant constancy of quantum Steenrod operations
• Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves
• Number theory casting a look at the mirror
• Fixed-point Localization for ${\mathbb{RP}}^{2m}\subset {\mathbb{CP}}^{2m}$
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians
• Mirror symmetry for concavex vector bundles on projective spaces
• A reconstruction of Euler data
• The fundamental solution matrix and relative stable maps