Stable envelopes for critical loci

TL;DR

This work develops critical stable envelopes for symmetric GIT quotients with potentials within critical cohomology and critical K-theory, enabling geometric actions of shifted quantum groups on critical spaces. It constructs stable envelope correspondences, proves their existence and uniqueness under self-duality, and demonstrates compatibility with dimensional reduction, specialization, and Hall operations. The authors establish a triangle lemma for both cohomology and K-theory, relate stable envelopes to Hall envelopes and R-matrices, and obtain explicit formulas in key cases, including recovering Nakajima quiver envelopes via dimensional reduction for tripled quivers with canonical cubic potentials. The framework connects to qKZ equations and shifted quantum (super)groups, providing foundational tools for applications in geometric representation theory and enumerative geometry on symmetric quiver varieties.

Abstract

This is the first in a sequence of papers devoted to stable envelopes in critical cohomology and critical $K$-theory for symmetric GIT quotients with potentials and related geometries, and their applications to geometric representation theory and enumerative geometry. In this paper, we construct critical stable envelopes and establish their general properties, including compatibility with dimensional reductions, specializations, Hall products, and other geometric constructions. In particular, for tripled quivers with canonical cubic potentials, the critical stable envelopes reproduce those on Nakajima quiver varieties. These set up foundations for applications in subsequent papers.

Theorems & Definitions (262)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Definition 2.2