Cherednik algebras, \(W\)-algebras and the equivariant cohomology of the moduli space of instantons on \(\mathbb A^2\)
type: Item
instance of: scholarly article
title: Cherednik algebras, \(W\)-algebras and the equivariant cohomology of the moduli space of instantons on \(\mathbb A^2\)
zbMATH Open document ID: 1284.14008
published in: Publications Mathématiques
publication date: 2013-11-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.2756
review text: The authors observe a striking relation between \(N=2\) supersymmetric gauge theory in dimension four and two-dimensional Conformal Field theory. On one hand, there is the conformal blocks of Liouville theory associated with a punctured Riemann surface and the group \(U\), and on the other hand, there is the instanton part of the Nekrasov partition function for a suitable four-dimentional gauge theory associated with the group \(U_r\). Parts of this correspondence are obtained in the physics literature. In mathematical terms, a consequence of the AGT conjecture is the existence of a representation of the affine \(W\)-algebra of \(G\) on the equivariant intersection cohomology of the moduli space of \(G^L\)-instantons on \(\mathbb R^4\) satisfying some extra properties. \(G\) and \(G^L\) denotes a pair of complex reductive groups which are dual to each other in the sense of Langlands. This paper studies this action in detail for the gauge group \(G=G^L=\text{GL}_r\). A detailed construction is given, based on degenerate double affine Hecke algebras. The main result is described more precisely in what follows: let \(M_r=\bigsqcup_{n>0}M_{r,n}\) be the moduli space of rank \(r\) torsion free sheaves on \(\mathbb P^2\), equipped with a framing along \(\mathbb P^1_\infty\subset\mathbb P^2\). For fixed \(n\), \(M_{r,n}\) is a smooth symplectic variety, \(\dim M_{r,n}=2rn\), acted upon by a \(r+2\)-dimensional torus \(\tilde D=(\mathbb C^\times)^2\times D\) where \((\mathbb C^\times)^2\) acts on \(\mathbb P^2\) and \(D=(\mathbb C^\times)^r\) acts on the framing. In the particular case when \(r=1\), the moduli space \(M_{1,n}\) is isomorphic to the Hilbert scheme \(\text{Hilb}_n\) of \(n\) points on \(\mathbb C^2\). This article is in some sense guided by Nakajima's construction of a representation of the rank one Heisenberg algebra on the space \(\tilde L^{(1)}=\bigoplus_{n\geq 0} H_\ast(\text{Hilb}_n)\). The construction is by geometric methods, identifying it with the st…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 14D21
Mathematics Subject Classification ID: 81T30
zbMATH DE Number: 6233894
zbMATH Keywords: \(W\)-algebra
zbMATH Keywords: Hecke algebra
zbMATH Keywords: AGT conjecture
zbMATH Keywords: gauge theory
zbMATH Keywords: equivariant cohomology
zbMATH Keywords: Hecke correspondence
zbMATH Keywords: Heisenberg subalgebra
zbMATH Keywords: Virasoro subalgebra
zbMATH Keywords: Sekiguchi operators
author: Olivier Schiffmann
author: Eric Vasserot
MaRDI profile type: Publication
OpenAlex ID: W2964131193
arXiv ID: 1202.2756
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DOI: 10.1007/S10240-013-0052-3
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description: scientific article; zbMATH DE number 6233894