Rational Curves on Hypersurfaces (after A. Givental)

We describe here a remarkable relationship studied by Givental between hypergeometric series and the quantum cohomology of hypersurfaces in pro-jective space [G1]. As the quantum product involves genus 0 Gromov-Witten invariants, a connection between hypergeometric series and the geometry of rational curves on the hypersurfaces is made. While the most general context for such relationships has not yet been understood, analogous results for complete intersections in smooth toric varieties and flag varieties have been pursued by Givental [G2] and Kim [Ki]. The first discovery in this subject was the startling prediction from Mirror symmetry by Candelas, de la Ossa, Green, and Parkes [COGP] of the numbers of rational curves on quintic 3-folds in P 4. We recount an equivalent form of their original prediction. Let I i (t) be defined by: 3 i=0