Symmetric quiver varieties and critical stable envelopes

TL;DR

The paper develops the theory of symmetric quiver varieties with potentials, showing they behave like universally deformed Nakajima quiver varieties and admit well behaved affine reductions and symplectic resolutions. It then proves the existence of critical stable envelopes for these varieties and provides a sheaf theoretic interpretation using Braden's hyperbolic restriction on the affinization, from which a triangle lemma follows by associativity. The work connects geometric representation theory and enumerative geometry in the critical setting, and it establishes a robust framework linking stable envelopes, vanishing cycles, and perverse sheaves on affine moduli spaces. These results yield new tools for studying quantum multiplication, fixed point phenomena, and categorified structures in the realm of quiver varieties with potentials.

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.

Theorems & Definitions (62)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 2.5
• proof : Proof of Theorem \ref{['main thm']}
• Example 3.2