Rational connectedness implies finiteness of quantum K-theory

TL;DR

The paper establishes a link between the rational connectedness of Gromov-Witten varieties in flag varieties and the finiteness of quantum K-theory structure constants, revealing deep geometric constraints.

Contribution

It proves that rational connectedness of Gromov-Witten varieties implies the finiteness of quantum K-theory structure constants for large degrees.

Findings

Rational connectedness of Gromov-Witten varieties leads to vanishing of quantum K-theory constants

The structure constants of quantum K-theory vanish for large degrees under certain geometric conditions

Provides a geometric criterion for finiteness in quantum K-theory

Abstract

Let X be any generalized flag variety with Picard group of rank one. Given a degree d, consider the Gromov-Witten variety of rational curves of degree d in X that meet three general points. We prove that, if this Gromov-Witten variety is rationally connected for all large degrees d, then the structure constants of the small quantum K-theory ring of X vanish for large degrees.