旗流形的某些3点K-理论Gromov-Witten不变量的显式描述
Explicit description of certain 3-point K-theoretic Gromov-Witten invariants for flag manifolds
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中文总结 AI 辅助
该研究借助量子Bruhat图，给出了全旗流形的3点亏格0的K-理论Gromov-Witten不变量的显式描述，是前期量子K-理论除子公理的部分推广，证明用到了量子K-理论环中Chevalley公式的推广形式。
中文摘要 AI 辅助
我们借助量子Bruhat图，给出了全旗流形$X=G/B$的（环面等变）3点亏格0的K-理论Gromov-Witten不变量$\nlangle \n\n\no(- \n\nepsilon), \n\no^{w}, \n\no_{u} \n\nrangle_{d}$的显式描述，其中$\n\no(- \n\nepsilon)$表示与位于极小基本权$\n\ni$的Weyl群轨道中的权$\n\nepsilon \n\nin \n\nW \n\nvarpi_i$相关联的、$X$的线丛$\n\no_{X}(- \n\nepsilon) = G \n\times_{B} \n\nmathbb{C}_{\n\nepsilon}$在$X$的（环面等变）K-理论环$K_{T}(X)$中的类；$\n\no_{u}$、$\n\no^{w}$是$u, w \n\nin \n\nW$对应的$K_{T}(X)$中的Schubert类和反Schubert类。该结果可视为我们前期工作中得到的量子K-理论除子公理的部分推广；证明利用了$X$的（环面等变）量子K-理论环$QK_{T}(X)$中Chevalley公式的推广形式，该公式用于计算与上述权$\n\nepsilon$相关联的线丛类$\n\no(- \n\nepsilon)$的量子乘积。
英文摘要
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 $\langle \mathcal{O}(- λ), \mathcal{O}^{w}, \mathcal{O}_{u} \rangle_{d}$ for the (full) flag manifold $X = G/B$, where $\mathcal{O}(- λ)$ denotes the class in the (torus-equivariant) $K$-theory ring $K_{T}(X)$ of $X$ of the line bundle $\mathcal{O}_{X}(- λ) = G \times_{B} \mathbb{C}_λ$ over $X = G/B$ associated to a weight $λ\in W \varpi_i$ 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 $QK_{T}(X)$ of $X$, which computes the quantum product with the line bundle class $\mathcal{O}(- λ)$ associated to the weight $λ$ above.
