On deformation quantizations of hypertoric varieties
type: Item
instance of: scholarly article
title: On deformation quantizations of hypertoric varieties
zbMATH Open document ID: 1259.14002
author: Gwyn Bellamy
author: Toshiro Kuwabara
published in: Pacific Journal of Mathematics
publication date: 2013-01-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1005.4645
full work available at URL: p02.xhtml
review text: Let \(X\) be complex analytic manifold with \(\mathcal O_X\) the sheaf of regular, holomorphic functions. Denote by \(\mathcal D_X\) the sheaf of differential operators, by \(\mathbf{k}=\mathbb C((\hbar))\) the field of formal Laurent series in \(\hbar\), and by \(\mathbf{k}(0)\) the subring \(\mathbb C\hbar\) of formal functions on \(\mathbb C\). Given \(m\in\mathbb Z\), define \(\mathcal W_{T^\ast\mathbb C^n}(m)\) to be the sheaf of formal power series \(\sum_{i\geq -m}\hbar^ia_i\in\mathbb O_{T^\ast\mathbb C^n}\) on the cotangent bundle \(T^\ast\mathbb C^n\) on \(\mathbb C^n\). Fix coordinates \(x_1,\dots,x_n\) on \(\mathbb C^n\) and dual coordinates \(\xi_1,\dots,\xi_n\) on \((\mathbb C^n)^\ast\). Set \(\mathcal W_{T^\ast\mathbb C^n}=\cup_{m\in\mathbb Z}W_{T^\ast\mathbb C^n}(m)\). Then \(W_{T^\ast\mathbb C^n}\) is given a structure of a sheaf of (noncommutative) \(\mathbf{k}\)-algebras on \(T^\ast\mathbb C^n\), and \(\mathcal O_{T^\ast\mathbb C^n}\) is a sheaf of Poisson algebras with a particular given Poisson bracket. Assume that \(X\) is a symplectic manifold with holomorphic 2-form \(\omega_X\). A \(W\)-algebra on \(X\) is a sheaf of \(\mathbf{k}\)-algebras \(\mathcal W_X\) together with a \(\mathbf{k}(0)\)-subalgebra \(\mathcal W_X(0)\) such that for each point \(x\in X\) there exists an open neighborhood \(U\) of \(x\in X\), a symplectic map \(f:U\rightarrow V\subset T^\ast\mathbb C^n\) and a \(\mathbf{k}\)-algebra isomorphism \(r:f^{-1}(\mathcal W_{T^\ast\mathbb C^n}|_V)\overset\sim\rightarrow\mathcal W|_U\) such that these work well together. Kontsevich and Polesello and Schapira have shown that one can construct a stack of \(W\)-algebroids on any symplectic manifold. This provides a quantization of the sheaf of holomorphic functions on the manifold. In certain cases, these stacks of \(W\)-algebroids are the algebroids associated to a sheaf of \(W\)-algebras. When the manifold is the Hamiltonian reduction of a space equipped with a sheaf of \(W\)-alge…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 16S80
Mathematics Subject Classification ID: 18E35
zbMATH DE Number: 6121947
zbMATH Keywords: deformation quantixation
zbMATH Keywords: hypertoric varieties
zbMATH Keywords: quantum localization
zbMATH Keywords: Cherednik algebra
zbMATH Keywords: \(W\)-algebras
zbMATH Keywords: \(W\)-algebroids
MaRDI profile type: Publication
OpenAlex ID: W1964547573
arXiv ID: 1005.4645
DOI: 10.2140/PJM.2012.260.89
description: scientific article