Symmetric quiver varieties and critical stable envelopes

TL;DR

This paper explores symmetric quiver varieties with potentials, demonstrating their similarities to Nakajima quiver varieties and establishing the existence and properties of critical stable envelopes through geometric and sheaf-theoretic methods.

Contribution

It introduces a new proof for the existence of critical stable envelopes on symmetric quiver varieties and provides a sheaf-theoretic interpretation using hyperbolic restriction.

Findings

Symmetric quiver varieties behave like universally deformed Nakajima quiver varieties.

Existence of critical stable envelopes is established with a new proof.

Hyperbolic restriction leads to a sheaf-theoretic interpretation and the triangle lemma.

Abstract

Symmetric quiver varieties with potentials are natural generalizations of Nakajima quiver varieties, and their equivariant critical cohomologies provide more flexible settings for geometric representation theory and enumerative geometry. In this paper, we study their geometric properties and show that they behave like universally deformed Nakajima quiver varieties. Based on this, we provide a new proof of the existence of critical stable envelopes on them. Following an idea of Nakajima, we give a sheaf theoretic interpretation of critical stable envelopes by the hyperbolic restriction in the affinization of symmetric quiver varieties. The associativity of hyperbolic restrictions implies the triangle lemma of critical stable envelopes.