Explicit description of certain 3-point K-theoretic Gromov-Witten invariants for flag manifolds

Abstract

We give an explicit description, in terms of the quantum Bruhat graph, of the (torus-equivariant) 3-point, genus 0, $K$-theoretic Gromov-Witten invariants $⟨\mathcal{O}(-\lambda ),{\mathcal{O}}^w,{\mathcal{O}}_u{⟩}_d$ for the (full) flag manifold $X=G/B$, where $\mathcal{O}(-\lambda )$ denotes the class in the (torus-equivariant) $K$-theory ring $K_T(X)$ of $X$ of the line bundle ${\mathcal{O}}_X(-\lambda )=G{\times }_B{\mathbb{C}}_{\lambda }$ over $X=G/B$ associated to a weight $\lambda \in W{\varpi }_i$ lying in the Weyl group orbit of a minuscule fundamental weight ${\varpi }_i$, and ${\mathcal{O}}_u$, ${\mathcal{O}}^w$ are the Schubert and opposite Schubert classes in $K_T(X)$ for $u,w\in W$. This result can be thought of as a partial generalization of the quantum $K$-theoretic divisor axiom, which we obtained in our previous work; our proof utilizes a generalization of the Chevalley formula in the (torus-equivariant) quantum $K$-theory ring $Q{K}_T(X)$ of $X$, which computes the quantum product with the line bundle class $\mathcal{O}(-\lambda )$ associated to the weight $\lambda$ above.