Mirror construction of Hecke correspondence between Nakajima quiver varieties
Siu-Cheong Lau, Ju Tan
TL;DR
The paper constructs a symplectic realization of Nakajima-style Hecke correspondences by embedding them as holomorphic Lagrangian subvarieties arising from localized mirror symmetry applied to framed Lagrangian branes in a plumbing of $T^*bS^2$. It develops a comprehensive framework linking Floer theory, Maurer–Cartan deformation spaces, and a noncommutative Maurer–Cartan quiver algebra $bA_{bL}$ to Nakajima quiver varieties and their Hecke correspondences, including a precise identification of Hecke kernels with $ ext{HF}^2$-supports and a detailed monadic holomorphic Lagrangian construction. For non-ADE graphs, the localized mirror functor is shown to be fully faithful, and for ADE surfaces a local Homological Mirror Symmetry equivalence is established between a Fukaya subcategory generated by the universal Lagrangian and the category of perfect modules over $bA_{bL}$. The results illuminate a deep bridge between symplectic Floer theory and geometric representation theory, offering a path to categorify Kac–Moody actions via mirror-symmetric, holomorphic-Lagrangian correspondences and to understand Calabi–Yau properties of the localized mirrors in this context.
Abstract
Nakajima constructed geometric representations of a deformed Kac-Moody Lie algebra using Hecke correspondences between quiver varieties. In this paper, we show that Hecke correspondences, which are holomorphic Lagrangians in products of Nakajima quiver varieties, can be obtained by applying the localized mirror construction to the morphism spaces between families of framed Lagrangian branes supported on the core of a plumbing of two-spheres. Moreover, for a non-ADE quiver, we show that the localized mirror functor is fully-faithful.