Moyal planes are spectral triples
type: Item
instance of: scholarly article
title: Moyal planes are spectral triples
zbMATH Open document ID: 1084.58008
published in: Communications in Mathematical Physics
publication date: 2004-12-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/hep-th/0307241
review text: Axioms for nonunital spectral triples, extending those introduced in the unital case by \textit{A. Connes} [Commun. Math. Phys. 182, 155--176 (1996; Zbl 0881.58009)], are given. A \textit{compact noncommutative spin manifold} consists of a \textit{spectral triple} \((\mathcal A, \mathcal H, D)\), where \(\mathcal A\) is a unital algebra, represented on a Hilbert space \(\mathcal H\), together with a distinguished self-adjoint operator, the abstract Dirac operator \(D\), whose resolvent is completely continuous, such that each operator \([D, a]\) for \(a \in \mathcal A\) is bounded. The authors show how to build noncompact noncommutative spin geometries: The indispensable commutative example of noncompact manifolds is considered. Then using a suitably chosen Dirac operator, the geometry associated to the Moyal product is laid out. As a consequence, the classical background of present-day noncommutative field theory is recast in the framework of the rigorous Connes formalism for geometrical noncommutative spaces. The spaces \(\mathbb R^{2N}\) endowed with Moyal products are investigated, and some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes-Lott functional action, are given for these noncommutative hyperplanes. The main results have been summarized in [\textit{V. Gayral}, Lett. Math. Phys. 65, 147--157 (2003; Zbl 1043.46051)].
reviewed by: Choonkil Park
Mathematics Subject Classification ID: 58J42
Mathematics Subject Classification ID: 81R60
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 46L60
zbMATH DE Number: 2123146
zbMATH Keywords: Moyal plane
zbMATH Keywords: spectral triple
zbMATH Keywords: noncompact noncommutative spin geometry
zbMATH Keywords: Moyal product
author: V. Gayral
author: José M. Gracia-Bondía
author: Bruno Iochum
author: Thomas Schücker
author: Joseph C. Várilly
MaRDI profile type: Publication
arXiv ID: hep-th/0307241
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description: scientific article