The conformal limit for Nakajima quiver varieties

TL;DR

This paper introduces a conformal limit construction for Nakajima quiver varieties, establishing a biholomorphic correspondence between certain Lagrangian submanifolds and exploring related conjectures.

Contribution

It defines and analyzes a conformal limit for Nakajima quiver varieties, linking different varieties through biholomorphic maps and discussing conjectural completeness properties.

Findings

The conformal limit forms a limit of a one-parameter family within a quiver variety.

It provides a biholomorphic map between holomorphic Lagrangian submanifolds.

The paper discusses an analog of Simpson's conjecture on submanifold completeness.

Abstract

Inspired by Gaiotto's conformal limit construction for Higgs bundles we define and study a conformal limit construction for Nakajima quiver varieties. We prove that the conformal limit is indeed a limit of a one parameter family of points inside a specified quiver variety and that it gives a biholomorphic map between holomorphic Lagrangian submanifolds foliating two different quiver varieties. In the last part of the paper we discuss the analog of Simpson's conjecture on the completeness of these holomorphic Lagrangian submanifolds.