Shifted quantum groups via critical stable envelopes

TL;DR

The paper develops shifted quantum groups acting on the critical cohomology of framed quiver varieties with potentials, linking Reshetikhin-type and Drinfeld-type Yangians through R-matrices derived from critical stable envelopes. It establishes a PBW-type framework via a Lie superalgebra $\mathfrak{g}_{Q,\mathsf W}$ and constructs coproducts and shift maps that organize both zero and antidominant shifts, with explicit divisor-quantum multiplication formulas expressed through Casimir operators. The work provides a detailed, example-rich realization of shifted Yangians (including trivial, Jordan, and tripled Jordan quivers), and connects these structures to Maulik–Okounkov Yangians and BPS/Davison–Meinhardt Lie algebras, while offering a robust auxiliary-data formalism that ensures independence from choices. Overall, the results offer a geometric, representation-theoretic framework for shifted quantum groups with concrete computational tools and broad applicability to symmetric quivers with potentials and Hilbert schemes like points on C^3.

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.

Theorems & Definitions (296)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Definition 2.2
• Definition 2.3