Wreath Macdonald polynomials, quiver varieties, and quasimap counts

TL;DR

The paper develops a $K$-theoretic, quasimap-theoretic framework for cyclic Nakajima quiver varieties, focusing on $Hilb^{m}([\,\mathbb{C}^{2}/\mathbb{Z}_{l}\,])$ and its identification with wreath symmetric functions. By combining wreath Macdonald polynomials, the quantum toroidal algebra of type $A$, and the ABRR/fusion-operator formalism, the authors derive an explicit generating function for capped vertex functions with descendants given by exterior powers of the $0$th tautological bundle, valid for empty $l$-core sectors and expressed as a global exponential in colored power sums. They establish a refined large framing vanishing result, discuss global integrality and potential wall-crossing phenomena, and connect capped-vertex data to tensor-product structures and Calabi–Yau reductions via the capping operator. The work suggests integrality and wall-crossing conjectures for capped vertex functions and provides new evaluations and specializations (notably $q=\hbar$) that link to quantized Coulomb branches and the representation theory of quantum toroidal algebras. Overall, the results deepen the bridge between enumerative geometry of quiver varieties, wreath Macdonald theory, and quantum algebra actions, with concrete formulas and conjectural directions for broader wall-crossing and integrality phenomena.

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.

Figures (1)

• Figure 1: The Maya diagram for the partition $(3,2,2,1,1,1)$. The $3$-core is shaded.

Theorems & Definitions (63)

• Theorem 1.1: Theorem \ref{['mainthm']}
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6: Theorem \ref{['lfv']}
• Conjecture 1.7
• Proposition 1.8
• Conjecture 1.9
• Conjecture 1.10