Noncommutative geometry of the quantum disk

These notes provide an introduction to the noncommutative matrix geometry which arises within matrix models of Yang-Mills type. Starting from basic examples of compact fuzzy spaces, a general notion of embedded noncommutative spaces (branes) is formulated, and their effective Riemannian geometry is elaborated. This class of configurations is preserved under small deformations, and is therefore appropriate for matrix models. A realization of generic 4-dimensional geometries is sketched, and the relation with spectral geometry and with noncommutative gauge theory is explained. In a second part, dynamical aspects of these matrix geometries are considered. The one-loop effective action for the maximally supersymmetric IKKT or IIB matrix model is discussed, which is well-behaved on 4-dimensional branes.

The paper presents applications of Toeplitz algebras in Noncommutative Geometry. As an example, a quantum Hopf fibration is given by gluing trivial U(1) bundles over quantum discs (or, synonymously, Toeplitz algebras) along their boundaries. The construction yields associated quantum line bundles over the generic Podles spheres which are isomorphic to those from the well-known Hopf fibration of quantum SU(2). The relation between these two versions of quantum Hopf fibrations is made precise by giving an isomorphism in the category of right U(1)-comodules and left modules over the C*-algebra of the generic Podles spheres. It is argued that the gluing construction yields a significant simplification of index computations by obtaining elementary projections as representatives of K-theory classes.

Part 1 Mathematical aspects of quantum groups theory and non-commutative geometry: Hopf algebra and Poisson structure of Lie groups and algebras quantum groups, algebras and their duality non-commutative spaces and quantum groups invariant differential calculi elements of quantum groups representations theory tensor products of representations q-tensors, q-vectors, q-scalars. Part 2 Deformation of harmonic oscillators: q-deformation of single-harmonic oscillator different forms of commutation relations representations q real and roots of unity cases algebraic maps from non-deformed to deformed oscillators path integral quantization of q-oscillator. Part 3 Q-deformation of space-time symmetries: relativistic space-time symmetries Poincare group as a classical deformation of Galilei group its representations multiparametric q-deformation of linear groups, twisted groups and algebras q-Poincare group as a q-subgroup of q-conformal one and its induced representations. Part 4 Non-commutative geometry and unification models: unified models, the problem of natural introduction of Higgs fields unified models of Higgs fields in the frame of non-commutative geometry and others.

The conference Conceptual and Technical Challenges for Quantum Gravity at Sapienza University of Rome, from 8 to 12 September 2014, has provided a beautiful opportunity for an encounterbetween different approaches and different perspectives on the quantum-gravity problem. It contributedto a higher level of shared knowledge among the quantum-gravity communities pursuing each specific research program.There were plenary talks on many different approaches, including in particular string theory, loop quantum gravity, spacetime noncommutativity, causal dynamical triangulations, asymptotic safety and causal sets. Contributions from the perspective of philosophy of science were also welcomed.In addition several parallel sessions were organized. The present volume collects contributions from the Noncommutative Geometry and Quantum Gravity parallel session4, with additional invited contributions from specialists in the field.Noncommutative geometry in its many incarnations appears at the crossroad of many researches in theoretical and mathematical physics:• from models of quantum space-time (with or without breaking of Lorentz symmetry) to loop gravity and string theory,• from early considerations on UV-divergencies in quantum field theory to recent models of gauge theories on noncommutative spacetime,• from Connes description of the standard model of elementary particles to recent Pati-Salam like extensions.This volume provides an overview of these various topics, interesting for the specialist as well as accessibleto the newcomer. 4partially funded by CNRS PEPS /PTI ‘‘Metric aspect of noncommutative geometry: from Monge to Higgs’’

It is for the most part expected that dark matter is important to clarify the rotation of the galaxy, It has effectively been seen that the non-commutative geometry background can achieve this objective similarly. The objective of this study is to investigate a relationship between non-commutative geometry and certain aspect of dark matter. We are relying on a basic mathematical expression argument that indicates that the appearance of dark matter in galaxies and galaxy clusters with regard to flat rotation curves is similarly a result of non commutative geometry.

Quantum Harmonic Analysis and Geometric Invariants

Given a unital $C^*$ -algebra and a faithful trace, we prove that the topology on the associated density space induced by the $C^*$ -norm is finer than the Bures metric topology. We also provide an example when this containment is strict. Next, we provide a metric on the density space induced by a quantum metric in the sense of Rieffel and prove that the induced topology is the same as the topology induced by the Bures metric and $C^*$ -norm when the $C^*$ -algebra is assumed to be finite dimensional. Finally, we provide an example where the Bures metric and induced quantum metric are not metric equivalent. Thus, we provide a bridge between these aspects of quantum information theory and noncommutative metric geometry.

After discussing several aspects of non-commutative geometry from a rather subjective point of view, algebraic techniques are shown to offer a powerful tool for studying specific manifolds in the realm of commutative geometry, with possible generalization to infinite dimensions.

0.1. Plan of the paper. This paper consists of two sections discussing various aspects of commutative and non-commutative geometry of tori and abelian varieties.

A Dirac operator is presented that will yield a $1^+$-summable regular even spectral triple for all noncommutative compact surfaces defined as subalgebras of the Toeplitz algebra. Connes’ conditions for noncommutative spin geometries are analyzed and it i