2017/05/02 by Thomas Lam, Lam, Thomas, Nicolas Templier +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1705.00758
openalex publication_date 2017/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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-Ngô-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.