Leonardo Mihalcea : Quantum-K theory of the Grassmannians
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. In joint work with Anders Buch, we shown that 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. We therefore extended - and also reproved - the "quantum=classical" phenomenon earlier discovered by Buch-Kresch-Tamvakis in the case of the usual Gromov-Witten invariants. Further, we obtained a Pieri and a Giambelli rule, which yield an effective algorithm to multiply any two classes in the quantum K algebra.
- Category: Algebraic Geometry
- Duration: 01:34:44
- Date: October 10, 2007 at 4:25 PM
- Tags: seminar, Algebraic Geometry Seminar