The $D_{\hslash }$-module mirror conjecture for flag varieties

Abstract

Rietsch constructed a candidate $T$-equivariant mirror LG model for any flag variety $G/P$. In this paper, we prove the following mirror symmetry prediction: the small $T\times {\mathbb{G}}_m$-equivariant quantum cohomology of $G/P$ equipped with quantum $\hslash$-connection is isomorphic as $D_{\hslash }$-modules to the Brieskorn lattice associated to the LG model equipped with Gauss-Manin $\hslash$-connection.