Monday, February 25th, 2008
1:30am - 11:30am, in Campus Center 3-3540

Leonardo Mihalcea

Duke University

Quantum K-theory of the Grassmannian

Abstract: If $X$ is a Grassmannian (or an arbitrary homogeneous space) the $3$-point, genus $0$, Gromov-Witten invariants count rational curves of degree $d$ satisfying certain incidence conditions - if this number is expected to be finite. If the number is infinite, Givental and Lee defined the K-theoretic Gromov-Witten invariants, which compute the sheaf Euler characteristic of the space of rational curves in question, embedded in Kontsevich's moduli space of stable maps. The resulting quantum cohomology theory - the quantum K-theory algebra - encodes the associativity relations satisfied by the K-theoretic Gromov-Witten invariants. The goal of this talk is to introduce the quantum cohomology and quantum K-theory algebras, and to explain how the products of Schubert classes in each algebra can be computed explicitly. The key to this computation is the (equivariant) K-theoretic version of the ``quantum = classical" phenomenon: the (equivariant) K-theoretic Gromov-Witten invariants for Grassmannians are equal to structure constants of the ordinary (equivariant) K-theory of certain two-step flag manifolds. This generalizes - and reproves - a result obtained earlier by Buch-Kresch-Tamvakis for the (ordinary, non-K) Gromov-Witten invariants. These results were obtained in joint work with Anders Buch.