Rationality of some Gromov-Witten varieties and application to quantum K-theory

TL;DR

This paper proves that Gromov-Witten varieties for certain homogeneous spaces are rational or empty, and uses this to relate quantum K-theoretic invariants to classical ones, advancing understanding in algebraic geometry.

Contribution

It establishes the rationality or emptiness of Gromov-Witten varieties for minuscule or cominuscule spaces and connects quantum K-theory invariants to classical invariants.

Findings

Gromov-Witten varieties are rational or empty for these spaces.

Quantum K-theoretic invariants equal classical invariants on auxiliary spaces.

Provides new tools for computations in quantum K-theory.

Abstract

We show that for any minuscule or cominuscule homogeneous space X, the Gromov-Witten varieties of degree d curves passing through three general points of X are rational or empty for any d. Applying techniques of A. Buch and L. Mihalcea to constructions of the authors together with L. Manivel, we deduce that the equivariant K-theoretic three points Gromov-Witten invariants are equal to classical equivariant K-theoretic invariants on auxilliary spaces.