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

TL;DR

This paper establishes a general structure theorem for quantum cohomology rings of Fano manifolds, applies it to compute quantum cohomology of Grassmannians, and proves a notable formula of Vafa and Intriligator for intersection numbers.

Contribution

It introduces a broad structure theorem for quantum cohomology rings and generalizes a key formula to all Fano manifolds with complete intersection cohomology.

Findings

Rigorous computation of quantum cohomology for Grassmannians

Proof of Vafa-Intriligator formula for intersection numbers

Generalization of the formula to all Fano manifolds with complete intersection cohomology

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.