Positivity determines the quantum cohomology of Grassmannians

Abstract

We prove that if $X$ is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring $\text{QH}\left(X\right)$ is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring $H^{\ast }\left(X\right)$ that multiplies with nonnegative structure constants. This implies that the (three point, genus zero) Gromov–Witten invariants of $X$ are uniquely determined by Witten’s presentation of $\text{QH}\left(X\right)$ and the fact that they are nonnegative. We conjecture that the same is true for any flag variety $X=G∕P$ of simply laced Lie type. For the variety of complete flags in ${ℂ}^n$, 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.