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We study the metric corresponding to a three-dimensional coset space SO(4)/SO(3) in the lattice setting. With the use of three integers n 1 , n 2 , and n 3 , and a length scale, l μ , the continuous metric is transformed into a discrete space. The numerical outcomes are compared with the continuous ones. The singularity of the black hole is explored and different domains are studied. [ABSTRACT FROM AUTHOR]

We give an overview of the applications of noncommutative geometry to physics. Our focus is entirely on the conceptual ideas, rather than on the underlying technicalities. Starting historically from the Heisenberg relations, we will explain how in general noncommutativity yields a canonical time evolution, while at the same time allowing for the coexistence of discrete and continuous variables. The spectral approach to geometry is then explained to encompass two natural ingredients: the line element and the algebra. The relation between these two is dictated by so-called higher Heisenberg relations, from which both spin geometry and non-abelian gauge theory emerges. Our exposition indicates some of the applications in physics, including Pati–Salam unification beyond the Standard Model, the criticality of dimension 4, second quantization and entropy. [ABSTRACT FROM AUTHOR]

We study numerically the curvature tensor in a three-dimensional discrete space. Starting from the continuous metric of a three-sphere, we transformed it into a discrete space using three integers n 1 , n 2 , and n 3 . The numerical results are compared with the expected values in the continuous limit. We show that as the number of cells in the lattice increases, the continuous limit is recovered. [ABSTRACT FROM AUTHOR]

We focus on studying, numerically, the scalar curvature tensor in a two-dimensional discrete space. The continuous metric of a two-sphere is transformed into that of a lattice using two possible slicings. In the first, we use two integers, while in the second we consider the case where one of the coordinates is ignorable. The numerical results of both cases are then compared with the expected values in the continuous limit as the number of cells of the lattice becomes very large. [ABSTRACT FROM AUTHOR]

The 1.5 formalism played a key role in the discovery of supergravity and it has been used to prove the invariance of essentially all supergravity theories under local supersymmetry. It emerged from the gauging of the super Poincaré group to find supergravity. We review both of these developments as well as the auxiliary fields for simple supergravity and its most general coupling to matter using the tensor calculus. [ABSTRACT FROM AUTHOR]

We formulate a supersymmetric version of gravity with mimetic dark matter. The coupling of a constrained chiral multiplet to N = 1 supergravity is made locally supersymmetric using the rules of tensor calculus. The chiral multiplet is constrained with a Lagrange multiplier multiplet that could be either a chiral multiplet or a linear multiplet. We obtain the fully supersymmetric Lagrangians in both cases. It is then shown that the system consisting of the supergravity multiplet, the chiral multiplet and the Lagrange multiplier multiplet can break supersymmetry spontaneously leading to a model of a graviton, massive gravitino and two scalar fields representing mimetic dark matter. The combination of the chiral multiplet and Lagrange multiplier multiplet can act as the hidden sector breaking local N = 1 supersymmetry. [ABSTRACT FROM AUTHOR]

We assume that the points in volumes smaller than an elementary volume (which may have a Planck size) are indistinguishable in any physical experiment. This naturally leads to a picture of a discrete space with a finite number of degrees of freedom per elementary volume. In such discrete spaces, each elementary cell is completely characterized by displacement operators connecting a cell to the neighboring cells and by the spin connection. We define the torsion and curvature of the discrete spaces and show that in the limiting case of vanishing elementary volume the standard results for the continuous curved differentiable manifolds are completely reproduced. [ABSTRACT FROM AUTHOR]

Quantization of the noncommutative geometric spectral action has so far been performed on the final component form of the action where all traces over the Dirac matrices and symmetry algebra are carried out. In this work, in order to preserve the noncommutative geometric structure of the formalism, we derive the quantization rules for propagators and vertices in matrix form. We show that the results in the case of a product of a four-dimensional Euclidean manifold by a finite space, could be cast in the form of that of a Yang–Mills theory. We illustrate the procedure for the toy electroweak model. [ABSTRACT FROM AUTHOR]

We compute the information theoretic von Neumann entropy of the state associated to the fermionic second quantization of a spectral triple. We show that this entropy is given by the spectral action of the spectral triple for a specific universal function. The main result of our paper is the surprising relation between this function and the Riemann zeta function. It manifests itself in particular by the values of the coefficients c (d) by which it multiplies the d dimensional terms in the heat expansion of the spectral triple. We find that c (d) is the product of the Riemann xi function evaluated at - d by an elementary expression. In particular c (4) is a rational multiple of ζ (5) and c (2) a rational multiple of ζ (3) . The functional equation gives a duality between the coefficients in positive dimension, which govern the high energy expansion, and the coefficients in negative dimension, exchanging even dimension with odd dimension. [ABSTRACT FROM AUTHOR]

This is a tribute to Abdus Salam's memory whose insight and creative thinking set for me a role model to follow. In this contribution I show that the simple requirement of volume quantization in spacetime (with Euclidean signature) uniquely determines the geometry to be that of a noncommutative space whose finite part is based on an algebra that leads to Pati-Salam grand unified models. The Standard Model corresponds to a special case where a mathematical constraint (order one condition) is satisfied. This provides evidence that Salam was a visionary who was generations ahead of his time. [ABSTRACT FROM AUTHOR]

In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold decomposes into disconnected spheres, which will represent quanta of geometry. We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected spin manifolds with large quantized volume are then obtained as solutions. The two algebras M2(H) and M4(C) are obtained, which are the exact constituents of the standard model. Using the two maps from M4 to S4 the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. We give several physical applications of this scheme such as quantization of the cosmological constant, mimetic dark matter, and area quantization of black holes. [ABSTRACT FROM AUTHOR]

Abstract: We extend inner fluctuations to spectral triples that do not fulfill the first-order condition. This involves the addition of a quadratic term to the usual linear terms. We find a semi-group of inner fluctuations, which only depends on the involutive algebra and which extends the unitary group of . This has a key application in noncommutative spectral models beyond the Standard Model, of which we consider here a toy model. [Copyright &y& Elsevier]

Abstract: Background/aims: Reduced activity of the antioxidant glutathione peroxidase-1 (GPx1) correlates with increased risk of cardiovascular events in patients with coronary artery disease. However, it remains unclear whether this imbalance in antioxidant capacity directly contributes to activation of vascular cells. In response to oxidative stress, smooth muscle cells (SMCs) secrete the pro-inflammatory immunomodulator cyclophilin A (CyPA). We hypothesized that reduction in vascular cell GPx1 activity causes secretion of CyPA and paracrine-mediated activation of NF-κB and proliferation of SMCs. Methods/results: Using a murine model of GPx1 deficiency (GPx1+/−), we found elevated levels of hydrogen peroxide levels and increased secretion of CyPA in both arterial segments and cultured SMCs as compared to wild type (WT). Conditioned media from GPx1+/− SMCs caused increased NF-κB activation of quiescent WT SMCs, and this was inhibited by the antioxidant N-acetyl-l-cysteine or by cyclosporine A (CsA). In co-culture experiments, SMCs derived from GPx1+/− aorta caused increased proliferation of WT SMCs, which was also inhibited by CsA. Conclusions: Reduction in vascular cell GPx1 activity and the associated increase in oxidative stress cause CyPA-mediated paracrine activation of SMCs. These findings identify a novel mechanism by which an imbalance in antioxidant capacity may contribute to vascular disease. [Copyright &y& Elsevier]

Abstract: We study small perturbations around the supersymmetric CVMN monopole solution of the gauged supergravity in . We find that the perturbation spectrum contains an infinite tower of Coulomb-type bound states both in the bosonic and fermionic parts of the supergravity multiplet. Due to supersymmetry, the eigenvalues are the same for the two bosonic parity sectors, as well as for the fermionic sector. We also find that the fermion scattering on the monopole is accompanied by isospin flip. This is analogous to the Rubakov–Callan effect of monopole catalysis of proton decay and suggests that there could be a similar effect of catalysis for decay of fermionic systems in supergravity. [Copyright &y& Elsevier]

Abstract: We study spectral action for Riemannian manifolds with boundary, and then generalize this to noncommutative spaces which are products of a Riemannian manifold times a finite space. We determine the boundary conditions consistent with the hermiticity of the Dirac operator. We then define spectral triples of noncommutative spaces with boundary. In particular we evaluate the spectral action corresponding to the noncommutative space of the standard model and show that the Einstein–Hilbert action gets modified by the addition of the extrinsic curvature terms with the right sign and coefficient necessary for consistency of the Hamiltonian. We also include effects due to the addition of a dilaton field. [ABSTRACT FROM AUTHOR]

Noncommutative geometry has been slowly emerging as a new paradigm of geometry which starts from quantum mechanics. One of its key features is that the new geometry is spectral in agreement with the physical way of measuring distances. In this paper we present a detailed introduction with an overview on the study of the quantum nature of space-time using the tools of noncommutative geometry. In particular we examine the suitability of using the spectral action as an action functional for the theory. To demonstrate how the spectral action encodes the dynamics of gravity we examine the accuracy of the approximation of the spectral action by its asymptotic expansion in the case of the round sphere S3. We find that the two terms corresponding to the cosmological constant and the scalar curvature term already give the full result with remarkable accuracy. This is then applied to the physically relevant case of S3 × S1, where we show that the spectral action in this case is also given, for any test function, by the sum of two terms up to an astronomically small correction, and in particular all higher order terms a2 n vanish. This result is confirmed by evaluating the spectral action using the heat kernel expansion where we check that the higher order terms a4 and a6 both vanish due to remarkable cancelations. We also show that the Higgs potential appears as an exact perturbation when the test function used is a smooth cutoff function. [ABSTRACT FROM AUTHOR]

Abstract: The Standard Model is based on the gauge invariance principle with gauge group and suitable representations for fermions and bosons, which are begging for a conceptual understanding. We propose a purely gravitational explanation: space-time has a fine structure given as a product of a four-dimensional continuum by a finite noncommutative geometry . The raison d’être for is to correct the -theoretic dimension from four to ten (modulo eight). We classify the irreducible finite noncommutative geometries of -theoretic dimension six and show that the dimension (per generation) is a square of an integer . Under an additional hypothesis of quaternion linearity, the geometry which reproduces the Standard Model is singled out (and one gets ) with the correct quantum numbers for all fields. The spectral action applied to the product delivers the full Standard Model, with neutrino mixing, coupled to gravity, and makes predictions (the number of generations is still an input). [Copyright &y& Elsevier]

Abstract: We prove in the general framework of noncommutative geometry that the inner fluctuations of the spectral action can be computed as residues and give exactly the counterterms for the Feynman graphs with fermionic internal lines. We show that for geometries of dimension less than or equal to four the obtained terms add up to a sum of a Yang–Mills action with a Chern–Simons action. [Copyright &y& Elsevier]

The arbitrary mass scale in the spectral action for the Dirac operator is made dynamical by introducing a dilaton field. We evaluate all the low-energy terms in the spectral action and determine the dilaton couplings. These results are applied to the spectral action of the noncommutative space defined by the standard model. We show that the effective action for all matter couplings is scale invariant, except for the dilaton kinetic term and Einstein-Hilbert term. The resulting action is almost identical to the one proposed for making the standard model scale invariant as well as the model for extended inflation and has the same low-energy limit as the Randall-Sundrum model. Remarkably, all desirable features with correct signs for the relevant terms are obtained uniquely and without any fine tuning. [ABSTRACT FROM AUTHOR]

The idea of applying the gauge principle to formulate the general theory of relativity started with Utiyama in 1956. In this article, various applications of the gauge principle applied to different aspects of the gravitational interactions, are reviewed. [ABSTRACT FROM AUTHOR]

Reactive oxygen species (ROS) derived from vascular NADPH oxidase are important in normal and pathological regulation of vessel growth and function. Cell-specific differences in expression and function of the catalytic subunit of NADPH oxidase may contribute to differences in vascular cell response to NADPH oxidase activation. We examined the functional expression of gp91[sup phox] on NADPH oxidase activity in vascular smooth muscle cells (SMC) and fibroblasts (FB). As measured by dihydroethidium fluorescence in situ, superoxide (O[sup -, sub 2]·) levels were greater in adventitial cells compared with medial SMC in wild-type aorta. In contrast, there was no difference in O[sup -, sub 2]· levels between adventitial cells and medial SMC in aorta from gp91[sup phox]-deficient (gp91[sup phox] KO) mice. Adventitial-derived FB and medial SMC were isolated from the aorta of wild-type and gp91[sup phox] KO mice and grown in culture. Consistent with the observations in situ, basal and stimulated ROS levels were reduced in FB isolated from aorta of gp91[sup phox] KO compared with FB from wild-type aorta, whereas ROS levels were similar in SMC derived from gp91[sup phox] KO and wild-type aorta. There were no differences in expression of superoxide dismutase between gp91[sup phox] KO and wild-type FB to account for these observations. Because gp91[sup phox] is associated with membranes, we examined NADPH-stimulated O[sup -, sub 2]· production in membrane-enriched fractions of cell lysate. As measured by chemiluminescence, NADPH oxidase activity was markedly greater in wild-type FB compared with gp91[sup phox] KO FB but did not differ among the SMCs. Confirming functional expression of gp91[sup phox] in FB, antisense to gp91[sup phox] decreased ROS levels in wild-type FB. Finally, deficiency of gp91[sup phox] did not alter expression of the gp91[sup phox] homolog NOX4 in isolated FB. We conclude that the neutrophil subunit gp91[sup phox] contributes to NADPH oxidase function in vascular FB, but not SMC. [ABSTRACT FROM AUTHOR]

An action for a massless graviton interacting with a massive tensor field is proposed. The model is based on coupling the metric tensor to an SP(4) gauge theory spontaneously broken to SL(2,C). The symmetry breaking is achieved by using a Higgs multiplet containing a scalar field and a vector field related by a constraint. We show that in the non-unitary gauge and for the Fierz–Pauli form of the mass term, the six degrees of freedom of the massive tensor are identified with two tensor helicities, two vector helicities of the Goldstone vector, and two scalars present in the Goldstone multiplet. The propagators of this system are well behaved, in contrast to the system consisting of two tensors. [Copyright &y& Elsevier]

Abstract: We present solutions describing spatially closed, open, or flat cosmologies in the massive gravity theory within the recently proposed tetrad formulation. We find that the effect of the graviton mass is equivalent to introducing to the Einstein equations a matter source that can consist of several different matter types – a cosmological term, quintessence, gas of cosmic strings, and non-relativistic cold matter. [Copyright &y& Elsevier]

The recently proposed theory of "Asymptotically Free Mimetic Gravity" is extended to the general non‐homogeneous, spatially non‐flat case. We present a modified theory of gravity which is free of higher derivatives of the metric. In this theory asymptotic freedom of gravity implies the existence of a minimal black hole with vanishing Hawking temperature. Introducing a spatial curvature dependent potential, we moreover obtain non‐singular, bouncing modifications of spatially non‐flat Friedmann and Bianchi universes. The recently proposed theory of "Asymptotically Free Mimetic Gravity" is extended to the general non‐homogeneous, spatially non‐at case. The authors present a modified theory of gravity which is free of higher derivatives of the metric. In this theory asymptotic freedom of gravity implies the existence of a minimal black hole with vanishing Hawking temperature. Introducing a spatial curvature dependent potential, non‐singular, bouncing modifications of spatially non‐at Friedmann and Bianchi universes are obtained. [ABSTRACT FROM AUTHOR]

We show that the scalar field of mimetic gravity could be used to construct diffeomorphism invariant models that reduce to Hořava gravity in the synchronous gauge. The gradient of the mimetic field provides a timelike unit vector field that allows to define a projection operator of four-dimensional tensors to three-dimensional spatial tensors. Conversely, it also enables us to write quantities invariant under space diffeomorphisms in fully covariant form without the need to introduce new propagating degrees of freedom. [ABSTRACT FROM AUTHOR]

We show that in asymptotically free mimetic gravity with limiting curvature the black hole singularity can be resolved and replaced by a static patch of de Sitter space. As a result of Hawking evaporation of these non-singular black holes, there remain stable remnants with vanishing Hawking temperature. [ABSTRACT FROM AUTHOR]

The idea of "asymptotically free" gravity is implemented using a constrained mimetic scalar field. The effective gravitational constant is assumed to vanish at some limiting curvature. As a result singularities in spatially flat Friedmann and Kasner universes are avoided. Instead, the solutions in both cases approach a de Sitter metric with limiting curvature. We show that quantum metric fluctuations vanish when this limiting curvature is approached. [ABSTRACT FROM AUTHOR]

The mass of the graviton can be generated using a Brout-Englert-Higgs mechanism with four scalar fields. We show that when one of these fields is costrained as in mimetic gravity, the massive gravity obtained is ghost free and consistent. The mass term is not of the Fierz-Pauli type. There are only five degrees of freedom and the sixth degree of freedom associated with the Boulware-Deser ghost is constrained and replaced by mimetic matter to all orders. The van Dam-Veltman-Zakharov discontinuity is also absent. [ABSTRACT FROM AUTHOR]

We present a theory of ghost-free massive gravity where the mass of the graviton is generated through the Brout-Englert-Higgs (BEH) mechanism and one of the four scalar fields used is that of mimetic gravity. The mass term is not of the Fierz-Pauli type and the constraint eliminates the Boulware-Deser ghost which is absent to all orders. We perform a detailed analysis using the methods of cosmological perturbation theory and consider quantum fluctuations of the degrees of freedom of massive graviton and mimetic matter. It is shown that for three of the degrees of freedom of the graviton of mass m the nonlinear corrections become comparable to the linear terms already at a length scale of order m−12. Thus, at smaller scales they become strongly coupled and the graviton remains with two transverse degrees of freedom which get strongly coupled only at Planck scale. The mimetic field behaves as cold particles of half of the graviton mass and could well explain the source of dark matter in our universe. In the weakly coupled domain mimetic matter is completely decoupled from the massive graviton. [ABSTRACT FROM AUTHOR]

We describe a novel approach to dimensional reduction in classical field theory. Inspired by ideas from noncommutative geometry, we introduce extended algebras of differential forms over space-time, generalized exterior derivatives, and generalized connections associated with the 'geometry' of space-times with discrete extra dimensions. We apply our formalism to theories of gauge- and gravitational fields and find natural geometrical origins for an axion- and a dilaton field, as well as a Higgs field. [ABSTRACT FROM AUTHOR]

Objective: Vascular injury causes neointimal hypertrophy, which is characterized by redox-mediated matrix degradation and smooth muscle cell (SMC) migration and proliferation. We hypothesized that, as compared to the adjacent medial SMCs, neointimal SMCs produce increased superoxide via NADPH oxidase, which induces redox-sensitive intracellular signaling to activate matrix metalloproteinase-9 (MMP-9). Methods and Results: Two weeks after balloon injury, rat aorta developed a prominent neointima, containing increased expression of NADPH oxidase and reactive oxygen species (ROS) as compared to the medial layer. Next, SMCs were isolated from either the neointima or the media and studied in culture. Neointimal-derived SMCs exhibited increased Nox1 expression and ROS levels as compared to medial SMCs. Neointimal SMCs had higher cell growth rates than medial SMCs. ROS-dependent ERK1/2 phosphorylation was greater in neointimal SMCs. MMP-9 activity, as detected by gel zymography, was greater in neointimal SMCs under resting and stimulated conditions and was prevented by expression of an antisense to Nox1 or treatment with an ERK1/2 inhibitor. Conclusions: Following vascular injury, the increased expression of Nox1 in SMCs within the neointima initiates redox-dependent phosphorylation of ERK1/2 and subsequent MMP-9 activation. Copyright © 2012 S. Karger AG, Basel [ABSTRACT FROM AUTHOR]