Derivations of the Moyal Algebra and Noncommutative Gauge Theories

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 ${\mathbb{Z}}_2$-graded unital involutive algebras. We show, in the case of the Moyal algebra or some related ${\mathbb{Z}}_2$-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 ${\varphi }^4$-model with harmonic term and a gauge theory model. Some possible consequences of this are briefly discussed.

arXiv comment25 pages, 1 figure. Based on a talk given at the XVIIth International Colloquium on Integrable Systems and Quantum Symmetries, June 19-22, 2008, Prague

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• Quantum instability of gauge theories on $\kappa$-Minkowski space
• Gradient expansion of the non-Abelian gauge-covariant Moyal star-product
• Gauge theory models on $\kappa$-Minkowski space: Results and prospects