Generalized quantum difference equations for quiver varieties

Curve counts inside certain varieties are constrained by an equation called the quantum difference equation. Okounkov and Smirnov described the quantum difference equation for Nakajima quiver varieties in terms of a quantum group. There is a natural modification of their description, which leads to a collection of “exotic” quantum difference equations, one for each alcove in a certain hyperplane arrangement. Our main result is an enumerative-geometric description of the fundamental solution of these exotic equations. I will give an overview of these ideas, illustrating the results for the example of the Hilbert scheme of points in the complex plane.