Wreath Macdonald polynomials, quiver varieties, and quasimap counts
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2024
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| author | Ayers, Jeffrey Dinkins, Hunter |
| author_facet | Ayers, Jeffrey Dinkins, Hunter |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_07399 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wreath Macdonald polynomials, quiver varieties, and quasimap counts Ayers, Jeffrey Dinkins, Hunter Algebraic Geometry High Energy Physics - Theory Mathematical Physics Combinatorics Representation Theory 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. |
| title | Wreath Macdonald polynomials, quiver varieties, and quasimap counts |
| topic | Algebraic Geometry High Energy Physics - Theory Mathematical Physics Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2410.07399 |