Quantum $K$-theoretic divisor axiom for flag manifolds

TL;DR

This paper establishes a new identity for torus-equivariant 3-point genus 0 $K$-theoretic Gromov-Witten invariants of flag manifolds, facilitating their computation and extending the divisor axiom to quantum $K$-theory.

Contribution

It introduces a type-independent identity for these invariants, leveraging the Chevalley formula and quantum Bruhat graph for computations.

Findings

Enables computation of invariants with Schubert class insertions.

Provides a new divisor axiom analogue in quantum $K$-theory.

Uses Chevalley formula and quantum Bruhat graph for proof.

Abstract

We prove an identity for (torus-equivariant) 3-point, genus 0, $K$-theoretic Gromov-Witten invariants of flag manifolds $G/P$, which can be thought of as a replacement for the ``divisor axiom'' in their (torus-equivariant) quantum $K$-theory. This identity enables us to compute these invariants when two insertions are Schubert classes and the other a Schubert divisor class. Our type-independent proof utilizes the Chevalley formula for the (torus-equivariant) quantum $K$-theory ring of flag manifolds, which computes multiplications by Schubert divisor classes in terms of the quantum Bruhat graph.