Mirror Construction for Nakajima Quiver Varieties

TL;DR

This paper constructs mirror objects for Nakajima quiver varieties using Lagrangian sphere deformations, establishing a functor from Fukaya categories to derived categories of sheaves, with applications to ALE spaces.

Contribution

It introduces a mirror construction for Nakajima quiver varieties via Lagrangian immersions and develops a mirror functor connecting Fukaya categories to sheaf categories.

Findings

Constructed mirror objects for Nakajima quiver varieties.

Established a mirror functor from Fukaya categories to sheaf categories.

Linked moduli of Lagrangian immersions to ALE space sheaves.

Abstract

In this paper, we construct the ADHM quiver representations and the corresponding sheaves as the mirror objects of formal deformations of the framed immersed Lagrangian sphere decorated with flat bundles. More generally, we construct Nakajima quiver varieties as localized mirrors of framed nodal unions of Lagrangian spheres in dimension two. This produces a mirror functor from the Fukaya category of a framed plumbing of surfaces to the dg category of complexes of bundles over the corresponding Nakajima quiver varieties. For affine ADE quivers in specific multiplicities, the corresponding (unframed) Lagrangian immersions are homological tori, whose moduli of stable deformations are asymptotically locally Euclidean (ALE) spaces. We show that framed stable Lagrangian branes are transformed into monadic complexes of framed torsion-free sheaves over the ALE spaces. A main ingredient is the notion of framed Lagrangian immersions and their Maurer-Cartan deformations. Moreover, using the formalism of quiver algebroid stacks, we find isomorphisms between the moduli of stable Lagrangian immersions and that of special Lagrangian fibers of an SYZ fibration in the affine $A_n$ cases.