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We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple ( A , H , D ) where D is no longer required to be self-adjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative geometry with boundary. In particular, we derive the Hochschild character theorem in this setting. We give a detailed study of Dirac operators with Dirichlet boundary conditions on domains in R d , d ≥ 2 .

We introduce a model of noncommutative geometry that gives rise to the uncertainty relations recently derived from the discussion of a quantum clock. We investigate the dynamics of a free particle in this model from the point of view of doubly special relativity and discuss the geodesic motion in a Schwarzschild background., 7 pages, version accepted for publication on Phys. Lett. A. A discussion of the motion of a particle in a Schwarzschild background has been added

In noncommutative geometry, Connes's spectral distance is an extended metric on the state space of a C*-algebra generalizing Kantorovich's dual formula of the Wasserstein distance of order 1 from optimal transport. It is expressed as a supremum. We present a dual formula - as an infimum - generalizing Beckmann's "dual of the dual" formulation of the Wasserstein distance. We then discuss some examples with matrix algebras, where such a dual formula may be useful to obtain upper bounds for the distance., 11 pages

In noncommutative geometry (NCG) the spectral action principle predicts the standard model (SM) particle masses by constraining the scalar and Yukawa couplings at some heavy scale, but gives an inconsistent value for the Higgs mass. Nevertheless, the scalar sector in the NCG approach to the standard model, is in general composed of two Higgs doublets and its phenomenology remains unexplored. In this work, we present a type-II two-Higgs-doublet model in NCG, with a SM-like Higgs mass compatible with the 125~GeV experimental value and extra scalars within the alignment limit without decoupling with masses from 350 GeV., Comment: 19 pages, 3 figures

This paper studies the K-homology of a crossed product of a discrete group acting smoothly on a manifold, with a better understanding of the noncommutative geometry of the crossed-product as the primary goal, and the Baum-Connes apparatus as the main tool. Examples suggest that the correct notion of the `Dirac class' of such a noncommutative space is the image under the equivalence determined by Baum-Connes of the fibre of the fibration of the Borel space associated to the action and a smooth model for the classifying space of the group. We give a systematic study of such fibre, or `Dirac classes,' with applications to the construction of interesting spectral triples and computation of their K-theory functionals, and we prove in particular that both the well-known deformation of the Dolbeault operator on the noncommutative torus, and the class of the boundary extension of a hyperbolic group, are both Dirac classes in this sense and therefore can be treated topologically in the same way.

A general framework of non-perturbative quantum field theory on a curved background is proposed. A quantum field theory is in this setting characterised by an embedding of a space of field configurations into a Hilbert space over R ∞ . This embedding, which is only local up to a scale that we interpret as the Planck scale, coincides in the local and flat limit with the plane wave expansion known from canonical quantisation. We identify a universal Bott–Dirac operator acting in the Hilbert space over R ∞ and show that it gives rise to the free Hamiltonian both in the case of a scalar field theory and in the case of a Yang–Mills theory. These theories come with a canonical fermionic sector for which the Bott–Dirac operator also provides the Hamiltonian. We prove that Hilbert space representations of algebras of observables exist non-perturbatively for a real scalar theory and for a gauge theory, both with or without the fermionic sectors, and show that the free theories are given by semi-finite spectral triples over the respective configuration spaces. Finally, we propose a class of quantum field theories whose interactions are generated by inner fluctuations of the Bott–Dirac operator.

Gravitational collapse of a shell of dust in noncommutative geometry is probed by the renormalized geodesic length, which is dual to probe the thermalization by the two-point correlation function in the dual conformal field theory. We find that larger the noncommutative parameter is, longer the thermalization time is, which implies that the large noncommutative parameter delays the thermalization process. We also investigate how the noncommutative parameter affects the thermalization velocity and thermalization acceleration., Comment: some materials have been deleted

This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This involves a notion of Poincar\'e duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa-Witten, in the sense of an explicit control of the Vafa-Witten bound under conformal changes of metric. This result has several noncommutative manifestations for conformal deformations of ordinary spectral triples, spectral triples associated to conformal weights on noncommutative tori, and spectral triples associated to duals of torsion-free discrete cocompact subgroups satisfying the Baum-Connes conjecture., Comment: Final version. 38 pages

Let N be the set of all natural numbers and l ∞ = l ∞ ( N ) (resp., L ∞ = L ∞ ( 0 , ∞ ) ) be the Banach space of all bounded sequences x = ( x 1 , x 2 , … ) (resp., (classes of) Lebesgue measurable bounded functions on ( 0 , ∞ ) ) with the uniform norm. We work with subsets of the set of all normalized positive functionals on l ∞ and L ∞ which are used in noncommutative geometry to define various classes of Dixmier traces (singular positive traces) on the ideal M 1 , ∞ of compact operators on an infinite-dimensional Hilbert space H with logarithmic divergence of the partial sums of their singular values. These classes of traces are of importance in noncommutative geometry and with each such a class (say B ) of traces, a subset M 1 , ∞ B ⊂ M 1 , ∞ on which all traces from B coincide is linked. The main objective of the present article is to give new characterizations of M 1 , ∞ B for commonly occurring classes of Dixmier traces. We also answer a question from Benameur and Fack (2006) [4] concerning conditions on a Dixmier trace under which Lidskii-type formulae from that article hold.

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 $\theta$ and temperature $T$ will correspond to that in the commutative background with temperature $T(1+kTm\theta)^{-1}$. Such a correspondence implies that the ideal gas energy will asymptotically approach to a finite limiting value as that on commutative geometry at $T_\theta= (km\theta)^{-1}$. 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., Comment: Add analysis about thermodynamics of squeezed coherent state, Latex 12 pages

Recently Verlinde proposed that gravity can be described as an emergent phenomena arising from changes in the information associated with the positions of material bodies. By using noncommutative geometry as a way to describe the microscopic, microstructure of quantum spacetime, we derive modified Friedmann equation in this setup and study the entropic force modifications to the inflationary dynamics of early universe., 14 pages, 5 figuers, Accepted for Publication in Phys. Lett. B

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.

In this work, we find exact wormhole solutions in the context of noncommutative geometry, and further explore their physical properties and characteristics. The energy density of these wormhole geometries is a smeared and particle-like gravitational source, where the mass is diffused throughout a region of linear dimension $\sqrt{\alpha}$ due to the intrinsic uncertainty encoded in the coordinate commutator. Furthermore, we also analyze these wormhole geometries considering that the equation governing quantum fluctuations behaves as a backreaction equation. In particular, the energy density of the graviton one loop contribution to a classical energy in a traversable wormhole background and the finite one loop energy density is considered as a self-consistent source for these wormhole geometries. Interesting solutions are found for an appropriate range of the parameters, validating the perturbative computation introduced in this semi-classical approach., Comment: 8 pages, 4 figures. Final version to appear in PLB. References added and one figure

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 is an endomorphism bundle, we construct a PSU(N)-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., Comment: 14 pages

Recent developments in superstring theory and noncommutative geometry are deeply related to the idea of Eguchi-Kawai reduction in large N gauge theories which dates back to early 80s. After a general review on this subject including revived interests in solving planar QCD, we present some results in the superstring matrix model suggesting the dynamical generation of 4d space-time due to the collapse of the eigenvalue distribution. We then discuss interesting dynamical properties of field theories in noncommutative geometry, which have been revealed by Monte Carlo simulations of twisted reduced models. We conclude with a comment on the recent proposal for a lattice construction of supersymmetric gauge theories based on reduced models., Comment: 14 pages, 12 figures, Lattice2003(plenary)

In this Paper, we investigate the effects of space noncommutativity and the generalized uncertainty principle on the stability of circular orbits of particles in a central force potential. We show that, up to first order in noncommutativity parameter, an angular momentum dependent term will be appear in the equations of the particle orbits which affects the stability of circular orbits. In the case of high angular momentum, the condition for stability of circular orbits will change considerably relative to commutative case.

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 $SO(d,d|{\mathbb Z})$-duality, an elementary discussion of instantons and noncommutative orbifolds. 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., v. 3: section 8.2 expanded, typos corrected, two references added. The second part of this review is now available as a preprint hep-th/0107251. Both parts will be published together supplemented by the English translation of A. Connes' 1980 seminal paper "C*algebras and differential geometry", Compt. Rend. Acad. Sci. (Ser.I Math.) A290 (1980) 599-604. This translation is also available as hep-th/0101093

A non technical discussion of some aspects and uses of noncommutative geometry in physics, using the words Spectral Geometry, part of the title of the workshop, as a guide.

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 the finite commutative case which corresponds to a metric on a finite set, and also give some examples of computations in both commutative and noncommutative cases., Comment: 27 pages, 2 figures

The “trace formula” of Chazarain, Duistermaat, and Guillemin expresses that the singularities of the distribution trace of the wave group on a compact Riemannian manifoldXis included in the set of periods of the geodesic flow restricted toS*X. Most of the objects involved in this trace formula have analogues in Connes' Noncommutative Geometry. This paper shows, on several significant examples of Noncommutative Geometry, that Connes' definition of geodesic flow leads to statements analogous to the classical trace formula of Chazarain, Duistermaat, and Guillemin.

Supergravity on $AdS_3\times S^3\times {\bf T}^4$ has a dual description as a conformal sigma-model with the target space being the moduli space of instantons on the noncommutative torus. We derive the precise relation between the parameters of this noncommutative torus and the parameters of the near-horizon geometry. We show that the low energy dynamics of the system of $D1D5$ branes wrapped on the torus of finite size is described in terms of the noncommutative geometry. As a byproduct, we give a prediction on the dependence of the moduli space of instantons on the noncommutative ${\bf T}^4$ on the metric and the noncommutativity parameter. We give a compelling evidence that the moduli space of stringy instantons on ${\bf R}^4$ with the $B$ field does not receive $\alpha'$-corrections. We also study the relation between the $D1D5$ sigma-model instantons and the supergravity instantons., Comment: latex, 45 pp., 2 figures, references and explanations added

The structure of S-duality in U(1) gauge theory on a 4-manifold M is examined using the formalism of noncommutative geometry. A noncommutative space is constructed from the algebra of Wilson-'t Hooft line operators which encodes both the ordinary geometry of M and its infinite-dimensional loop space geometry. S-duality is shown to act as an inner automorphism of the algebra and arises as a consequence of the existence of two independent Dirac operators associated with the spaces of self-dual and anti-selfdual 2-forms on M. The relations with the noncommutative geometry of string theory and T-duality are also discussed., 13 pages LaTeX, no figures

We analyse in detail the quantization of a simple noncommutative model of spontaneous symmetry breaking in zero dimensions taking into account the noncommutative setting seriously. The connection to the counting argument of Feyman diagrams of the corresponding theory in four dimensions is worked out explicitly. Special emphasis is put on the motivation as well as the presentation of some well-known basic notions of quantum field theory which in the zero-dimensional theory can be studied without being spoiled by technical complications due to the absence of divergencies., 111 pages, 6 figures, Habilitationsschrift

A. Connes and A. Chamseddine have proposed a new geometric version of the standard model including a noncommutative relativity action. We present a systematic analysis of the relations among masses and coupling constants in these approaches. At the tree level, for a given top mass, the Higgs mass mH is constrained to lie in an interval. Moreover, playing with the noncommutative gauge couplings, we compare the influence of the Higgs mass renormalization in these effective theories. The existing intersection is mH = 188–201 GeV.

We consider new D = 2 nonrelativistic model of classical mechanics providing via the Noether theorem the (2 + 1)-Galilean symmetry algebra with two central charges: the mass m and the coupling constant k of a Chern-Simons-like term. We discuss the interpretation of k as describing the noncommutativity of D = 2 space coordinates. We quantize the model and show that it describes the superposition of a free motion in noncommutative D = 2 space as well as the “internal” oscillator modes. We add a suitably chosen class of velocity-dependent two-particle interactions, which is described by local potentials in D = 2 noncommutative space. We show that the indefinite metric due to the third-order time derivative term in the field equations, even in the presence of interactions, can be eliminated by the imposition of a subsidiary condition.

A supersymmetric theory in two-dimensions has enough data to define a noncommutative space thus making it possible to use all tools of noncommutative geometry. In particular, we apply this to the N=1 supersymmetric non-linear sigma model and derive an expression for the generalized loop space Dirac operator, in presence of a general background, using canonical quantization. The spectral action principle is then used to determine a spectral action valid for the fluctuations of the string modes., Latex file, 13 pages. Correction to equation 47, which should read Tr| |^2 and not |Tr |^2. Final form to appear in Physics Letters B

We apply noncommutative geometry to a system of N parallel D-branes, which is interpreted as a quantum space. The Dirac operator defining the quantum differential calculus is identified to be the supercharge for strings connecting D-branes. As a result of the calculus, Connes' Yang-Mills action functional on the quantum space reproduces the dimensionally reduced U(N) super Yang-Mills action as the low energy effective action for D-brane dynamics. Several features that may look ad hoc in a noncommutative geometric construction are shown to have very natural physical or geometric origin in the D-brane picture in superstring theory., 16 pages, Latex, typos corrected and minor modification made

We consider a new D=2 nonrelativistic classical mechanics model providing via the Noether theorem the (2+1)-Galilean symmetry algebra with two central charges: mass m and the coupling constant k of a Chern-Simons-like term. In this way we provide the dynamical interpretation of the second central charge of the (2+1)-dimensional Galilean algebra. We discuss also the interpretation of k as describing the noncommutativity of D=2 space coordinates. The model is quantized in two ways: using the Ostrogradski-Dirac formalism for higher order Lagrangians with constraints and the Faddeev-Jackiw method which describes constrained systems and produces nonstandard symplectic structures. We show that our model describes the superposition of a free motion in noncommutative D=2 space as well as the "internal" oscillator modes. We add a suitably chosen class of velocity-dependent two-particle interactions, which is descrobed by local potentials in D=2 noncommutative space. We treat, in detail, the particular case of a harmonic oscillator and describe its quantization. It appears that the indefinite metric due to the third order time derivative term in the field equations, even in the presence of interactions, can be eliminated by the imposition of a subsidiary condition., small changes to make the paper look identical to the published version

We study a class of noncommutative geometries that give rise to dimensionally reduced Yang-Mills theories. The emerging geometries describe sets of copies of an even dimensional manifold. Similarities to the D-branes in string theory are discussed., Comment: 12 pages, Latex, minor corrections

We present a derivation of the general form of the scalar potential in Yang-Mills theory of a non-commutative space which is a product of a four-dimensional manifold times a discrete set of points. We show that a non-trivial potential without flat directions is obtained after eliminating the auxiliary fields only if constraints are imposed on the mass matrices utilised in the Dirac operator. The constraints and potential are related to a prepotential function., Comment: 8 pages, Tex file

We study the form of the continuous time heat kernel for a second order discrete Laplacian on a weighted graph. The analysis is shown to be closely related to the theory of symmetric Markov semigroups on noncommutative L p spaces and to the noncommutative geometry of Connes. The paper obtains better pointwise upper bounds on the heat kernels than those previously known, by the use of a novel metric on the graph. In certain cases it is shown that the new estimates are optimal of their type. The metric is investigated at some length and compared with other known metrics on graphs.

The present report contains an introduction to some elementary concepts in noncommutative differential geometry. The material extends upon ideas first presented by Dimakis and Mueller-Hoissen. In particular, stochastic calculus and the Ito formula are shown to arise naturally from introducing noncommutativity of functions (0-forms) and differentials (1-forms). The abstract construction allows for the straightforward generalization to lattice theories for the direct implementation of numerical models. As an elementary demonstration of the formalism, the standard Black-Scholes model for option pricing is reformulated.