Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus

Understand

Derivations of a noncommutative algebra can be used to construct differential calculi, the so-called derivation-based differential calculi.

• We apply this framework to a version of the Moyal algebra ${\cal{M}}$.
• We show that the differential calculus, generated by the maximal subalgebra of the derivation algebra of ${\cal{M}}$ that can be related to infinitesimal symplectomorphisms, gives rise to a natural construction of Yang-Mills-Higgs models on ${\cal{M}}$ and a natural interpretation of the covariant coordinates as Higgs fields.
• We also compare in detail the main mathematical properties characterizing the present situation to those specific of two other noncommutative geometries, namely the finite dimensional matrix algebra $M_n({\mathbb{C}})$ and the algebra of matrix valued functions $C^\infty(M)\otimes M_n({\mathbb{C}})$.