Quantum cohomology of the Hilbert scheme of points in the plane
type: Item
instance of: scholarly article
title: Quantum cohomology of the Hilbert scheme of points in the plane
zbMATH Open document ID: 1198.14054
published in: Inventiones Mathematicae
publication date: 2010-02-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0411210
review text: The Hilbert scheme of points Hilb\(_n(\mathbb{C}^2)\) is the crepant resolution of the orbifold quotient \((\mathbb{C}^2)^n/S_n\) containing the configurations of \(n\) distinct points in \(\mathbb{C}^2\). Its geometry and algebraic invariants have been a subject of intense study in the last decade due to connections with the string theory. In particular, the ring structure of its torus equivariant quantum cohomology has been established (the natural action of the complex torus on \((\mathbb{C}^2)^n/S_n\) extends to the Hilbert scheme). In the paper under review, the authors give an explicit formula for \(M_D\), the operator of multiplication by the first Chern class \(D\) of the Hilbert scheme in small quantum cohomology. The matrix elements of \(M_D\) give counts of rational curves meeting three given subvarieties of the scheme, and an associated differential equation is an integrable non-stationary deformation of the Calogero-Sutherland equation for quantum particles on a torus. The formula implies that \(D\) generates the small quantum cohomology ring, and completes establishing the four-way correspondence between the quantum cohomology of Hilb\(_n(\mathbb{C}^2)\), the equivariant orbifold cohomology of \((\mathbb{C}^2)^n/S_n\), and the Gromov-Witten and Donaldson-Thomas invariants of \(\mathbb{P}^1\times\mathbb{C}^2\).
reviewed by: Sergiy Koshkin
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 14C05
zbMATH DE Number: 5675040
zbMATH Keywords: Hilbert scheme of points
zbMATH Keywords: small quantum cohomology
zbMATH Keywords: Calogero-Sutherland operator
Wikidata QID: Q112631868
author: Andrei Okounkov
author: Rahul Pandharipande
MaRDI profile type: Publication
OpenAlex ID: W2020734693
arXiv ID: math/0411210
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DOI: 10.1007/S00222-009-0223-5
description: scientific article