An introductory review to emergent noncommutative gravity within Yang–Mills matrix models is presented. Spacetime is described as a noncommutative brane solution of the matrix model, i.e. as a submanifold of . Fields and matter on the brane arise as fluctuations of the bosonic resp. fermionic matrices around such a background, and couple to an effective metric interpreted in terms of gravity. Suitable tools are provided for the description of the effective geometry in the semi-classical limit. The relation to non-commutative gauge theory and the role of UV/IR mixing are explained. Several types of geometries are identified, in particular 'harmonic' and 'Einstein' types of solutions. The physics of the harmonic branch is discussed in some detail, emphasizing the non-standard role of vacuum energy. This may provide a new approach to some of the big puzzles in this context. The IKKT model with D = 10 and close relatives are singled out as promising candidates for quantum theory of fundamental interactions including gravity.

Emergent geometry and gravity from matrix models: an introduction

Determinations of the motion of the sun with respect to the extra-galactic nebulae have involved a K term of several hundred kilometers which appears to be variable. Explanations of this paradox have been sought in a correlation between apparent radial velocities and distances, but so far the results have not been convincing. The present paper is a re-examination of the question, based on only those nebular distances which are believed to be fairly reliable.

Distances of extra-galactic nebulae depend ultimately upon the application of absolute-luminosity criteria to involved stars whose types can be recognized. These include, among others, Cepheid variables, novae, and blue stars involved in emission nebulosity. Numerical values depend upon the zero point of the period-luminosity relation among Cepheids, the other criteria merely check the order of the distances. This method is restricted to the few nebulae which are well resolved by existing instruments. A study of these nebulae, together with those in which any stars at all can be recognized, indicates the probability of an approximately uniform upper limit to the absolute luminosity of stars, in the late-type spirals and irregular nebulae at least, of the order of M (photographic) = −6.3.1 The apparent luminosities of the brightest stars in such nebulae are thus criteria which, although rough and to be applied with caution, furnish reasonable estimates of the distances of all extra-galactic systems in which even a few stars can be detected.

Finally, the nebulae themselves appear to be of a definite order of absolute luminosity, exhibiting a range of four or five magnitudes about an average value M (visual) = −15.2.1 The application of this statistical average to individual cases can rarely be used to advantage, but where considerable numbers are involved, and especially in the various clusters of …

A Relation Between Distance and Radial Velocity Among Extra-Galactic Nebulae

The QCD running coupling

We review the present knowledge for $\alpha_s$, the fundamental coupling underlying the interactions of quarks and gluons in QCD. The dependence of $\alpha_s(Q^2)$ on momentum transfer $Q$ encodes the underlying dynamics of hadron physics -from color confinement in the infrared domain to asymptotic freedom at short distances. We review constraints on $\alpha_s(Q^2)$ at high $Q^2$, as predicted by perturbative QCD, and its analytic behavior at small $Q^2$, based on models of nonperturbative dynamics. In the introductory part of this review, we explain the phenomenological meaning of $\alpha_s$, the reason for its running, and the challenges facing a complete understanding of its analytic behavior in the infrared domain. In the second, more technical, part of the review, we discuss the behavior of $\alpha_s(Q^2)$ in the high $Q^2$ domain of QCD. We review how $\alpha_s$ is defined, including its renormalization scheme dependence, the definition of its renormalization scale, the utility of effective charges, as well as Commensurate Scale Relations which connect the various definitions of $\alpha_s$ without renormalization-scale ambiguity. We also report recent measurements and theoretical analyses which have led to precise QCD predictions at high energy. In the last part of the review, we discuss the challenge of understanding the analytic behavior $\alpha_s(Q^2)$ in the infrared domain. We also review important methods for computing $\alpha_s$, including lattice QCD, the Schwinger-Dyson equations, the Gribov-Zwanziger analysis and light-front holographic QCD. After describing these approaches and enumerating their conflicting predictions, we discuss the origin of these discrepancies and how to remedy them. Our aim is not only to review the advances in this difficult area, but also to suggest what could be an optimal definition of $\alpha_s$ in order to bring better unity to the subject.

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The QCD Running Coupling

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.

Topics in Noncommutative Geometry Inspired Physics

Matrix models of the Yang-Mills type lead to an emergent gravity theory, which does not require fine-tuning of a cosmological constant. We find cosmological solutions of the Friedmann-Robertson-Walker type. They generically have a big bounce, and an early inflationlike phase with graceful exit. The mechanism is purely geometrical; no ad hoc scalar fields are introduced. The solutions are stabilized through vacuum fluctuations and are thus compatible with quantum mechanics. This leads to a Milne-like universe after inflation, which appears to be in remarkably good agreement with observation and may provide an alternative to standard cosmology.

Cosmological solutions of emergent noncommutative gravity.

Fermions coupled to Yang-Mills matrix models are studied from the point of view of emergent gravity. We show that the simple matrix model action provides an appropriate coupling for fermions to gravity, albeit with a non-standard spin connection. Integrating out the fermions in a nontrivial geometrical background induces indeed the Einstein-Hilbert action and a dilaton-like term, at least for on-shell geometries. This explains and precisely reproduces the UV/IR mixing for fermions in noncommutative gauge theory, extending recent results for scalar fields. It also explains why some UV/IR mixing remains even in supersymmetric models, except in the N = 4 case.

Fermions and emergent noncommutative gravity

We study fermions coupled to Yang-Mills matrix models from the point of view of emergent gravity. The matrix model Dirac operator provides an appropriate coupling for fermions to the effective gravitational metric for general branes with nontrivial embedding, albeit with a non-standard spin connection. This generalizes previous results for 4-dimensional matrix models. Integrating out the fermions in a nontrivial geometrical background induces indeed the Einstein-Hilbert action of the effective metric, as well as additional terms which couple the Poisson tensor to the Riemann tensor, and a dilaton-like term.

Fermions and noncommutative emergent gravity II: curved branes in extra dimensions

In this paper, we explore the appealing possibility that the strong suppression of large-size QCD instantons — as evident from lattice data — is due to a surviving conformal space-time inversion symmetry. This symmetry is both suggested from the striking invariance of high-quality lattice data for the instanton size distribution under inversion of the instanton size ρ → ρ2/ρ and from the known validity of space-time inversion symmetry in the classical instanton sector. We project the instanton calculus onto the four-dimensional surface of a five-dimensional sphere via conformal stereographic mapping, before investigating conformal inversion. This projection to a compact, curved geometry is both to avoid the occurence of divergences and to introduce the average instanton size ρ→ρ2/ρ from the lattice data as a new length scale. The average instanton size is identified with the radius b of this 5d-sphere and acts as the conformal inversion radius. For b = ρ, our corresponding results are almost perfectly symmetric under space-time inversion and in good qualitative agreement with the lattice data. For ρ/b → 0 we recover the familiar results of instanton perturbation theory in flat 4d-space. Moreover, we illustrate that a (weakly broken) conformal inversion symmetry would have significant consequences for QCD beyond instantons. As a further successful test for inversion symmetry, we present striking implications for another instanton dominated lattice observable, the chirality-flip ratio in the QCD vacuum.

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Daniela Klammer

Papers

Cosmological solutions of emergent noncommutative gravity.

Fermions and emergent noncommutative gravity

Fermions and noncommutative emergent gravity II: curved branes in extra dimensions

QCD-Instantons and Conformal Space-Time Inversion Symmetry

QCD-instantons and conformal inversion symmetry