# Journal of Noncommutative Geometry

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**Volume 5, Issue 1, 2011, pp. 39–67**

**DOI: 10.4171/JNCG/69**

Published online: 2010-12-19

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

Eric Cagnache^{[1]}, Thierry Masson

^{[2]}and Jean-Christophe Wallet

^{[3]}(1) Univ. Paris-Sud 11, Orsay

(2) Univ. Paris-Sud 11, Orsay

(3) Univ. Paris-Sud 11, Orsay

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 ℳ. We show that the differential calculus, generated by the maximal subalgebra of the derivation algebra of ℳ that can be related to infinitesimal symplectomorphisms, gives rise to a natural construction of Yang–Mills–Higgs models on 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`^{∞}(`M`) ⊗ `M`_{n}(ℂ). The UV/IR mixing problem of the resulting Yang–Mills–Higgs models is also discussed.

*Keywords: *Derivation-based differential calculus, noncommutative gauge theories, Higgs–Yang–Mills theories

Cagnache Eric, Masson Thierry, Wallet Jean-Christophe: Noncommutative Yang–Mills–Higgs actions from derivation-based differential calculus. *J. Noncommut. Geom.* 5 (2011), 39-67. doi: 10.4171/JNCG/69