Derivations of the Moyal Algebra and Noncommutative Gauge Theories

Understand

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.