Quiver varieties and quantum Knizhnik--Zamolodchikov equation

TL;DR

This paper connects the geometry of quiver varieties with quantum algebra, demonstrating how their equivariant volumes relate to vertex operators and satisfy the quantum Knizhnik--Zamolodchikov equation in certain cases.

Contribution

It establishes a link between tensor product quiver varieties and vertex operators of Yangians, providing new insights into their geometric and algebraic structures.

Findings

Equivariant volumes of tensor product quiver varieties are expressed via vertex operators.

These volumes satisfy the rational, level 1, quantum Knizhnik--Zamolodchikov equation in some cases.

The work bridges geometric representation theory and quantum integrable systems.

Abstract

We show how equivariant volumes of tensor product quiver varieties of type A are given by matrix elements of vertex operators of centrally extended doubles of Yangians, and how they satisfy in some cases the rational, level 1, quantum Knizhnik--Zamolodchikov equation.