Introduction to Dubois-Violette's noncommutative differential geometry

We present a detailed review of the Dubois-Violette approach to noncommutative differential calculus. The noncommutative differential geometry of matrix algebras and the noncommutative Poisson structures are treated in some detail. We also present the analog of Maxwell's theory and new models of Yang-Mills-Higgs theories that can be constructed in this framework. In particular, some simple models are compared with the standard model. Finally, we discuss some perspectives and open questions.     

Riemannian manifolds in noncommutative geometry

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c${}^c$ manifolds; and conversely, in the presence of a spin^c${}^c$ structure. We also show how to obtain an analogue of Kasparov’s fundamental class for a Riemannian manifold, and the associated notion of Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.     

On noncommutative and pseudo-Riemannian geometry

We introduce the notion of a pseudo-Riemannian spectral triple which generalizes the notion of spectral triple and allows for a treatment of pseudo-Riemannian manifolds within a noncommutative setting. It turns out that the relevant spaces in noncommutative pseudo-Riemannian geometry are not Hilbert spaces any more but Krein spaces, and Dirac operators are Krein-selfadjoint. We show that the noncommutative tori can be endowed with a pseudo-Riemannian structure in this way. For the noncommutative tori as well as for pseudo-Riemannian spin manifolds the dimension, the signature of the metric, and the integral of a function can be recovered from the spectral data.     

On the variational noncommutative poisson geometry

We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in the associative words. We state the basic properties of the variational Schouten bracket and derive an interesting criterion for (non)commutative differential operators to be Hamiltonian (and thus determine the (non)commutative Poisson structures). We place the noncommutative jet-bundle construction at hand in the context of the quantum string theory.     

Inflation mechanism in asymptotic noncommutative geometry

The possibility of having an inflationary epoch within a noncommutative geometry approach to unifying gravity and the Standard Model is demonstrated. This inflationary phase occurs without the need to introduce ad hoc additional fields or potentials, rather it is a consequence of a nonminimal coupling between the geometry and the Higgs field.     

Boundary conditions of the RGE flow in the noncommutative geometry approach to particle physics and cosmology

We investigate the effect of varying boundary conditions on the renormalization group flow in a recently developed noncommutative geometry model of particle physics and cosmology. We first show that there is a sensitive dependence on the initial conditions at unification, so that, varying a parameter even slightly can be shown to have drastic effects on the running of the model parameters. We compare the running in the case of the default and the maximal mixing conditions at unification. We then exhibit explicitly a particular choice of initial conditions at the unification scale, in the form of modified maximal mixing conditions, which have the property that they satisfy all the geometric constraints imposed by the noncommutative geometry of the model at unification, and at the same time, after running them down to lower energies with the renormalization group flow, they still agree in order of magnitude with the predictions at the electroweak scale.     

Lattice gauge fields and noncommutative geometry

Conventional approaches to lattice gauge theories do not properly consider the topology of spacetime or of its fields. In this paper, we develop a formulation which tries to remedy this defect. It starts from a cubical decomposition of the supporting manifold (compactified space-time or spatial slice) interpreting it as a finite topological approximation in the sense of Sorkin. This finite space is entirely described by the algebra of cochains with the cup product. The methods of Connes and Lott are then used to develop gauge theories on this algebra and to derive Wilson's actions for the gauge and Dirac fields therefrom which can now be given geometrical meaning. We also describe very natural candidates for the QCD θ-term and Chern-Simons action suggested by this algebraic formulation. Some of these formulations are simpler than currently available alternatives. The paper treats both the functional integral and Hamiltonian approaches.     

Charged thin-shell gravastars in noncommutative geometry

In this paper we construct a charged thin-shell gravastar model within the context of noncommutative geometry. To do so, we choose the interior of the nonsingular de Sitter spacetime with an exterior charged noncommutative solution by cut-and-paste technique and apply the generalized junction conditions. We then investigate the stability of a charged thin-shell gravastar under linear perturbations around the static equilibrium solutions as well as the thermodynamical stability of the charged gravastar. We find the stability regions, by choosing appropriate parameter values, located sufficiently close to the event horizon.     

Introduction to M(atrix) theory and noncommutative geometry

Noncommutative geometry is based on an idea that an associative algebra can be regarded as “an algebra of functions on a noncommutative space”. The major contribution to noncommutative geometry was made by A. Connes, who, in particular, analyzed Yang–Mills theories on noncommutative spaces, using important notions that were introduced in his papers (connection, Chern character, etc). It was found recently that Yang–Mills theories on noncommutative spaces appear naturally in string/M-theory; the notions and results of noncommutative geometry were applied very successfully to the problems of physics.

In this paper we give a mostly self-contained review of some aspects of M(atrix) theory, of Connes’ noncommutative geometry and of applications of noncommutative geometry to M(atrix) theory. The topics include introduction to BFSS and IKKT matrix models, compactifications on noncommutative tori, a review of basic notions of noncommutative geometry with a detailed discussion of noncommutative tori, Morita equivalence and -duality, an elementary discussion of noncommutative orbifolds, noncommutative solitons and instantons. The review is primarily intended for physicists who would like to learn some basic techniques of noncommutative geometry and how they can be applied in string theory and to mathematicians who would like to learn about some new problems arising in theoretical physics.

The second part of the review (Sections 10–12) devoted to solitons and instantons on noncommutative Euclidean space is almost independent of the first part.     

Topologically nontrivial field configurations in noncommutative geometry

In the framework of noncommutative geometry we describe spinor fields with nonvanishing winding number on a truncated (fuzzy) sphere. The corresponding field theory actions conserve all basic symmetries of the standard commutative version (space isometries and global chiral symmetry), but due to the noncommutativity of the space the fields are regularized and they contain only a finite number of modes.Part of the Project P8916-PHY of the Fonds zur Förderung der wissenschaftlichen Forschung in Österreich.     

Coverings,correspondences, and noncommutative geometry

We construct an additive category where objects are embedded graphs in the 3-sphere and morphisms are geometric correspondences given by 3-manifolds realized in different ways as branched covers of the 3-sphere, up to branched cover cobordisms. We consider dynamical systems obtained from associated convolution algebras endowed with time evolutions defined in terms of the underlying geometries. We describe the relevance of our construction to the problem of spectral correspondences in noncommutative geometry.     

Finite quantum field theory in noncommutative geometry

We describe a self-interacting scalar field on a truncated sphere and perform the quantization using the functional (path) integral approach. The theory possesses full symmetry with respect to the isometries of the sphere. We explicitly show that the model is finite and that UV regularization automatically takes place.     

Distances in finite spaces from noncommutative geometry

Following the general principles of noncommutative geometry, it is possible to define a metric on the space of pure states of the noncommutative algebra generated by the coordinates. This metric generalizes the usual Riemannian one. We investigate some general properties of this metric in finite commutative cases corresponding to a metric on a finite set, and also compute explicitly some distances associated to commutative or noncommutative algebras.     

Thermodynamics on noncommutative geometry in coherent state formalism

The thermodynamics of ideal gas on the noncommutative geometry in the coherent state formalism is investigated. We first evaluate the statistical interparticle potential and see that there are residual “attraction (repulsion) potential” between boson (fermion) in the high temperature limit. The characters could be traced to the fact that, the particle with mass m in noncommutative thermal geometry with noncommutativity θ and temperature T will correspond to that in the commutative background with temperature T(1+kTmθ)⁻¹. Such a correspondence implies that the ideal gas energy will asymptotically approach to a finite limiting value as that on commutative geometry at T_θ=(kmθ)⁻¹. We also investigate the squeezed coherent states and see that they could have arbitrary mean energy. The thermal properties of those systems are calculated and compared to each other. We find that the heat capacity of the squeezed coherent states of boson and fermion on the noncommutative geometry have different values, contrast to that on the commutative geometry.     

Does noncommutative geometry predict nonlinear Higgs mechanism?

It is argued that the noncommutative geometry construction of the standard model may predict a nonlinear symmetry-breaking mechanism rather than the orthodox Higgs mechanism if there is much heavy generation in addition to the lightest generations. Such models have experimentally verifiable consequences.     

Distances on a one-dimensional lattice from noncommutative geometry

In this Letter, we continue the work of Bimonte, Lizzi and Sparano on distances on a one-dimensional lattice. We succeed in analytically proving the exact formulae for such distances. We find that the distance to an even point on the lattice is the geometrical average of the predecessor and successor distances to the neighbouring odd points.