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

TL;DR

This paper derives explicit formulas for capped descendent vertex functions of the Hilbert scheme of points on the plane, revealing their rational dependence on the quantum parameter and connecting to Macdonald polynomials.

Contribution

It provides the first explicit formulas for these vertex functions, introducing a one-parameter deformation related to Macdonald polynomials.

Findings

Capped vertex functions are rational functions of the quantum parameter.

The formulas establish a deformation linking to Macdonald polynomial generating functions.

Explicit expressions facilitate further computations in enumerative geometry.

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.