Noncommutative spectral geometry: A guided tour for theoretical physicists

We consider aspects of the noncommutative approach to the standard model based on the spectral action principle. We show that as a consequence of the incorporation of the Clifford structures in the formalism, the spectral action contains an extended scalar sector, with respect to the minimal standard model. This may have interesting phenomenological consequences. Some of these new scalar fields carry both weak isospin and color indices. We calculate the new terms in spectral action due to the presence of these fields. Our analysis demonstrates that the fermionic doubling in the noncommutative geometry is not just a presence of spurious degrees of freedom, but it is an interesting and peculiar property of the formalism, which leads to physically valuable conclusions. Some of the new fields do not contribute to the physical fermionic action, but they appear in the bosonic spectral action. Their contributions to the Dirac operator correspond to couplings with the spurious fermions, which are projected out.

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 geometric spaces with boundary: Spectral action

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.

We report the results obtained in the study of Alain Connes noncommutative spectral geometry construction focusing on its essential ingredient of the algebra doubling. We show that such a two-sheeted structure is related with the gauge structure of the theory, its dissipative character and carries in itself the seeds of quantization. From the algebraic point of view, the algebra doubling process has the same structure of the deformed Hops algebra structure which characterizes quantum field theory.

We consider the spectral action within the context of a 4-dimensional manifold with torsion and show that, in the vacuum case, the equations of motion reduce to Einstein's equations, securing the linear stability of the theory. To subsequently investigate the nonvacuum case, we consider the spectral action of an almost commutative torsion geometry and show that the Hamiltonian is bounded from below, a result which guarantees the linear stability of the theory.

This brief Letter demonstrates that effects from a noncommutative spacetime geometry will measurably affect the value of (g−2)μ inferred from the decay of the muon to an electron plus two neutrinos. If the scale of noncommutativity is O (TeV), the alteration of the V–A structure of the lepton–lepton–W vertex is sufficient to shift the inferred value of (g−2)μ to one part in 108. This may account for the recently reported 2.6σ discrepancy between the BNL measurement aexpt=11659202(14)(6)×10−10 and the Standard Model prediction aSM=11659159.6(6.7)×10−10.

We start by briefly reviewing the description of gravity theories as gauge theories in four dimensions. More specifically we recall the procedure leading to the results of General Relativity and Weyl Gravity in a gauge-theoretic manner. Then, after a brief reminder of the formulation of gauge theories on noncommutative spaces, we review our recent work, where gravity is constructed as a gauge theory on the fuzzy \(dS_4\).

We have obtained the supersymmetric extension of a spectral triple that specifies a noncommutative geometry. We assume that the functional space |$\mathcal {H}$| consists of wave functions of matter fields and their superpartners included in the minimum supersymmetric standard model (MSSM). We introduce the internal fluctuations of the Dirac operator on the finite space as well as on the manifold by elements of the algebra |$\mathcal {A}$| in the triple. So, we obtain not only the vector supermultiplets that mediate |${\rm SU}(3)\otimes {\rm SU}(2)\otimes {\rm U}(1)_Y$| gauge degrees of freedom but also Higgs supermultiplets that appear in the MSSM from the same standpoint. According to the supersymmetric version of the spectral action principle, we calculate the square of the fluctuated total Dirac operator and verify that the Seeley–DeWitt coefficients give the correct action of the vector and Higgs supermultiplets. We also verify that the relation between the coupling constants of |${\rm SU}(3)$|⁠, |${\rm SU}(2)$|⁠, and |${\rm U}(1)_Y$| is same as that of SU(5) unification theory.

We investigate cosmological predictions on the early universe based on the noncommutative geometry (NCG) models of gravity coupled to matter. Using the renormalization group analysis for the standard model with right-handed neutrinos and Majorana mass terms, which is the particle
\nphysics content of the most recent NCG models, we analyze the behavior of the coefficients of the gravitational and cosmological terms in the Lagrangian derived from the asymptotic expansion of the spectral action functional of NCG. We find emergent Hoyle–Narlikar and conformal
\ngravity at the see-saw scales and a running effective gravitational constant, which affects the propagation of gravitational waves and the evaporation law of primordial black holes and provides Linde models of negative gravity in the early universe. The same renormalization group
\nanalysis also governs the running of the effective cosmological constant of the model. The model also provides a Higgs-based slow-roll inflationary mechanism, for which one can explicitly compute the slow-roll parameters.
\nThe particle physics content allows for dark matter models based on sterile neutrinos with Majorana mass terms.

Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M_2(H) and M_4(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the "particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication.

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.

We present a model of (modified) gravity on spacetimes with fractal structure based on packing of spheres, which are (Euclidean) variants of the packed swiss cheese cosmology models. As the action functional for gravity we consider the spectral action of noncommutative geometry, and we compute its expansion on a space obtained as an Apollonian packing of three-dimensional spheres inside a four-dimensional ball. Using information from the zeta function of the Dirac operator of the spectral triple, we compute the leading terms in the asymptotic expansion of the spectral action. They consist of a zeta regularization of the divergent sum of the leading terms of the spectral actions of the individual spheres in the packing. This accounts for the contribution of points 1 and 3 in the dimension spectrum (as in the case of a 3-sphere). There is an additional term coming from the residue at the additional point in the real dimension spectrum that corresponds to the packing constant, as well as a series of fluctuations coming from log-periodic oscillations, created by the points of the dimension spectrum that are off the real line. These terms detect the fractality of the residue set of the sphere packing. We show that the presence of fractality influences the shape of the slow-roll potential for inflation, obtained from the spectral action. We also discuss the effect of truncating the fractal structure at a certain scale related to the energy scale in the spectral action.

Spacetime geometry is twisted (deformed) into noncommutative spacetime geometry, where functions and tensors are now star-multiplied. Consistently, spacetime diffeomorhisms are twisted into noncommutative diffeomorphisms. Their deformed Lie algebra structure and that of infinitesimal Poincaré transformations is defined and explicitly constructed.This allows to construct a noncommutative theory of gravity.This article is based on common work with Christian Blohmann, Marija Dimitrijević, Frank Meyer, Peter Schupp and Julius Wess.

A short historical review is made of some recent literature in the field of noncommutative geometry, especially the efforts to add a gravitational field to noncommutative models of space-time and to use it as an ultraviolet regulator. An extensive bibliography has been added containing reference to recent review articles as well as to part of the original literature.