Quantum cohomology of orthogonal Grassmannians

Let V be a vector space with a non-degenerate symmetric form and OG be the orthogonal Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH*(OG) and show that its product structure is determined by the ring of (P) over tilde -polynomials. A 'quantum Schubert calculus is formulated, which includes quantum Pieri and Giambelli formulas, as well as algorithms for computing Gromov-Witten invariants. As an application, we show that the table of three-point, genus zero Gromov-Witten invariants for OG coincides with that for a corresponding Lagrangian Grassmannian LG, up to an involution.