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Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus. (English) Zbl 1226.81279

The authors demonstrate that most of the classical properties of noncommutative (NC) gauge theories on the Moyal space have a natural interpretation within the framework of differential calculus based on derivations. Indeed, this framework underlies the first prototypes of NC matrix-valued field theories. By considering a natural modification of the minimal differential calculus generated by the “spatial derivations”, which underlies most of the works in the physical literature, the authors demonstrate that this new differential calculus, which is generated by the maximal subalgebra of the derivation algebra of \(\mathcal M\) whose elements are related to infinitesimal symplectomorphisms, creates the possibility of constructing NC gauge theories that can be interpreted as Yang-Mills-Higgs models on \(\mathcal M\), the covariant coordinates of the physics literature being interpreted as Higgs fields due to the existence of a gauge-invariant canonical connection. The authors consider models invariant under \(U(1)\) or \(U(n)\) gauge transformations. They also compare in detail the main mathematical properties characterizing the present situation to two other NC geometries, that is, the finite dimensional matrix algebra \(M_n(C)\) and the algebra of matrix-valued functions \(C^\infty (M)\otimes M_n(C)\). They also discuss the topics of a phenomenon called ultraviolet/infrared (UV/IR) mixing.

MSC:

81T75 Noncommutative geometry methods in quantum field theory
81T13 Yang-Mills and other gauge theories in quantum field theory
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
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