Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 013, 25 pages      arXiv:0811.3850      https://doi.org/10.3842/SIGMA.2009.013
Contribution to the Proceedings of the XVIIth International Colloquium on Integrable Systems and Quantum Symmetries

Derivations of the Moyal Algebra and Noncommutative Gauge Theories

Jean-Christophe Wallet
Laboratoire de Physique Théorique, Bât. 210, CNRS, Université Paris-Sud 11, F-91405 Orsay Cedex, France

Received October 29, 2008, in final form January 17, 2009; Published online January 30, 2009

Abstract
The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital algebras with involution. We extend this framework to the case of Z2-graded unital involutive algebras. We show, in the case of the Moyal algebra or some related Z2-graded version of it, that the derivation based differential calculus is a suitable framework to construct Yang-Mills-Higgs type models on Moyal (or related) algebras, the covariant coordinates having in particular a natural interpretation as Higgs fields. We also exhibit, in one situation, a link between the renormalisable NC φ4-model with harmonic term and a gauge theory model. Some possible consequences of this are briefly discussed.

Key words: noncommutative geometry; noncommutative gauge theories.

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