The mirror conjecture for minuscule flag varieties
The mirror conjecture for minuscule flag varieties
复制标题
DOI：
10.1215/00127094-2024-0007
复制
发表时间：
2017-05
影响因子：
2.5
通讯作者：
T. Lam;Nicolas Templier
中科院分区：
文献类型：
--
作者：
T. Lam;Nicolas Templier
我们证明了 Rietsch 的镜像猜想，即小旗品种的杜布罗文量子联系与 Berenstein-Kazhdan 几何晶体的特征 D 模同构。这个想法是将量子联系视为伽罗瓦，并将几何晶体视为自守。我们揭示了与 Frenkel-Gross、Heinloth-Ngo-Yun 和 Zhu 在 Kloosterman 滑轮上的作品的惊人关系。同构来自全局刚性结果，其中赫克特征轮由其局部分支决定。作为推论，我们获得了有理曲线计数的组合恒等式以及小量子上同调环的 Peterson 簇表示。
We prove Rietsch's mirror conjecture that the Dubrovin quantum connection for minuscule flag varieties is isomorphic to the character D-module of the Berenstein-Kazhdan geometric crystal. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. We reveal surprising relations with the works of Frenkel-Gross, Heinloth-Ngo-Yun and Zhu on Kloosterman sheaves. The isomorphism comes from global rigidity results where Hecke eigensheaves are determined by their local ramification. As corollaries we obtain combinatorial identities for counts of rational curves and the Peterson variety presentation of the small quantum cohomology ring.