Okounkov's conjecture via BPS Lie algebras

TL;DR

The paper establishes a precise bridge between Maulik–Okounkov Lie algebras and BPS Lie algebras by constructing nonabelian stable envelopes that intertwine their representations on Nakajima quiver varieties and the critical CoHA framework. By proving an isomorphism between the MA and BPS Lie algebras, it derives Okounkov's conjecture, equating the graded dimensions of MO Lie algebras with coefficients of Kac polynomials through a cohomological Hall algebra perspective. It further describes MO Lie algebras as generalized Kac–Moody algebras with Cartan data given by intersection cohomology of singular Nakajima quiver varieties, and shows that Maulik–Okounkov Yangians arise from the MO Lie algebras via a PBW-type isomorphism, orchestrated through stable envelopes and dimension reduction in the 3-Calabi–Yau setting. The work introduces critical stable envelopes as a natural generalization to vanishing cycle cohomology, providing a robust framework for comparing CoHAs, Kac polynomials, and quantum groups in a unified geometric language. Overall, the results illuminate deep connections between enumerative DT/BPS theory, representation theory of generalized Kac–Moody algebras, and the geometric construction of MO Yangians, with consequences for how stability and critical phenomena govern algebraic structures in quiver settings.

Abstract

Let $Q$ be an arbitrary finite quiver. We use nonabelian stable envelopes to relate representations of the Maulik-Okounkov Lie algebra $\mathfrak{g}^{MO}_Q$ to representations of the BPS Lie algebra associated to the tripled quiver $\tilde Q$ with its canonical potential. We use this comparison to provide an isomorphism between the Maulik-Okounkov Lie algebra and the BPS Lie algebra. Via this isomorphism we prove Okounkov's conjecture, equating the graded dimensions of the Lie algebra $\mathfrak{g}^{MO}_Q$ with the coefficients of Kac polynomials. Via general results regarding cohomological Hall algebras in dimensions two and three we furthermore give a complete description of $\mathfrak{g}^{MO}_Q$ as a generalised Kac-Moody Lie algebra with Cartan datum given by intersection cohomology of singular Nakajima quiver varieties, and prove a conjecture of Maulik and Okounkov, stating that their Lie algebra is obtained from a Lie algebra defined over the rationals, by extension of scalars. Finally, we explain how our results suggest the correct definition of critical stable envelopes in vanishing cycle cohomology.

Theorems & Definitions (131)

• Conjecture 1.1: Okounkov OConj
• Theorem A
• Theorem B
• Corollary 1.2
• Corollary 1.3
• Corollary 1.4
• Theorem C: Theorem \ref{['NakBPS_prop']}
• Remark 2.1
• Remark 2.2
• Example 2.3