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This article surveys the noncommutative-geometric (NCG) approach to fundamental physics, in which geometry is encoded spectrally by a generalized Dirac operator and where dynamics arise from the spectral action. I review historically how the simple idea of marrying a Riemannian manifold to a two point space, progressed to lead to the uniqueness of the Standard Model and beyond. I explain how inner fluctuations of the Dirac operator reconstruct the full gauge-Higgs sector of the Standard Model on an almost-commutative space, fixing representations and hypercharges and naturally accommodating right-handed neutrinos and the see-saw mechanism. On the gravitational side, the heat-kernel expansion of the spectral action yields the cosmological constant, Einstein–Hilbert term, and higher-curvature corrections, with volumequantized variants clarifying the status of ƒ. I discuss the renormalization-group interpretation of the spectral action as a high-scale boundary condition, phenomenological implications for Higgs stability and neutrino masses. I present generalized Heisenberg equation leading to identify the NCG space at unification. I conclude by emphasizing that NCG provides a unified, testable, and geometrically principled quantum framework, linking matter, gauge fields, and gravity. 

We investigate the coupling of mimetic dark matter to the Gauss-Bonnet topological term in addition to the Einstein action. We show that such interactions can lead to a slight modification to the power law expansion of.2/3 in the dust dominated era for small coupling constant. Alternatively, and for very large coupling constant 5 such interaction leads to the generation ofmatter in the formof radiation with a slight 6 modification to the power law of .1/2. 

A<sc>bstract</sc> We investigate the coupling of mimetic dark matter to the Gauss-Bonnet topological term in addition to the Einstein-Hilbert action. We demonstrate that such interactions can naturally give rise to mimetic dark matter during the inflationary stage of the universe’s evolution. By choosing an appropriate coupling between the mimetic field and the Gauss-Bonnet term, we find that at the end of inflation, the correct amount of dust- like dark matter is produced, with its energy density expressible in terms of the Hubble parameter at the end of inflation. Furthermore, depending on the form of the coupling, the post matter-radiation equality behavior of mimetic dark matter can experience slight modifications. 

Abstract A model is presented that provides an explanation for the presence of (Dark Matter and) Dark Energy in the universe. A key idea is to express the volume form of the Lorentzian metric on space–time in terms of a positive function of a new scalar field multiplying a certain four‐form given by the wedge product of the differential of the mimetic scalar field and a certain closed three‐form. An ansatz for this three‐form related to one commonly used to determine the winding number of a map from a three‐dimensional hypersurface to a three‐sphere is discussed. An action functional depending on the space–time metric, the new scalar field, the mimetic scalar and the three‐form is proposed, and the field equations  are derived. Special solutions of these equations for a Friedmann–Lemaître universe are presented. 

A formulation of discrete gravity was recently proposed based on defining a lattice and a shift operator connecting the cells. Spinors on such a space will have rotational SO(d) invariance which is taken as the fundamental symmetry. Inspired by lattice QCD, discrete analogues of curvature and torsion were defined that go smoothly to the corresponding tensors in the continuous limit. In this paper, we show that the absence of diffeomorphism invariance could be replaced by requiring translational invariance in the tangent space by enlarging the tangent space from SO(d) to the inhomogeneous Lorentz group ISO(d) to include translations. We obtain the ISO(d) symmetry by taking instead the Lie group SO(d+ 1) and perform on it Inonu-Wigner contraction. We show that, just as for continuous spaces, the zero torsion constraint converts the translational parameter to a diffeomorphism parameter, thus explaining the effectiveness of this formulation. 

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 n1, n2, and n3, and a length scale, lμ, 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. 

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 n1, n2, and n3. 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 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 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. 

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.