Framed cohomological Hall algebras and cohomological stable envelopes

TL;DR

This paper introduces a new cohomological Hall algebra for framed quiver representations, linking it to stable envelopes of Nakajima varieties and providing a formula for their computation.

Contribution

It defines the framed CoHA and shows its relation to stable envelopes, offering a new algebraic framework and explicit formulas for calculations.

Findings

The equivariant cohomology of Nakajima varieties forms a subalgebra of the framed CoHA.

Stable envelopes are identified with the algebra multiplication restricted to this subalgebra.

An explicit inductive formula for stable envelopes in terms of tautological classes is derived.

Abstract

There are multiple conjectures relating the cohomological Hall algebras (CoHAs) of certain substacks of the moduli stack of representations of a quiver $Q$ to the Yangian $Y^{Q}_{MO}$ by Maulik-Okounkov, whose construction is based on the notion of stable envelopes of Nakajima varieties. In this article, we introduce the cohomological Hall algebra of the moduli stack of framed representations of a quiver $Q$ (framed CoHA) and we show that the equivariant cohomology of the disjoint union of the Nakajima varieties $\mathcal{M}_Q(\text{v},\text{w})$ for all dimension vectors $\text{v}$ and framing vectors $\text{w}$ has a canonical structure of subalgebra of the framed CoHA. Restricted to this subalgebra, the algebra multiplication is identified with the stable envelope map. As a corollary, we deduce an explicit inductive formula to compute stable envelopes in terms of tautological classes.