Exts and vertex operators
type: Item
instance of: scholarly article
title: Exts and vertex operators
zbMATH Open document ID: 1256.14010
DOI: 10.1215/00127094-1593380
author: Erik Carlsson
author: Andrei Okounkov
published in: Duke Mathematical Journal
publication date: 2012-07-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/0801.2565
full work available at URL: https://projecteuclid.org/euclid.dmj/1338987169
reviewed by: Sergiy Koshkin
Mathematics Subject Classification ID: 14D21
zbMATH DE Number: 6053750
zbMATH Keywords: Hilbert scheme of points
zbMATH Keywords: Nakajima operators
zbMATH Keywords: Ext-groups
zbMATH Keywords: Nekrasov instanton partition function
zbMATH Keywords: Jack symmetric functions
Wikidata QID: Q112631867
MaRDI profile type: Publication
OpenAlex ID: W3099515180
arXiv ID: 0801.2565
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review text: The cohomology of the Hilbert scheme of points \(\mathrm{Hilb}_n(S)\) of non-singular quasi-projective surfaces \(S\) has been described in terms of vertex operators on \(\mathcal{F}:=\bigoplus_nH^*(\mathrm{Hilb}_n(S,\mathbb{Q}))\) introduced by Nakajima and Grojnowski. In this paper the authors introduce another natural set of vertex operators on \(\mathcal{F}\) that depends on a line bundle \(\mathcal{L}\) over \(S\). These operators are defined in terms of the Chern classes of the alternating Ext-groups between two ideal sheaves twisted by \(\mathcal{L}\), and can be characterized by their commutation relations with the Nakajima operators. Moreover, the authors give an explicit formula for them in terms of the latter.NEWLINENEWLINEFor the \((\mathbb{C}^*)^2\)--equivariant cohomology of \(\mathrm{Hilb}_n(\mathbb{C}^2)\) the trace of these vertex operators becomes the Nekrasov instanton partition function with matter in the adjoint representation. Equivariant localization reduces the above mentioned formula to a Pieri-type formula for the Jack symmetric functions.
description: scientific article; zbMATH DE number 6053750