Wreath Macdonald polynomials, quiver varieties, and quasimap counts

TL;DR

This paper explores the $K$-theoretic enumerative geometry of cyclic Nakajima quiver varieties, connecting wreath Macdonald polynomials with vertex functions and proposing conjectures on their integrality and wall-crossing behaviors.

Contribution

It provides an explicit formula for capped vertex functions of Hilbert schemes on cyclic quotient surfaces and refines large framing vanishing results, linking algebraic and geometric structures.

Findings

Derived explicit generating functions for capped vertex functions.

Identified conditions under which vertex functions are purely classical.

Proposed conjectures on integrality and wall-crossing of vertex functions.

Abstract

We study the $K$-theoretic enumerative geometry of cyclic Nakajima quiver varieties, with particular focus on $\text{Hilb}^{m}([\mathbb{C}^{2}/\mathbb{Z}_{l}])$, the equivariant Hilbert scheme of points on $\mathbb{C}^2$. The direct sum over $m$ of the equivariant $K$-theories of these varieties is known to be isomorphic to the ring symmetric functions in $l$ colors, with structure sheaves of torus fixed points identified with wreath Macdonald polynomials. Using properties of wreath Macdonald polynomials and the recent identification of the Maulik-Okounkov quantum affine algebra for cyclic quivers with the quantum toroidal algebras of type $A$, we derive an explicit formula for the generating function of capped vertex functions of $\text{Hilb}^{m}([\mathbb{C}^{2}/\mathbb{Z}_{l}])$ with descendants given by exterior powers of the $0$th tautological bundle. We also sharpen the large framing vanishing results of Okounkov, providing a class of descendants and cyclic quiver varieties for which the capped vertex functions are purely classical. Our results also suggest certain integrality and wall-crossing conjectures for capped vertex functions.