Stable envelopes for critical loci

TL;DR

This paper introduces stable envelopes in critical cohomology and K-theory for symmetric GIT quotients with potentials, establishing their properties and connections to geometric representation theory.

Contribution

It constructs critical stable envelopes, proves their key properties, and links them to Nakajima quiver varieties, laying groundwork for future research.

Findings

Critical stable envelopes are constructed and their properties are established.

Compatibility with geometric operations like dimensional reductions and Hall products is proven.

Connections to Nakajima quiver varieties are demonstrated.

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.