Cohomological Hall algebras and stable envelopes of Nakajima varieties

Abstract: 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 Maulik-Okounkov Yangian coincides with the Drinfel’d double of the CoHA. Motivated by this conjecture, in this talk I will show how to interpret the stable envelopes themselves in terms of the appropriate CoHA, together with some applications to the associated Yangians and difference equations. Time permitting, I will also discuss connections with Cherkis bow varieties in relation to 3d Mirror symmetry, which is object of ongoing project with Richard Rimanyi.