Graphs, spectral triples and Dirac zeta functions

Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes' non-commutative spin geometry encompassing non-commutative Riemannian, symplectic, complex-Hermitian and (Hyper-)Kaehler geometry. A general framework for non-commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples. In particular, the non-commutative torus and the non-commutative 3-sphere are studied in some detail.

The principal object in noncommutative geometry is the spectral triple consisting of an algebra , a Hilbert space and a Dirac operator . Field theories are incorporated in this approach by the spectral action principle, which sets the field theory action to , where f is a real function such that the trace exists and Λ is a cutoff scale. In the low-energy (weak-field) limit, the spectral action reproduces reasonably well the known physics including the standard model. However, not much is known about the spectral action beyond the low-energy approximation. In this paper, after an extensive introduction to spectral triples and spectral actions, we study various expansions of the spectral actions (exemplified by the heat kernel). We derive the convergence criteria. For a commutative spectral triple, we compute the heat kernel on the torus up to the second order in gauge connection and consider limiting cases.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker’s 75th birthday devoted to ‘Applications of zeta functions and other spectral functions in mathematics and physics’.

Noncommutative geometry and the BV formalism: Application to a matrix model

Metric approximations of spectral triples on the Sierpiński gasket and other fractal curves

This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, and the comparison of Connes distance with the minimal length that emerges in various models of quantum spacetime. Links with other areas of mathematics are studied, in particular the horizontal distance in sub-Riemannian geometry. The interpretation of Connes distance as a noncommutative version of the Monge-Kantorovich metric in optimal transport is also discussed.

In this review we present some of the fundamental mathematical structures\nwhich permit to define noncommutative gauge field theories. In particular, we\nemphasize the theory of noncommutative connections, with the notions of\ncurvatures and gauge transformations. Two different approaches to\nnoncommutative geometry are covered: the one based on derivations and the one\nbased on spectral triples. Examples of noncommutative gauge field theories are\ngiven to illustrate the constructions and to display some of the common\nfeatures.\n

Morita equivalence and spectral triples on noncommutative orbifolds

Noncommutative geometric spaces with boundary: Spectral action

Compact κ-deformation and spectral triples

Zeta functions, excision in cyclic cohomology and index problems

The class of a fibre in noncommutative geometry

The generalization of the Chamseddine–Connes spectral triples action to its (left and right) fractional counterpart is constructed within the context of the Riemann–Liouville and Erdelyi–Kober (left and right) fractional operators. In the fractional approach, the Dirac operators [Formula: see text] is approximated by [Formula: see text] and the spectral triple [Formula: see text] is replaced by its fractional equivalent [Formula: see text], [Formula: see text], [Formula: see text], 0 < α < 1. When the (left) fractional action is applied to the noncommutative space defined by the spectrum of the Standard Model, one obtains many attractive characteristics including time-dependent gauge couplings constants ([Formula: see text]), a time-dependent cosmological constant (Λ cos ), a time-dependent scalar Ricci curvature (R), a time-dependent Newton's coupling constant, and a time-dependent Higgs square mass [Formula: see text]. Furthermore, [Formula: see text], Λ cos , R, and [Formula: see text] were found to be nonsingulars at the Planck's time. When the (left and right) fractional bosonic action is taken into account, all the previous functions are found to be complexified, including gravity. Many additional interesting features are discussed and explored in some details.

As we have learned in Sect. 1.6 a given spectral triple \((\mathscr {A,H,D})\) ought to be considered as a representative of the entire family of triples \((\mathscr {A},\mathscr {H},\mathscr {D}_{\mathbb {A}})\), which yield equivalent geometries. It is therefore of utmost importance to understand how the spectral action is affected by the fluctuations of geometry. We explore the meromorphic structure of the fluctuated zeta function and, for regular spectral triples with simple dimension spectra, we provide a few formulae for the noncommutative integrals. Finally, we sketch the method of operator perturbations.

We compute Zero Point Energy in a spherically symmetric background with the help of the Wheeler-DeWitt equation. This last one is regarded as a Sturm-Liouville problem with the cosmological constant considered as the associated eigenvalue. The graviton contribution, at one loop is extracted with the help of a variational approach together with Gaussian trial functionals. The divergences handled with a zeta function regularization are compared with the results obtained using a Noncommutative Geometry (NCG) and Modified Dispersion Relations (MDR). In both NCG and MDR no renormalization scheme is necessary to remove infinities in contrast to what happens in conventional approaches. Effects on photon propagation are briefly discussed.KeywordsCosmological ConstantZero Point EnergyNoncommutative GeometryModify Dispersion RelationWave FunctionalThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We construct spectral triples in a sense of noncommutative differential geometry, associated with a Riemannian foliation on a compact manifold, and describe its dimension spectrum.