Capped Vertex Functions for ${\text{Hilb}}^n({\mathbb{C}}^2)$

Abstract

We obtain explicit formulas for capped descendent vertex functions of ${\text{Hilb}}^n({\mathbb{C}}^2)$ for descendents given by the exterior algebra of the tautological bundle. This formula provides a one-parametric deformation of the generating function for normalized Macdonald polynomials. In particular, we show that the capped vertex functions are rational functions of the quantum parameter.

Various expository changes, updated proofs and references