Positivity determines the quantum cohomology of Grassmannians

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 H*(X) 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 QH(X) 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.