[Paper Review] The mirror conjecture for minuscule flag varieties

Thomas Lam, Nicolas Templier|arXiv (Cornell University)|May 2, 2017

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper proves Rietsch's mirror conjecture for minuscule flag varieties by establishing an isomorphism between the quantum D-module of $G^{ullet}/P^{ullet}$ and the character D-module of the Berenstein–Kazhdan geometric crystal. The proof leverages global rigidity of Hecke eigensheaves and connects the quantum connection to Galois theory and automorphic forms, revealing deep links to Kloosterman sheaves and Frenkel–Gross connections, with applications to quantum cohomology and combinatorial curve counts.

ABSTRACT

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.

Motivation & Objective

• To prove Rietsch's mirror conjecture for minuscule flag varieties, establishing an isomorphism between the quantum D-module of $G^{ullet}/P^{ullet}$ and the character D-module of a geometric crystal.
• To extend this isomorphism to the equivariant setting with a parameter $\hbar$, recovering both the quantum mirror theorem and the equivariant Peterson isomorphism as special cases.
• To reveal unexpected connections between quantum cohomology, geometric crystals, and automorphic forms via the theory of Hecke eigensheaves and global rigidity.
• To provide new combinatorial formulas for counts of rational curves and a presentation of the small quantum cohomology ring via the Peterson isomorphism.
• To unify disparate constructions—Frenkel–Gross, Heinloth–Ngô–Yun, Zhu—under a common framework of D-modules and mirror symmetry.

Proposed method

• Recognize the Dubrovin quantum connection on $G^{ullet}/P^{ullet}$ as a Galois D-module via global rigid local systems.
• Realize the geometric crystal as an automorphic D-module through the Langlands dual group and representation theory.
• Apply global rigidity theorems to show that Hecke eigensheaves are uniquely determined by their local ramification data.
• Construct an explicit isomorphism between the quantum D-module and the character D-module of the geometric crystal using the $\hbar$-deformation framework.
• Use the theory of $D_{\hbar}$-modules and filtered $D$-modules to handle the semiclassical limit ($\hbar \to 0$) and recover the Peterson isomorphism.
• Leverage the Frenkel–Gross connection and Kloosterman $D$-module as key examples in the mirror duality diagram, showing their isomorphism to the quantum and geometric crystal D-modules respectively.

Experimental results

Research questions

• RQ1 Is the quantum D-module of a minuscule flag variety $G^{ullet}/P^{ullet}$ isomorphic to the character D-module of the associated geometric crystal?
• RQ2 How does the $\hbar$-deformation of the quantum connection relate to the equivariant quantum cohomology and the semiclassical limit?
• RQ3 Can the global rigidity of Hecke eigensheaves be used to prove mirror symmetry for minuscule flag varieties?
• RQ4 What is the precise relationship between the Frenkel–Gross connection, the Kloosterman $D$-module, and the quantum D-module in the context of geometric Langlands?
• RQ5 How do the results yield new combinatorial identities for rational curve counts and presentations of the small quantum cohomology ring?

Key findings

• The quantum D-module of $G^{ullet}/P^{ullet}$ is isomorphic to the character D-module of the geometric crystal for any minuscule flag variety.
• The isomorphism is established via global rigidity of Hecke eigensheaves, where local ramification data uniquely determine the global system.
• The $\hbar$-deformation framework unifies the quantum mirror theorem and the equivariant Peterson isomorphism as special cases at $\hbar = 1$ and $\hbar \to 0$.
• The Frenkel–Gross connection and the Kloosterman $D$-module are shown to be isomorphic to the quantum and geometric crystal D-modules, respectively.
• The isomorphism yields a new presentation of the small quantum cohomology ring of $G^{ullet}/P^{ullet}$ via the Peterson isomorphism.
• Combinatorial identities for counts of rational curves are derived from the mirror isomorphism, generalizing known results for Grassmannians.