Derivations of the Moyal Algebra and Noncommutative Gauge Theories

TL;DR

This paper reviews derivation-based differential calculus for noncommutative algebras, extends it to graded cases, and demonstrates its application in constructing noncommutative gauge theories and linking them to renormalizable models.

Contribution

It extends the derivation-based calculus framework to graded algebras and applies it to develop noncommutative gauge theories with Higgs fields, connecting to renormalizable models.

Findings

Derivation-based calculus is suitable for noncommutative gauge theories.

Higgs fields can be interpreted as covariant coordinates.

Link established between NC $oldsymbol{}$-model and gauge theory.

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 $\phi^4$-model with harmonic term and a gauge theory model. Some possible consequences of this are briefly discussed.