Shifted quantum groups via critical stable envelopes

TL;DR

This paper develops a geometric framework for shifted Yangians acting on critical cohomologies of quiver varieties with potentials, connecting different algebraic structures and providing explicit formulas for quantum multiplication.

Contribution

It introduces a new geometric construction of shifted Yangians via critical stable envelopes and relates them to cohomological Hall algebras, extending known results to the critical setting.

Findings

Explicit formulas for quantum multiplication by divisors in symmetric quiver varieties.

Connections established between Reshetikhin and Drinfeld type Yangians.

Detailed computations for specific quivers like Jordan and tripled Jordan.

Abstract

Given a symmetric quiver with potential, we develop a geometric construction of shifted Yangians acting on the critical cohomologies of antidominantly framed quiver varieties with extended potentials, using the $R$-matrices constructed from critical stable envelopes. We relate such Reshetikhin type Yangians to Drinfeld type Yangians arising from critical cohomological Hall algebras. Several detailed examples, including the trivial, Jordan, and tripled Jordan quivers are explicitly computed. For symmetric quiver varieties with potentials, by using the smallness property of their affinization maps, we derive explicit formulas for quantum multiplication by divisors in terms of Casimir elements of the associated Lie (super)algebras, extending results from Nakajima quiver varieties to the critical setting. A similar formula in the antidominantly framed case is also obtained, which includes Hilbert schemes of points on $\mathbb C^3$ as examples.