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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]

In this paper, we define and investigate a notion of parallel transport on finite projective modules over finite matrix algebras. Given a derivation-based differential calculus on the algebra and a connection on the module, we construct for every derivation X a module parallel transport, which is a lift to the module of the one-parameter group of algebra automorphisms generated by X. This parallel transport morphism is determined uniquely by an ordinary differential equation depending on the covariant derivative along X. Based on these parallel transport morphisms, we define a basic set of gauge invariant observables, i.e. functions from the space of connections to the complex numbers. For modules equipped with a Hermitian structure, we prove that this set of observables is separating on the space of gauge equivalence classes of Hermitian connections. This solves the gauge copy problem for fuzzy gauge theories. [ABSTRACT FROM AUTHOR]

We refine the reconstruction theorem for almost-commutative spectral triples to a result for real almost-commutative spectral triples, clarifying in the process both concrete and abstract definitions of real commutative and almost-commutative spectral triples. In particular, we find that a real almost-commutative spectral triple algebraically encodes the commutative *-algebra of the base manifold in a canonical way, and that a compact oriented Riemannian manifold admits real (almost-)commutative spectral triples of arbitrary KO-dimension. Moreover, we define a notion of smooth family of real finite spectral triples and of the twisting of a concrete real commutative spectral triple by such a family, with interesting KK-theoretic and gauge-theoretic implications. [ABSTRACT FROM AUTHOR]

The U(1) gauge theory on a space with Lie type noncommutativity is constructed. The construction is based on the group of translations in Fourier space, which in contrast to space itself is commutative. In analogy with lattice gauge theory, the object playing the role of flux of field strength per plaquette, as well as the action, is constructed. It is observed that the theory, in comparison with ordinary U(1) gauge theory, has an extra gauge field component. This phenomena is reminiscent of similar ones in formulation of (N) gauge theory in space with canonical noncommutativity, and also appearance of gauge field component in discrete direction of Connes' construction of the Standard Model. [ABSTRACT FROM AUTHOR]

We present a study of D = 4 supersymmetric Yang-Mills matrix models with SO(3) mass terms based on the Monte Carlo method. In the bosonic models we show the existence of an exotic first-/second-order transition from a phase with a well defined background geometry (the fuzzy sphere) to a phase with commuting matrices with no geometry in the sense of Connes. At the transition point the sphere expands abruptly to infinite size then it evaporates as we increase the temperature (the gauge coupling constant). The transition looks first-order due to the discontinuity in the action whereas it looks second-order due to the divergent peak in the specific heat. The fuzzy sphere is stable for the supersymmetric models in the sense that the bosonic phase transition is turned into a very slow crossover transition. The transition point is found to scale to zero with N. We conjecture that the transition from the background sphere to the phase of commuting matrices is associated with spontaneous supersymmetry breaking. The eigenvalues distribution of any of the bosonic matrices in the matrix phase is found to be given by a nonpolynomial law obtained from the fact that the joint probability distribution of the four matrices is uniform inside a solid ball with radius R. The eigenvalues of the gauge field on the background geometry are also found to be distributed according to this nonpolynomial law. [ABSTRACT FROM AUTHOR]

We present a first numerical investigation of a noncommutative gauge theory defined via the spectral action for Moyal space with harmonic propagation. This action is approximated by finite matrices. Using Monte Carlo simulation we study various quantities such as the energy density, the specific heat density and some order parameters, varying the matrix size and the independent parameters of the model. We find a peak structure in the specific heat which might indicate possible phase transitions. However, there are mathematical arguments which show that the limit of infinite matrices can be quite different from the original spectral model. [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]

It is shown how Seiberg-Witten equations can be obtained by means of Fedosov deformation quantization of endomorphism bundle and the corresponding theory of equivalences of star products. In such setting, Seiberg-Witten map can be iteratively computed for arbitrary gauge group up to any given degree with recursive methods of Fedosov construction. Presented approach can be also considered as a generalization of Seiberg-Witten equations to Fedosov type of noncommutativity. [ABSTRACT FROM AUTHOR]

In this paper, we construct a model of spinor fields interacting with specific gauge fields on the fuzzy sphere and analyze the chiral symmetry of this "Schwinger model". In constructing the theory of gauge fields interacting with spinors on the fuzzy sphere, we take the approach that the Dirac operator Dq on the q-deformed fuzzy sphere $S_{qF}^2$ is the gauged Dirac operator on the fuzzy sphere. This introduces interaction between spinors and specific one-parameter family of gauge fields. We also show how to express the field strength for this gauge field in terms of the Dirac operators Dq and D alone. Using the path integral method, we have calculated the 2n-point functions of this model and show that, in general, they do not vanish, reflecting the chiral non-invariance of the partition function. [ABSTRACT FROM AUTHOR]

A covariant Poisson bracket and an associated covariant star product in the sense of deformation quantization are defined on the algebra of tensor-valued differential forms on a symplectic manifold, as a generalization of similar structures that were recently defined on the algebra of (scalar-valued) differential forms. A covariant star product of arbitrary smooth tensor fields is obtained as a special case. Finally, we study covariant star products on a more general Poisson manifold with a linear connection, first for smooth functions and then for smooth tensor fields of any type. Some observations on possible applications of the covariant star products to gravity and gauge theory are made. [ABSTRACT FROM AUTHOR]

In this review article we discuss some of the applications of noncommutative geometry in physics that are of recent interest, such as noncommutative many-body systems, noncommutative extension of Special Theory of Relativity kinematics, twisted gauge theories and noncommutative gravity. [ABSTRACT FROM AUTHOR]

We investigate a version of noncommutative QED where the interaction term, although natural, breaks the spin-statistics connection. We calculate e- + e- → e- + e- and γ + e- → γ + e- cross-sections in the tree approximation and explicitly display their dependence on θμν. Remarkably the zero of the elastic e- + e- → e- + e- cross-section at 90° in the center-of-mass system, which is due to Pauli principle, is shifted away as a function of θμν and energy. [ABSTRACT FROM AUTHOR]

A unified description of a symmetrized and anti-symmetrized Moyal star product of the noncommutative infinitesimal gauge transformations is presented and the corresponding Seiberg-Witten maps are derived. Moreover, the noncommutative covariant derivative, field strength tensor as well as gauge transformations are shown to be consistently constructed not on the enveloping but on the Lie and/or Poisson algebra. As an application, a pure geometric extension of the standard model is shown explicitly. [ABSTRACT FROM AUTHOR]

Some derivation-based differential calculi which have been used to construct models of noncommutative gauge theories are presented and commented. Some comparisons between them are made. [ABSTRACT FROM AUTHOR]

In this paper we review Klimčík's construction of noncommutative gauge theory on the fuzzy supersphere. This theory has an exact SUSY gauge symmetry with a finite number of degrees of freedom and thus in principle it is amenable to the methods of matrix models and Monte Carlo numerical simulations. We also write down in this paper a novel fuzzy supersymmetric scalar action on the fuzzy supersphere. [ABSTRACT FROM AUTHOR]

After a short introduction, we show how the canonical formalism is constructed in space-time noncommutative theories such that it allows defining the notion of first-class constraints and analyzing gauge symmetries. We use this formalism to noncommutatively deform the two-dimensional string gravity (also known as the Witten black hole). [ABSTRACT FROM AUTHOR]

In 1932 Ettore Majorana proposed an infinite-component relativistic wave equation for particles of arbitrary integer and half-integer spin. In the late 80s and early 90s it was found that the higher-derivative geometric particle models underlie the Majorana equation, and that its (2+1)-dimensional analogue provides with a natural basis for the description of relativistic anyons. We review these aspects and discuss the relationship of the equation to the exotic planar Galilei symmetry and noncommutative geometry. We also point out the relation of some Abelian gauge field theories with Chern-Simons terms to the Landau problem in the noncommutative plane from the perspective of the Majorana equation. [ABSTRACT FROM AUTHOR]

The idea of applying the gauge principle to formulate the general theory of relativity started with Utiyama in 1956. In this article, various applications of the gauge principle applied to different aspects of the gravitational interactions, are reviewed. [ABSTRACT FROM AUTHOR]

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]

We investigate through the Morita equivalence the problem of mapping, odd dimensional noncommutative lattice gauge theories onto suitable matrix models. We specialize our analysis to three-dimensional noncommutative QED (NCQED) and scalar QED (NCsQED), for which we explicitly build the corresponding matrix model. [ABSTRACT FROM AUTHOR]

We propose an action for gravity on a fuzzy sphere, based on a matrix model. We find striking similarities with an analogous model of two-dimensional gravity on a noncommutative plane, i.e. the solution space of both models is spanned by pure U(2) gauge transformations acting on the background solution of the matrix model, and there exist deformations of the classical diffeomorphisms which preserve the two-dimensional noncommutative gravity actions. [ABSTRACT FROM AUTHOR]

We introduce a matrix model for noncommutative gravity, based on the gauge group U(2)⊗U(2). The vierbein is encoded in a matrix Y[sub μ], having values in the coset space U(4)/(U(2)⊗U(2)), while the spin connection is encoded in a matrix X[sub μ], having values in U(2)⊗U(2). We show how to recover the Einstein equations from the θ→0 limit of the matrix model equations of motion. We stress the necessity of a metric tensor, which is a covariant representation of the gauge group in order to set up a consistent second order formalism. We finally define noncommutative gravitational instantons as generated by U(2)⊗U(2) valued quasi-unitary operators acting on the background of the matrix model. Some of these solutions have naturally self-dual or anti-self-dual spin connections. [ABSTRACT FROM AUTHOR]

Discretized non-Abelian gauge theories living on finite group spaces G are defined by means of a geometric action ∫ Tr F ∧ [sub *]F. This technique is extended to obtain discrete versions of the Born–Infeld action. The discretizations are in 1–1 correspondence with differential calculi on finite groups. A consistency condition for duality invariance of the discretized field equations is derived for discretized U(1) actions S[F] living on a four-dimensional Abelian G. Discretized electromagnetism satisfies this condition and therefore admits duality rotations. Yang–Mills and Born–Infeld theories are also considered on product spaces M[sup D]×G, and we find the corresponding field theories on M[sup D] after Kaluza–Klein reduction on the G discrete internal spaces. We examine in detail the case G=Z[sub N], and discuss the limit N→∞. A self-contained review on the noncommutative differential geometry of finite groups is included. [ABSTRACT FROM AUTHOR]

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]

In this paper, we apply Connes' noncommutative geometry and the Seiberg–Witten map to a discrete noncommutative space consisting of n copies of a given noncommutative space Rm. The explicit action functional of gauge fields on this discrete noncommutative space is obtained. [ABSTRACT FROM AUTHOR]

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]