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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]

κ -Poincaré invariant gauge theories on κ -Minkowski space-time, which are noncommutative analogs of the usual U(1) gauge theory, exist only in five dimensions. These are built from noncommutative twisted connections on a hermitian right module over the algebra coding the κ -Minkowski space-time. We show that twisting the action of this algebra on the hermitian module, assumed to be a copy of it, affects neither the value of the above dimension nor the noncommutative gauge group defined as the unitary automorphisms of the module leaving the hermitian structure unchanged. Only the hermiticity condition obeyed by the gauge potential becomes twisted. Similarities between the present framework and algebraic features of twisted spectral triples are exhibited. [ABSTRACT FROM AUTHOR]

In the context of noncommutative (NCG) gauge gravity, and using a cosmic time power law formula for the scale factor, a Friedman-Robertson-Walker (FRW) like metric is obtained. Within the fermions tunneling effect approach and depending on the various intervals of the power parameter, expressions of the apparent horizon are also derived. It is shown that in some regions of the parameter space, a pure NCG trapped horizon does exist leading to new interpretation of the role played by the noncommutativity of the space-time. [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]

Discretized Kaluza-Klein theory in spacetime can be constructed based on the concepts of noncommutative geometry. In this paper, we show that it is possible to incorporate the non-Abelian gauge fields in this framework. The generalized Hilbert-Einstein action is gauge invariant only in two cases. In the first case, the gauge group must be Abelian on one sheet of spacetime and non-Abelian on the other one. In the second case, the gauge group must be the same on two sheets of spacetime. Actually, the theories of electroweak and strong interactions can fit into these two cases. [ABSTRACT FROM AUTHOR]

Noncommutative geometry naturally emerges in low energy physics of Landau models as a consequence of level projection. In this work, we proactively utilize the level projection as an effective tool to generate fuzzy geometry. The level projection is specifically applied to the relativistic Landau models. In the first half of the paper, a detail analysis of the relativistic Landau problems on a sphere is presented, where a concise expression of the Dirac-Landau operator eigenstates is obtained based on algebraic methods. We establish 'gauge' transformation between the relativistic Landau model and the Pauli-Schrödinger nonrelativistic quantum mechanics. After the transformation, the Dirac operator and the angular momentum operators are found to satisfy the algebra. In the second half, the fuzzy geometries generated from the relativistic Landau levels are elucidated, where unique properties of the relativistic fuzzy geometries are clarified. We consider mass deformation of the relativistic Landau models and demonstrate its geometrical effects to fuzzy geometry. Super fuzzy geometry is also constructed from a supersymmetric quantum mechanics as the square of the Dirac-Landau operator. Finally, we apply the level projection method to real graphene system to generate valley fuzzy spheres. [ABSTRACT FROM AUTHOR]

We recall the emergence within noncommutative geometry of a generalized gauge theory from a real spectral triple for a unital C ⁎ -algebra A . This includes a gauge group determined by the unitaries in A and gauge fields arising from a so-called perturbation semigroup which is associated to A . Our main new result is the construction of an upper semi-continuous C ⁎ -bundle on a (Hausdorff) base space X whose space of continuous sections is isomorphic to A . The gauge group acts by vertical automorphisms on this C ⁎ -bundle and can (under some conditions) be identified with the space of continuous sections of a group bundle on X . Moreover, in some cases the gauge group is found to coincide with the group of inner automorphisms of A . We discuss two classes of examples of our construction: Yang–Mills theory and toric noncommutative manifolds and we show that they actually give rise to continuous C ⁎ -bundles. Moreover, in these examples the gauge groups coincide with the inner automorphism groups and can be realized as spaces of sections of group bundles that we explicitly determine. [ABSTRACT FROM AUTHOR]