On the finiteness of quantum K-theory of a homogeneous space

TL;DR

This paper proves that the quantum K-theory ring of a homogeneous space G/P involves only finitely many Novikov variables, using a novel approach based on finite difference modules and analysis of zastava space singularities.

Contribution

It introduces a new method leveraging finite difference modules to establish finiteness in quantum K-theory of homogeneous spaces, contrasting with previous approaches.

Findings

Finiteness of quantum K-theory product in G/P proven

Bound on the asymptotic growth of the J-function established

Equivalence between finiteness and quadratic growth condition shown

Abstract

We show that the product in the quantum K-ring of a generalized flag manifold $G/P$ involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the $J$-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.