Quantum $K$-theoretic divisor axiom for flag manifolds

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.

All but Appendix A are written by Cristian Lenart, Satoshi Naito, Daisuke Sagaki, and Weihong Xu; Appendix A is written by Leonardo C. Mihalcea and Weihong Xu