Quantum K-theory of Grassmannians

TL;DR

This paper establishes a connection between quantum K-theoretic Gromov-Witten invariants of Grassmannians and intersection theory on flag manifolds, providing formulas for quantum K-theory rings and extending to other homogeneous spaces.

Contribution

It generalizes previous results by relating Gromov-Witten invariants to intersection theory on flag varieties and derives Pieri and Giambelli formulas for quantum K-theory of Grassmannians.

Findings

Gromov-Witten invariants equal triple intersections in K-theory of flag manifolds

The Gromov-Witten variety of curves through three points is irreducible and rational

Formulas for quantum K-theory multiplication in Grassmannians

Abstract

We show that (equivariant) K-theoretic 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the ordinary (equivariant) K-theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the Gromov-Witten variety of curves passing through 3 general points is irreducible and rational. Our applications include Pieri and Giambelli formulas for the quantum K-theory ring of a Grassmannian, which determine the multiplication in this ring. Our formula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin.