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

Abstract

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 $\mathcal{M}$. We show that the differential calculus, generated by the maximal subalgebra of the derivation algebra of $\mathcal{M}$ that can be related to infinitesimal symplectomorphisms, gives rise to a natural construction of Yang–Mills–Higgs models on $\mathcal{M}$ 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 $M_n(C)$ and the algebra of matrix valued functions $C^{\infty }(M)\otimes M_n(C)$. The UV/IR mixing problem of the resulting Yang–Mills–Higgs models is also discussed.