Quantum $K$-theoretic divisor axiom for flag manifolds

TL;DR

The paper proves a type-free, torus-equivariant divisor-axiom-type identity for 3-point genus-zero KGW invariants on flag manifolds $G/P$, enabling computation when two insertions are Schubert classes and the third is a Schubert divisor. The authors develop a unified approach that combines the Chevalley formula in quantum $K$-theory, the quantum Bruhat graph, and the Peterson–Woodward comparison to reduce problems to $G/B$, where cancellation-free formulas from quantum $K$-Chevalley theory (via QLS paths) control the corrections. They give precise parabolic and refined two-point formulas (theorems parabolic and qk2p) and show vanishing of correction terms in key cases, yielding explicit, computable expressions. Positivity results follow from Brion’s theorem, giving alternating signs in non-equivariant limits and informing potential cancellations in the equivariant setting. The work thus provides a robust, combinatorial framework for quantum $K$-theory on flag manifolds and paves the way for new presentations and relations in $QK_T(G/P)$.

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.

Figures (1)

• Figure 1: Quantum Bruhat graph of type $G_{2}$. We omit the label $\beta$ of an edge $x \xrightarrow{\space\beta\space} y$ if $\langle \varpi_{i},\,\beta^{\vee} \rangle \in \{0,1\}$. The symbol $\spadesuit$ is $3\alpha_{1}+2\alpha_{2}$; the edge labeled by $\spadesuit$ is an edge in $\mathrm{QBG}_{(1/2)\varpi_{2}}(W)$. The symbols $\diamondsuit$ and $\clubsuit$ are $2\alpha_{1}+\alpha_{2}$ and $\alpha_{1}+\alpha_{2}$, respectively; the edge labeled by $\diamondsuit$ or $\clubsuit$ is an edge in $\mathrm{QBG}_{(1/3)\varpi_{2}}(W) = \mathrm{QBG}_{(2/3)\varpi_{2}}(W)$.

Theorems & Definitions (53)

• Conjecture 1.1: Buch-Mihalcea
• Proposition 1.3
• Definition 2.1
• Proposition 2.2: LNSSS1
• Theorem 2.3: see, for example, LNSSS1
• Definition 2.4: dual tilted Bruhat order
• Proposition 2.5: NOS
• Lemma 2.6
• proof
• Lemma 2.7: KNS