Positivity determines the quantum cohomology of Grassmannians

TL;DR

This paper proves that for Grassmannians, the quantum cohomology basis is uniquely characterized by positivity, determining Gromov-Witten invariants from classical data, and conjectures a similar property for all flag varieties.

Contribution

It establishes the uniqueness of the quantum cohomology basis for Grassmannians based on positivity and proposes a broader conjecture for flag varieties.

Findings

Quantum cohomology basis is uniquely positive for Grassmannians.

Gromov-Witten invariants are determined by classical data and positivity.

Supports conjecture for flag varieties of simply laced Lie type.

Abstract

We prove that if X is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring QH(X) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring of X that multiplies with non-negative structure constants. This implies that the (three point, genus zero) Gromov-Witten invariants of X are uniquely determined by Witten's presentation of QH(X) and the fact that they are non-negative. We conjecture that the same is true for any flag variety X = G/P of simply laced Lie type. For the variety GL(n)/B of complete flags, this conjecture is equivalent to Fomin, Gelfand, and Postnikov's conjecture that the quantum Schubert polynomials of type A are uniquely determined by positivity properties. Our proof for Grassmannians answers a question of Fulton.