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# Tommaso Botta, ETH Zurich – Solution of qKZB equations from the geometry of Nakajima quiver varieties

## January 20, 2023 @ 4:00 pm - 5:00 pm

Mode: In-Person

Title: **Solution of qKZB equations from the geometry of Nakajima quiver varieties**

Abstract:

The quantum Knizhnik–Zamolodchikov (qKZ) equations are an important family of difference equations, deeply related to the representation theory of affine quantum enveloping algebras (trigonometric quantum groups). Over the past years, Okounkov, Smirnov and their coauthors have succeeded in studying the qKZ equations via the geometry of Nakajima varieties and producing integral solutions through enumerative counts in K-theory.

The goal of this talk is to extend some of the above ideas to the elliptic setting. Firstly, I will exploit Aganagic-Okounkov’s theory of elliptic stable envelopes of Nakajima varieties to define a system of elliptic difference equations— the Knizhnik-Zamolodchikov-Bernard (qKZB) equations — for arbitrary quiver varieties. Then I will discuss how to produce integral presentations of their solutions. In this context, a Cohomological Hall algebra (CoHA) interpretation of the stable envelopes will replace Okounkov’s technology of enumerative counts. This talk is based on joint work in preparation with Felder and Wang.