On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator

Abstract

We observe a general structure theorem for quantum cohomology rings, a non-homogeneous version of the usual cohomology ring encoding information about (almost holomorphic) rational curves. An application is the rigorous computation of the quantum cohomology of Grassmannians. As purely algebraic consequence we prove a beautiful formula of Vafa and Intriligator for intersection numbers of certain compactifications of moduli spaces of maps from a Riemann surface (any genus) to G(k,n) which recently has excited many mathematicians. The formula generalizes to any Fano manifold whose cohomology ring can be presented as complete intersection.