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We review the present status of gauge theories built on various quantum space–times described by noncommutative space–times. The mathematical tools and notions underlying their construction are given. Different formulations of gauge theory models on Moyal spaces as well as on quantum spaces whose coordinates form a Lie algebra are covered, with particular emphasis on some explored quantum properties. Recent attempts aiming to include gravity dynamics within a noncommutative framework are also considered. [ABSTRACT FROM AUTHOR]

We analyze a U ( 2 ) -matrix model derived from a finite spectral triple. By applying the BV formalism, we find a general solution to the classical master equation. To describe the BV formalism in the context of noncommutative geometry, we define two finite spectral triples: the BV spectral triple and the BV auxiliary spectral triple. These are constructed from the gauge fields, ghost fields and anti-fields that enter the BV construction. We show that their fermionic actions add up precisely to the BV action. This approach allows for a geometric description of the ghost fields and their properties in terms of the BV spectral triple. [ABSTRACT FROM AUTHOR]

Abstract: We generalize to topologically non-trivial gauge configurations the description of the Einstein–Yang–Mills system in terms of a noncommutative manifold, as was done previously by Chamseddine and Connes. Starting with an algebra bundle and a connection thereon, we obtain a spectral triple, a construction that can be related to the internal Kasparov product in unbounded KK-theory. In the case that the algebra bundle has typical fiber , we construct a -principal bundle for which it is an associated bundle. The so-called internal fluctuations of the spectral triple are parametrized by connections on this principal bundle and the spectral action gives the Yang–Mills action for these gauge fields, minimally coupled to gravity. Finally, we formulate a definition for a topological spectral action. [Copyright &y& Elsevier]

Abstract: We present a method where derivations of ⋆-product algebras are used to build covariant derivatives for noncommutative gauge theory. We write down a noncommutative action by linking these derivations to a frame field induced by a nonconstant metric. An example is given where the action reduces in the classical limit to scalar electrodynamics on a curved background. We further use the Seiberg–Witten map to extend the formalism to arbitrary gauge groups. A proof of the existence of the Seiberg–Witten map for an Abelian gauge potential is given for the formality ⋆-product. We also give explicit formulas for the Weyl-ordered ⋆-product and its Seiberg–Witten maps up to second order. [Copyright &y& Elsevier]

The quantum statistical dynamics of a position coordinate x coupled to a reservoir requires theoretically two copies of the position coordinate within the reduced density matrix description. One coordinate moves forward in time while the other coordinate moves backward in time. It is shown that quantum dissipation induces, in the plane of the forward and backward motions, a noncommutative geometry. The noncommutative geometric plane is a consequence of a quantum dissipation induced phase interference which is closely analogous to the Aharonov–Bohm effect. [Copyright &y& Elsevier]

Abstract: It was argued in [Y. Maeda, A. Sako, Noncommutative deformation of instantons, J. Geom. Phys. 58 (2008) 1784] that the noncommutative deformation of instantons on a 4-torus should alter the instanton numbers for arbitrary noncommutativity parameter . We show that this is not the case for the theory discussed there. And we discuss the instanton numbers in general gauge theories on the noncommutative . [Copyright &y& Elsevier]