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We outline a brief description of noncommutative geometry and present some applications in string theory. We use the fuzzy torus as our guiding example. [ABSTRACT FROM AUTHOR]

We study the interconnection between the finite projective modules for a fuzzy sphere, determined in a previous paper, and the matrix model approach, making clear the physical meaning of noncommutative topological configurations. [ABSTRACT FROM AUTHOR]

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]

It is argued that the internal dynamics of a baryon, as a bound state of QCD-strings and quarks, may be captured by a theory of matrix coordinates. [ABSTRACT FROM AUTHOR]

We observe that the von Neumann envelop of (i.e. the smallest von Neumann algebra that contains) the quantum algebra of functions on the normalizer of the group SU(1,1)≅SL(2,ℝ) in ${\rm SL}(2,\mathbb C)$ via deformation quantization contains the von Neumann algebraic quantum normalizer of SU(1,1) in the frame work of Waronowicz–Korogodsky, see ([8, Introduction and Sec. 1, Definition 1]), i.e. the C*-envelop or von Neumann envelop (W*-envelop) $W^*_q\widetilde{{\rm SU}(1,1)}$ of the algebraic Hopf algebra. We then use the technique of reduction to the maximal subgroup to compute the K-theory, the periodic cyclic homology and the corresponding Chern–Connes character. [ABSTRACT FROM AUTHOR]

We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S3, the dihedral group D4 and the quaternion group Q. Poincaré duality holds in every case, and under some assumptions (essentially the existence of a top form) we find that it must hold in general. A short review of the bicovariant (noncommutative) differential calculus on finite G is given for selfconsistency. Exterior derivative, exterior product, metric, Hodge dual, connections, torsion, curvature, and biinvariant integration can be defined algebraically. A projector decomposition of the braiding operator is found, and used in constructing the projector on the space of two-forms. By means of the braiding operator and the metric a knot invariant is defined for any finite group. [ABSTRACT FROM AUTHOR]

We formulate Dirac–Kähler fermion action by introducing a new Clifford product with noncommutative differential form on a lattice. Hermiticity of the Dirac–Kähler action requires to choose the lattice structure having both orientabilities on a link. The Kogut–Susskind fermion and the staggered fermion actions are derived directly from the Dirac–Kähler fermion formulated by the Clifford product. The lattice QCD action with Dirac–Kähler matter fermion is also derived via an inner product defined by the Clifford product. [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 present a pedagogical review of particle physics models that are based on the noncommutativity of space–time, [formula], with specific attention to the phenomenology these models predict in particle experiments either in existence or under development. We summarize results obtained for high energy scattering such as would occur, for example, in a future e[sup +]e[sup -] linear collider with [formula], as well as low energy experiments such as those pertaining to elementary electric dipole moments and other CP violating observables, and finally comment on the status of phenomenological work in cosmology and extra dimensions. [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]

In space–time dimensions larger than 2, whenever a global symmetry G is spontaneously broken to a subgroup H, and G and H are Lie groups, there are Nambu–Goldstone modes described by fields with values in G/H. In two-dimensional space–times as well, models where fields take values in G/H are of considerable interest even though in that case there is no spontaneous breaking of continuous symmetries. We consider such models when the world sheet is a two-sphere and describe their fuzzy analogs for G = SU(N+1), H = S(U(N-1) ⊗ U(1)) ≃ U(N) and [formula]. More generally our methods give fuzzy versions of continuum models on S[sup 2] when the target spaces are Grassmannians and flag manifolds described by (N+1) × (N+1) projectors of rank ≤ (N+1)/2. These fuzzy models are finite-dimensional matrix models which nevertheless retain all the essential continuum topological features like solitonic sectors. They seem well suited for numerical work. [ABSTRACT FROM AUTHOR]

Connes' functional formula of the Riemannian distance is generalized to the Lorentzian case using the so-called Lorentzian distance, the d'Alembert operator and the causal functions of a globally-hyperbolic spacetime. As a step of the presented machinery, a proof of the almost-everywhere smoothness of the Lorentzian distance considered as a function of one of the two arguments is given. Afterwards, using a C*-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures giving rise to a Lorentzian version of part of Connes' noncommutative geometry. The generalized noncommutative spacetime consists of a direct set of Hilbert spaces and a related class of C*-algebras of operators. In each algebra a convex cone made of self-adjoint elements is selected which generalizes the class of causal functions. The generalized events, called loci, are realized as the elements of the inductive limit of the spaces of the algebraic states on the C*-algebras. A partial-ordering relation between pairs of loci generalizes the causal order relation in spacetime. A generalized Lorentz distance of loci is defined by means of a class of densely-defined operators which play the role of a Lorentzian metric. Specializing back the formalism to the usual globally-hyperbolic spacetime, it is found that compactly-supported probability measures give rise to a non-pointwise extension of the concept of events. [ABSTRACT FROM AUTHOR]

We study σ-models on noncommutative spaces, notably on noncommutative tori. We construct instanton solutions carrying a nontrivial topological charge q and satisfying a Belavin-Polyakov bound. The moduli space of these instantons is conjectured to consists of an ordinary torus endowed with a complex structure times a projective space [formula]. [ABSTRACT FROM AUTHOR]

We present a brief review of the fuzzy disc, the finite algebra approximating functions on a disc, which we have introduced earlier. We also present a comparison with recent papers of Balachandran, Gupta and Kürkçüoǧlu, and of Pinzul and Stern, aimed at the discussion of edge states of a Chern-Simons thoery. [ABSTRACT FROM AUTHOR]

We use the framework of noncommutative geometry to define a discrete model for fluctuating geometry. Instead of considering ordinary geometry and its metric fluctuations, we consider generalized geometries where topology and dimension can also fluctuate. The model describes the geometry of spaces with a countable number n of points. The spectral principle of Connes and Chamseddine is used to define dynamics. We show that this simple model has two phases. The expectation value 〈n〉, the average number of points in the universe, is finite in one phase and diverges in the other. Moreover, the dimension δ is a dynamical observable in our model, and plays the role of an order parameter. The computation of 〈δ〉 is discussed and an upper bound is found, 〈δ〉 < 2. We also address another discrete model defined on a fixed d = 1 dimension, where topology fluctuates. We comment on a possible spontaneous localization of topology. [ABSTRACT FROM AUTHOR]

We improve the study of the lack of perturbative unitarity of noncommutative space-time quantum field theories derived from open string theory in electric backgrounds, enforcing the universality of the mechanism by which a tachyonic branch cut appears when the Seiberg-Witten limit freezes the string in an unstable vacuum. The main example is realized in the context of the on-shell four-tachyon amplitude of the bosonic string, and the dependence of the phenomenon on the brane-worldvolume dimension is analysed. We discuss the possibility of a proof in superstring theory, and finally mention the NCOS limit in this framework. [ABSTRACT FROM AUTHOR]

We study the geometric aspects of two exotic bialgebras S03 and S14 introduced in math.QA/0206053. These bialgebras are obtained by the Faddeev–Reshetikhin–Takhtajan RTT prescription with non-triangular R-matrices which are denoted R[sub 03] and R[sub 14] in the classification of Hietarinta, and they are not deformations of either GL(2) or GL(1|1). We give the spectral decomposition which involves two, respectively, three, projectors. These projectors are then used to provide the baxterization procedure with one, respectively, two, parameters. Further, the projectors are used to construct the noncommutative planes together with the corresponding differentials following the Wess–Zumino prescription. In all these constructions there appear nonstandard features which are noted. Such features show the importance of systematic study of all bialgebras of four generators. [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]

We analytically calculate the topological charge of general noncommutative U (N) instantons using the Corrigan's identity and find that the result is exactly the instanton number k, which appears in the noncommutative ADHM construction as the dimension of the vector space V. This result coincides with the corresponding fact in the commutative case. [ABSTRACT FROM AUTHOR]

We have calculated the topological charge of U(N) instantons on nondegenerate noncommutative spacetime to be exactly the instanton number k in a previous paper.1 This paper, which deals with the degenerate ${\bf R}^2_{\rm NC}\times{\bf R}^2$ case, is the continuation of Ref. 1. We find that the same conclusion holds in this case, thus complete the answer to the problem of topological charge of noncommutative U(N) instantons. [ABSTRACT FROM AUTHOR]