Cohomological Hall algebras and stable envelopes of Nakajima varieties

Over the last years, two different approaches to construct symmetry algebras acting on the cohomology of Nakajima quiver varieties have been developed. The first one, due to Maulik and Okounkov, exploits certain Lagrangian correspondences, called stable envelopes, to generate R-matrices for an arbitrary quiver and hence, via the RTT formalism, an algebra called Yangian. The second one realises the cohomology of Nakajima varieties as modules over the cohomological Hall algebra (CoHA) of the preprojective algebra of the quiver Q. It is widely expected that these two approaches are equivalent, and in particular that the Maulik-Okounkov Yangian coincides with the Drinfel’d double of the CoHA.

Motivated by this conjecture, in this talk I will show how to identify the stable envelopes themselves with the multiplication map of a subalgebra of the appropriate CoHA. 

As an application, I will introduce explicit inductive formulas for the stable envelopes and use them to produce integral solutions of the elliptic quantum Knizhnik–Zamolodchikov–Bernard (qKZB) difference equation associated to arbitrary quiver (ongoing project with G. Felder and K. Wang). Time permitting, I will also discuss connections with Cherkis bow varieties in relation to 3d Mirror symmetry (ongoing project with R. Rimanyi).