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We recall a classical theory of torsion gravity with an asymmetric metric, sourced by a Nambu–Goto + Kalb–Ramond string [R. T. Hammond, Rep. Prog. Phys. 65, 599 (2002)]. We explain why this is a significant gravitational theory and in what sense classical general relativity is an approximation to it. We propose that a noncommutative generalisation of this theory (in the sense of Connes' noncommutative geometry and Adler's trace dynamics) is a "quantum theory of gravity." The theory is in fact a classical matrix dynamics with only two fundamental constants – the square of the Planck length and the speed of light, along with the two string tensions as parameters. The guiding symmetry principle is that the theory should be covariant under general coordinate transformations of noncommuting coordinates. The action for this noncommutative torsion gravity can be elegantly expressed as an invariant area integral and represents an atom of space–time–matter. The statistical thermodynamics of a large number of such atoms yields the laws of quantum gravity and quantum field theory, at thermodynamic equilibrium. Spontaneous localisation caused by large fluctuations away from equilibrium is responsible for the emergence of classical space–time and the field equations of classical general relativity. The resolution of the quantum measurement problem by spontaneous collapse is an inevitable consequence of this process. Quantum theory and general relativity are both seen as emergent phenomena, resulting from coarse graining of the underlying noncommutative geometry. We explain the profound role played by entanglement in this theory: entanglement describes interaction between the atoms of space–time–matter, and indeed entanglement appears to be more fundamental than quantum theory or space–time. We also comment on possible implications for black hole entropy and evaporation and for cosmology. We list the intermediate mathematical analysis that remains to be done to complete this programme. [ABSTRACT FROM AUTHOR]

We define bilinear functionals of vector fields and differential forms, the densities of which yield the metric and Einstein tensors on even-dimensional Riemannian manifolds. We generalise these concepts in non-commutative geometry and, in particular, we prove that for the conformally rescaled geometry of the noncommutative two-torus the Einstein functional vanishes. [ABSTRACT FROM AUTHOR]

This study is about the application of the noncommutativity on the DKP equation up to first-order in 𝜃 for the process of pair creation of spin-1 particles from vacuum in (1 + 1) curved space–time. The density of particles created in the vacuum can be calculated with the help of the Bogoliubov transformations. The noncommutative density of created particles is found to decrease as 1 / 𝜃 ∼ Λ NC , so that the rate of particle creation increases whenever a noncommutativity parameter is small and this corresponds to the spirit of quantum mechanics. [ABSTRACT FROM AUTHOR]

In the present contribution, the Hirota-Satsuma coupled KdV hierarchy on noncommutative phase-space is investigated using the noncommutative extension of Lax pair generating technique. In particular, the explicit representation of its associated Lax pair operators is constructed. It is shown that the obtained results for phase-space noncommutativity case reduce back to the standard commutative case when θ goes to 1/2 . [ABSTRACT FROM AUTHOR]

In this paper we have examined the influence of noncommutativity geometry on the two-center integrals for Slater-orbitals type in Hartree-Fock Formalism. Within the perturbation theory, we have evaluated analytically the correction due to noncommutativity in nuclear attraction integrals and electronic interaction integrals over Slater-Orbitals type (STO). It is shown that there is no correction due to the noncommutativity in both nuclear integrals for ns-state and interaction integrals between electrons of this quantum state. [ABSTRACT FROM AUTHOR]

We have considered the effects of space and momentum noncommutativity separately on the zitterbewegung (ZBW) phenomenon. In the space noncommutativity scenario, it has been expressed that, due to the conservation of momentum, the Fourier decomposition of the expectation value of position does not change. However, the noncommutative (NC) space corrections to the magnetic dipole moment of electron, that was traditionally perceived to come into play only in the first-order of perturbation theory, appear in the leading-order calculations with the similar structure and numerically the same order, but with an opposite sign. This result may explain why for large lumps of masses, the Zeeman effect due to the noncommutativity remains undetectable. Moreover, we have shown that the x- and y-components of the electron magnetic dipole moment, contrary to the commutative (usual) version, are nonzero and with the same structure as the z-component. In the momentum noncommutativity case, we have indicated that, due to the relevant external uniform magnetic field, the energy spectrum and also the solutions of the Dirac equation are changed in 3 + 1 dimensions. In addition, our analysis shows that in 2 + 1 dimensions, the resulted NC field makes electrons in the zero Landau level rotate not only via a cyclotron motion, but also through the ZBW motion with a frequency proportional to the field which doubles the amplitude of the rotation. In fact, this is a hallmark of the ZBW in graphene that provides a promising way to be tested experimentally. [ABSTRACT FROM AUTHOR]

In this paper, we explore the existence of wormhole solutions exhibiting spherical symmetry in a modified gravity namely f (R , T) theory by involving some aspects of noncommutative geometry. For this purpose, we consider the anisotropic matter contents along with the well-known Gaussian and Lorentizian distributions of string theory. For the sake of simplicity in analytic discussions, we take a specific form of f (R , T) function given by f (R , T) = R + λ T. For both these noncommutative distributions, we get exact solutions in terms of exponential and hypergeometric functions. By taking some suitable choice of free parameters, we investigate different interesting aspects of these wormhole solutions graphically. We also explored the stability of these wormhole models using equilibrium condition. It can be concluded that the obtained solutions are stable and physically viable satisfying the wormhole existence criteria. Lastly, we discuss the constraints for positivity of the active gravitational mass for both these distributions. [ABSTRACT FROM AUTHOR]

Abstract We introduce a model of noncommutative geometry that gives rise to the uncertainty relations recently derived from the discussion of a quantum clock. We investigate the dynamics of a free particle in this model from the point of view of doubly special relativity and discuss the geodesic motion in a Schwarzschild background. Highlights • The investigation of a quantum clock leads to generalized uncertainty relations. • These can be derived from a deformed Heisenberg algebra. • The deformation implies modifications of Lorentz transformations and kinematics of particles. [ABSTRACT FROM AUTHOR]

Abstract We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four dimensional space. [ABSTRACT FROM AUTHOR]

In this paper, we studied the string theory in its original form of Nambu-Goto. We found that the calculation of the variation of the Nambu-Goto action with respect to the metric of the word sheet leads to a nonlinear action which is not strictly equivalent to the Polyakov action. Then the equation of motion corresponding to this action is nonlinear. This was the reason why we decided to solve the nonlinear Nambu-Goto equation of motion directly. This has been done for a string interacting with a constant, antisymmetric tensor field B in the Nambu-Goto formulation, and we derive the noncommutativity among the string end points coordinates. [ABSTRACT FROM AUTHOR]

In this paper, we review some recent developments of compact quantum groups that arise as θ-deformations of compact Lie groups of rank at least two. A θ-deformation is merely a 2-cocycle deformation using an action of a torus of dimension higher than 2. Using the formula (Lemma 5.3) developed in [11], we derive the noncommutative 7-sphere in the sense of Connes and Landi [3] as the fixed-point subalgebra. [ABSTRACT FROM AUTHOR]

It is extended to twisted spectral triples the fluctuations of the metric as bounded perturbations of the Dirac operator that arises when a spectral triple is exported between Morita equivalent algebras, as well as gauge transformations which are obtained by the action of the unitary endomorphisms of the module implementing the Morita equivalence. It is firstly shown that the twisted-gauged Dirac operators, previously introduced to generate an extra scalar field in the spectral description of the standard model of elementary particles, in fact follow from Morita equivalence between twisted spectral triples. The law of transformation of the gauge potentials turns out to be twisted in a natural way. In contrast with the non-twisted case, twisted fluctuations do not necessarily preserve the self-adjointness of the Dirac operator. For a self-Morita equivalence, conditions are obtained in order to maintain self-adjointness that are solved explicitly for the minimal twist of a Riemannian manifold. [ABSTRACT FROM AUTHOR]

Beginning with a review of the existing literature on the computation of spectral distances on noncommutative spaces like Moyal plane and fuzzy sphere, adaptable to Hilbert–Schmidt operatorial formulation, we carry out a correction, revision and extension of the algorithm provided in [1] i.e. [F. G. Scholtz and B. Chakraborty, J. Phys. A, Math. Theor.46 (2013) 085204] to compute the finite Connes' distance between normal states. The revised expression, which we provide here, involves the computation of the infimum of an expression which involves the "transverse" (Δ ρ ⊥) component of the algebra element in addition to the "longitudinal" component Δ ρ of [1], identified with the difference of density matrices representing the states, whereas the expression given in [1] involves only Δ ρ and corresponds to the lower bound of the distance. This renders the revised formula less user-friendly, as the determination of the exact transverse component for which the infimum is reached remains a nontrivial task, but under rather generic conditions it turns out that the Connes' distance is proportional to the Hilbert-Schmidt norm of Δ ρ , leading to considerable simplification. In addition, we can determine an upper bound of the distance by emulating and adapting the approach of [P. Martinetti and L. Tomassini, Commun. Math. Phys.323 (2013) 107–141]. We then look for an optimal element for which the upper bound is reached. We are able to find one for the Moyal plane through the limit of a sequence obtained by finite-dimensional projections of the representative of an element belonging to a multiplier algebra, onto the subspaces of the total Hilbert space, occurring in the spectral triple and spanned by the eigen-spinors of the respective Dirac operator. This is in contrast with the fuzzy sphere, where the upper bound, which is given by the geodesic of a commutative sphere, is never reached for any finite n -representation of S U (2). Indeed, for the case of maximal noncommutativity (n = 1 / 2), the finite distance is shown to coincide exactly with the above-mentioned lower bound, with the transverse component playing no role. This, however, starts changing from n = 1 onwards and we try to improve the estimate of the finite distance and provide an almost exact result, using our revised algorithm. The contrasting features of these types of noncommutative spaces becomes quite transparent through the analysis, carried out in the eigen-spinor bases of the respective Dirac operators. [ABSTRACT FROM AUTHOR]

Abstract We review a sequence of papers in which we construct the λ ϕ 4 ⋆ 4 -model and the λ ϕ 2 , 4 , 6 ⋆ 3 -models on noncommutative Moyal space by a common method. Thereby we show that not only the Kontsevich model λ Φ 3 but also the λ Φ 4 4 -model is integrable in a certain scaling limit which corresponds to infinitely large Moyal deformation parameter. Surprisingly, this limit gives rise to Schwinger functions on commutative Euclidean space. Our explicit formulae permit us to discuss reflection positivity of these Schwinger functions. [ABSTRACT FROM AUTHOR]

We propose a way to understand the three fermion generations by the algebraic structures of noncommutative geometry, which is a promising framework to unify the standard model and general relativity. We make the tensor product extension and the quaternion extension on the framework. Each of the two extensions alone keeps the action invariant, and we consider them as the almost trivial structures of the geometry. We combine the two extensions, and show the corresponding physical effects, i.e. the emergence of three fermion generations and the mass relationships among those generations. We define the coordinate fiber space of the bundle of the manifold as the space in which the classical noncommutative geometry is expressed, then the tensor product extension explicitly shows the contribution of structures in the non-coordinate base space of the bundle to the action. The quaternion extension plays an essential role to reveal the physical effect of the structure in the non-coordinate base space. [ABSTRACT FROM AUTHOR]

We discuss exact multisoliton solutions of integrable hierarchies on noncommutative space-times in various dimensions. The solutions are represented by quasideterminants in compact forms. We study soliton scattering processes in the asymptotic region where the configurations can be real-valued. We find that the asymptotic configurations in the soliton scatterings can all be the same as commutative ones, i.e., the configuration of an N-soliton solution has N isolated localized lumps of energy, and each solitary wave-packet lump preserves its shape and velocity in the scattering process. The phase shifts are also the same as commutative ones. As new results, we present multisoliton solutions of the noncommutative anti-self-dual Yang-Mills hierarchy and discuss two-soliton scattering in detail. [ABSTRACT FROM AUTHOR]

Guided by ordinary quantum mechanics we introduce new fuzzy spheres S Λ d of dimensions d = 1 , 2 : we consider an ordinary quantum particle in D = d + 1 dimensions subject to a rotation invariant potential well V ( r ) with a very sharp minimum on a sphere of unit radius. Imposing a sufficiently low energy cutoff to ‘freeze’ the radial excitations makes only a finite-dimensional Hilbert subspace accessible and on it the coordinates noncommutative à la Snyder ; in fact, on it they generate the whole algebra of observables. The construction is equivariant not only under rotations – as Madore’s fuzzy sphere – , but under the full orthogonal group O ( D ) . Making the cutoff and the depth of the well dependent on (and diverging with) a natural number Λ , and keeping the leading terms in 1 ∕ Λ , we obtain a sequence S Λ d of fuzzy spheres converging to the sphere S d in the limit Λ → ∞ (whereby we recover ordinary quantum mechanics on S d ). These models may be useful in condensed matter problems where particles are confined on a sphere by an (at least approximately) rotation-invariant potential, beside being suggestive of analogous mechanisms in quantum field theory or quantum geometry. [ABSTRACT FROM AUTHOR]

It is well known that for any irrational rotation number α , the noncommutative torus A α must have representations π such that the generated von Neumann algebra π ( A α ) ″ is of type III. Therefore, it could be of interest to exhibit and investigate such kind of representations, together with the associated spectral triples whose twist of the Dirac operator and the corresponding derivation arises from the Tomita modular operator. In the present paper, we show that this program can be carried out, at least when α is a Liouville number satisfying a faster approximation property by rationals. In this case, we exhibit several type I I ∞ and II I λ , λ ∈ [ 0 , 1 ] , factor representations and modular spectral triples. The method developed in the present paper can be generalised to CCR algebras based on a locally compact abelian group equipped with a symplectic form. [ABSTRACT FROM AUTHOR]

Noncommutativity is an idea dating back to that early times of quantum mechanics and the string theory induced noncommutative (NC) geometry provided an effective framework to study the short distance spacetime dynamics. Also, string theory, a candidate for a consistent quantum theory of gravity, admits a variety of classical black hole solutions including black strings. In this paper, we study a NC geometry inspired rotating black string to cylindrical spacetime with a source given by a smeared distribution of mass. The resulting metric is a regular everywhere, i.e. curvature-singularity free rotating black string, that in large r limit interpolates Lemos black string. Thermodynamical properties of the black strings are also investigated and exact expressions for the temperature, the entropy and the heat capacity are obtained. Owing to the NC correction in the solution, the thermodynamic quantities have also been modified and that the NC geometry inspired black string is always thermodynamically stable. [ABSTRACT FROM AUTHOR]

We consider a noncommutative standard model with a minimal coupling scalar field and a dynamical deformation between the canonical momenta of its scale factor and scalar field, and a chameleon model with a non-minimally coupling scalar field. We indicate that there is a correspondence between these two models, more specific, actually between the noncommutative parameter and the chameleon coupling strength, and also between the matter density of the chameleon model and the noncommutative geometry. In addition, the analogy constrains the type of the matter field in the chameleon model to be nearly a cosmic string-like during the inflation. Thus, the scenario enables the evolution of the universe being described by one single scalar field. That is, the effects of the chameleon and the inflaton can be described by one single scalar field which plays the role of inflaton in the very early universe and then, acts as a chameleon field. Moreover, the proposed correspondence procedure not only sets some constraints on the noncommutative parameter and the chameleon coupling constant, but also nearly specifies functions of the scalar field and its potential. [ABSTRACT FROM AUTHOR]

We have made an initial investigation of an effective theory with Lorentz violation with Minimum Length. We investigate a scenario of anisotropy generated by a Lorentz symmetry breaking term which appears in Standard Model Extension (SME) on the even gauge sector with deformed Heisenberg algebra associated to the Minimum Length (ML). To study such possible correction of the gauge sector of SME, we analyze the dispersion relations of this model. [ABSTRACT FROM AUTHOR]

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tessellated affine manifold form the base, whereas the fibres are free associative algebras or, at a later stage, such algebras quotients over the linear relation of equivalence under cyclic shifts. The calculus of variations is developed on the infinite jet spaces over such noncommutative bundles. In the frames of such field-theoretic extension of the Kontsevich formal noncommutative symplectic (super)geometry, we prove the main properties of the Batalin–Vilkovisky Laplacian and Schouten bracket. We show as by-product that the structures which arise in the classical variational Poisson geometry of infinite-dimensional integrable systems do actually not refer to the graded commutativity assumption. [ABSTRACT FROM AUTHOR]

Noncommutative or quantum Riemannian geometry has been proposed as an effective theory for aspects of quantum gravity. Here the metric is an invertible bimodule map Ω 1 ⊗ A Ω 1 → A where A is a possibly noncommutative or ‘quantum’ spacetime coordinate algebra and ( Ω 1 , d ) is a specified bimodule of 1-forms or ‘differential calculus’ over it. In this paper we explore the proposal of a ‘quantum Koszul formula’ in Majid [ 12 ] with initial data a degree − 2 bilinear map ⊥ on the full exterior algebra Ω obeying the 4-term relations ( − 1 ) | η | ( ω η ) ⊥ ζ + ( ω ⊥ η ) ζ = ω ⊥ ( η ζ ) + ( − 1 ) | ω | + | η | ω ( η ⊥ ζ ) , ∀ ω , η , ζ ∈ Ω and a compatible degree − 1 ‘codifferential’ map δ . These provide a quantum metric, interior product and a canonical bimodule connection ∇ on all degrees. The theory is also more general than classically in that we do not assume symmetry of the metric nor that δ is obtained from the metric. We solve and interpret the ( δ , ⊥ ) data on the bicrossproduct model quantum spacetime [ r , t ] = λ r for its two standard choices of Ω . For the α -family calculus the construction includes the quantum Levi-Civita connection for a general quantum symmetric metric, while for the more standard β = 1 calculus we find the quantum Levi-Civita connection for a quantum ‘metric’ that in the classical limit is antisymmetric . This suggests to consider quantum Riemannian and symplectic geometry on a more equal footing than is currently the case. [ABSTRACT FROM AUTHOR]

We analyze noncommutative deformations of a higher dimensional anti-de Sitter-Einstein-Born-Infeld black hole. Two models based on noncommutative inspired distributions of mass and charge are discussed and their thermodynamical properties such as the equation of state are explicitly calculated. In the (3 + 1)-dimensional case the Gibbs energy function of each model is used to discuss the presence of phase transitions. [ABSTRACT FROM AUTHOR]

We investigate the geodesic motion of massless test particles in the background of a noncommutative geometry-inspired Schwarzschild black hole. The behavior of effective potential is analyzed in the equatorial plane and the possible motions of massless particles (i.e. photons) for different values of impact parameter are discussed accordingly. We have also calculated the frequency shift of photons in this space–time. Further, the mass parameter of a noncommutative inspired Schwarzschild black hole is computed in terms of the measurable redshift of photons emitted by massive particles moving along circular geodesics in equatorial plane. The strength of gravitational fields of noncommutative geometry-inspired Schwarzschild black hole and usual Schwarzschild black hole in General Relativity is also compared. [ABSTRACT FROM AUTHOR]

In this paper, we study noncommutative massive unquenched Chern–Simons matter theory using its gravity dual. We construct this novel background by applying a TsT-transformation on the known parent commutative solution. We discuss several aspects of this solution to the Type IIA supergravity equations of motion and, amongst others, check that it preserves 𝒩=1 supersymmetry. We then turn our attention to applications and investigate how dynamical flavor degrees of freedom affect numerous observables of interest. Our framework can be regarded as a key step toward the construction of holographic quantum Hall states on a noncommutative plane. [ABSTRACT FROM AUTHOR]

We study the Einstein-Hilbert gravitation with the deformed Heisenberg algebra leading to the minimum length, with the intention to find and estimate the corrections in this theory, clarifying whether or not it is possible to obtain, by means of the minimum length, a theory, in D=4, which is causal, unitary and provides a massive graviton. Therefore, we will calculate and analyze the dispersion relationships of the considered theory. [ABSTRACT FROM AUTHOR]

In this paper we will apply the Faddeev-Jackiw quantization methodology to the massive open strings in the D-brane background with a non-vanishing constant B-field.We shall work in the discrete version, and the reduced phase space is obtained directly by solving the mixed boundary conditions. The non-commutativity is extended to the noncommutativity of pace coordinates and momentum coordinates along the D-brane is reproduced in easy way.We feel that this method of obtaining the noncommutativity in string theory in the massive case is more elegant than previous approaches discussed in the literature. [ABSTRACT FROM AUTHOR]

We start from a set theoretical foundation of quantum mechanics and go all the way to giving a soap bubble minimal surface interpretation of quantum wave collapse. We outline first the quintessence of a fairly new quantum theory based on transfinite set theory representing a synthesis of the work of J. von Neumann's pointless continuous geometry, A. Connes' noncommutative quantum geometry, Menger-Urysohn dimensional theory, Sir R. Penrose's fractal tiling universe, 'tHooft's renormalon as well as D. Shectman quasi crystals and Einstein-Kaluza-Klein space-time theories. We conclude that the Aether is a real non-materialistic material and that is essentially what we commonly call space-time which can be accurately described as the empty five dimensional set in a Penrose quasi crystal universe. Seen in the light of the above, empty space-time can be the source of infinite clean energy and the realization of NASA's EM drive. [ABSTRACT FROM AUTHOR]

We study a first variation formula for the eigenvalues of the Laplacian evolving under the Ricci flow in a simple example of a noncommutative matrix geometry, namely a finite dimensional representation of a noncommutative torus. In order to do so, we first show that the Ricci flow in this matrix geometry is analytic. [ABSTRACT FROM AUTHOR]

We use DG categories to derive analogues of the Bianchi identities for the curvature of a connection in noncommutative differential geometry. We also revisit the Chern–Connes pairing but following the line of Chern’s original derivation. We show that a related DG category of extendable bimodule connections is a monoidal tensor category and in the metric compatible case obtain an analogue of a classical antisymmetry of the Riemann tensor. The monoidal structure implies the existence of a cup product on noncommutative sheaf cohomology. Another application shows that the curvature of a line module reduces to a 2-form on the base algebra. We illustrate the theory on the q -sphere, the permutation group S 3 and the bicrossproduct quantum spacetime [ r , t ] = λ r . [ABSTRACT FROM AUTHOR]

In this paper we consider semi-Riemannian time and space oriented manifolds of even dimension, and characterize the Lorentzian and antilorentzian signatures in terms of a time-orientation 1-form and a natural Krein product on spinor fields. It turns out that all the data available in Noncommutative Geometry (the algebra of functions, the Krein space of spinor fields, the representation of the algebra on it, the Dirac operator, charge conjugation and chirality), but nothing more, play a role in this characterization. It thus yields a possible definition extending Connes’ notion of even spectral triple to the Lorentzian setting. [ABSTRACT FROM AUTHOR]

We apply the orbifold grand unified theory (GUT) mechanism to the noncommutative Higgs model. An assignment of parity to the 'constituent fields' induces parity assignments of both the gauge and Higgs bosons, because these bosons are treated as some kind of composite fields in this formalism. [ABSTRACT FROM AUTHOR]

We propose a non-skew-symmetric generalization of the original definition of double Poisson Bracket by M. Van den Bergh. It allows one to explicitly construct a more general class of H 0 -Poisson structures on finitely generated associative algebras. We show that modified double Poisson brackets inherit certain major properties of the double Poisson brackets. [ABSTRACT FROM AUTHOR]

Building on the theory of noncommutative complex structures, the notion of a noncommutative Kähler structure is introduced. In the quantum homogeneous space case many of the fundamental results of classical Kähler geometry are shown to follow from the existence of such a structure, allowing for the definition of noncommutative Lefschetz, Hodge, Kähler–Dirac, and Laplace operators. Quantum projective space, endowed with its Heckenberger–Kolb calculus, is taken as the motivating example. The general theory is then used to show that the calculus has cohomology groups of at least classical dimension. [ABSTRACT FROM AUTHOR]

We present a new model of quantum gravity as a theory of random geometries given explicitly in terms of a multitrace matrix model. This is a generalization of the usual discretized random surfaces of two-dimensional quantum gravity which works away from two dimensions and captures a large class of spaces admitting a finite spectral triple. These multitrace matrix models sustain emergent geometry as well as growing dimensions and topology change. [ABSTRACT FROM AUTHOR]

By defining a noncommutative symplectic structure, we study thermodynamics of Schwarzschild black hole in a Snyder noncommutative phase space for the first time. Since natural cutoffs are the results of compactness of symplectic manifolds in phase space, the physics of black holes in such a space would be affected mainly by these cutoffs. In this respect, this study provides a basis for more deeper understanding of the black hole thermodynamics in a pure mathematical viewpoint. [ABSTRACT FROM AUTHOR]

We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with m vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction from the space of representations of the quiver. Three families of Poisson-commuting functions are constructed and written explicitly in suitable Darboux coordinates. The case m = 1 corresponds to the tadpole quiver and the Ruijsenaars–Schneider system and its variants, while for m > 1 we obtain new integrable systems that generalise the Ruijsenaars–Schneider system. These systems and their quantum versions also appeared recently in the context of supersymmetric gauge theory and cyclotomic DAHAs (Braverman et al. [32,34,35] and Kodera and Nakajima [36]), as well as in the context of the Macdonald theory (Chalykh and Etingof, 2013). [ABSTRACT FROM AUTHOR]

In the context of noncommutative (NCG) gauge gravity, and using a cosmic time power law formula for the scale factor, a Friedman-Robertson-Walker (FRW) like metric is obtained. Within the fermions tunneling effect approach and depending on the various intervals of the power parameter, expressions of the apparent horizon are also derived. It is shown that in some regions of the parameter space, a pure NCG trapped horizon does exist leading to new interpretation of the role played by the noncommutativity of the space-time. [ABSTRACT FROM AUTHOR]

We discuss a generalization of the Snyder model compatible with undeformed Lorentz symmetries, which we describe in terms of a large class of deformations of the Heisenberg algebra. The corresponding deformed addition of momenta, the twist and the R-matrix are calculated to first order in the deformation parameters for all models. In the particular case of the Snyder realization, an analytic formula for the twist is obtained. [ABSTRACT FROM AUTHOR]

We investigate the properties of the singular values of the quantised derivatives of essentially bounded functions on R d with d > 1 . The commutator i [ sgn ( D ) , 1 ⊗ M f ] of an essentially bounded function f on R d acting by pointwise multiplication on L 2 ( R d ) and the sign of the Dirac operator D acting on C 2 ⌊ d / 2 ⌋ ⊗ L 2 ( R d ) is called the quantised derivative of f . We prove the condition that the function x ↦ ‖ ( ∇ f ) ( x ) ‖ 2 d : = ( ( ∂ 1 f ) ( x ) 2 + … + ( ∂ d f ) ( x ) 2 ) d / 2 , x ∈ R d , being integrable is necessary and sufficient for the quantised derivative of f to belong to the weak Schatten d -class. This problem has been previously studied by Rochberg and Semmes, and is also explored in a paper of Connes, Sullivan and Telemann. Here we give new and complete proofs using the methods of double operator integrals. Furthermore, we prove a formula for the Dixmier trace of the d -th power of the absolute value of the quantised derivative. For real valued f , when x ↦ ‖ ( ∇ f ) ( x ) ‖ 2 d is integrable, there exists a constant c d > 0 such that for every continuous normalised trace φ on the weak trace class L 1 , ∞ we have φ ( | [ sgn ( D ) , 1 ⊗ M f ] | d ) = c d ∫ R d ‖ ( ∇ f ) ( x ) ‖ 2 d d x . [ABSTRACT FROM AUTHOR]

We analyze a U ( 2 ) -matrix model derived from a finite spectral triple. By applying the BV formalism, we find a general solution to the classical master equation. To describe the BV formalism in the context of noncommutative geometry, we define two finite spectral triples: the BV spectral triple and the BV auxiliary spectral triple. These are constructed from the gauge fields, ghost fields and anti-fields that enter the BV construction. We show that their fermionic actions add up precisely to the BV action. This approach allows for a geometric description of the ghost fields and their properties in terms of the BV spectral triple. [ABSTRACT FROM AUTHOR]

Using a fully-fledged formulation of gauge field theory deformed by the spacetime noncommutativity, we study its impact on relic neutrino direct detection, as proposed recently by the PTOLEMY experiment. The noncommutative background tends to influence the propagating neutrinos by providing them with a tree-level vector-like coupling to photons, enabling thus otherwise sterile right-handed (RH) neutrinos to be thermally produced in the early universe. Such a new component in the universe's background radiation has been switched today to the almost fully active sea of non-relativistic neutrinos, exerting consequently some impact on the capture on tritium at PTOLEMY. The peculiarities of our nonperturbative approach tend to reflect in the cosmology as well, upon the appearances of the coupling temperature, above which RH neutrinos stay permanently decoupled from thermal environment. This entails the maximal scale of noncommutativity as well, being of order of 10 − 4 M P l , above which there is no impact whatsoever on the capture rates at PTOLEMY. The latter represents an exceptional upper bound on the scale of noncommutativity coming from phenomenology. [ABSTRACT FROM AUTHOR]

We develop a sheaf theory approach to toric noncommutative geometry which allows us to formalize the concept of mapping spaces between two toric noncommutative spaces. As an application, we study the 'internalized' automorphism group of a toric noncommutative space and show that its Lie algebra has an elementary description in terms of braided derivations. [ABSTRACT FROM AUTHOR]

We investigate the classical Brownian motion of a particle in a two-dimensional noncommutative (NC) space. Using the standard NC algebra embodied by the symplectic Weyl-Moyal formalism we find that noncommutativity induces a nonvanishing correlation between both coordinates at different times. The effect stands out as a signature of spatial noncommutativity and thus could offer a way to experimentally detect the phenomena. We further discuss some limiting scenarios and the trade-off between the scale imposed by the NC structure and the parameters of the Brownian motion itself. [ABSTRACT FROM AUTHOR]

We study the noncommutative corrections on the time-dependent Aharonov–Bohm effect when both the coordinate–coordinate and momentum–momentum noncommutativities are considered. On the noncommutative phase space, while the ordinary gauge symmetry can be kept by the Seiberg–Witten map, but the Lorentz symmetry is broken. Therefore nontrivial noncommutative corrections are expected. We find there are three kinds of noncommutative corrections in general: (1) ξ -dependent correction which comes from the noncommutativity among momentum operators; (2) momentum-dependent correction which is rooted in the nonlocal interactions in the noncommutative extended model; (3) momentum-independent correction which emerges become of the gauge invariant condition on the nonlocal interactions in the noncommutative model. We proposed two dimensionless quantities, which are based on the distributions of the measured phase shift with respect to the external magnetic field and to the cross section enclosed by the particle trajectory, to extract the noncommutative parameters. We find that stronger (weaker) magnetic field strength can give better bounds on the coordinate–coordinate (momentum–momentum) noncommutative parameter, and large parameter space region can be explored by the time-dependent AB effect. [ABSTRACT FROM AUTHOR]

We propose to define𝔽1-algebras as integral bi-rings with the co-ring structure being the descent data fromℤto𝔽1. The coordinate bi-ring ofSpec(ℤ)∕𝔽1is then the co-ring of integral linear recursive sequences equipped with the Hadamard product. We associate a noncommutative moduli space to this setting and show that it is defined over𝔽1and has motive. [ABSTRACT FROM PUBLISHER]

In a previous paper we established Cwikel-type estimates on noncommutative tori and used them to get analogues in this setting of the Cwikel-Lieb-Rozenblum (CLR) and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators. In this paper, we focus on "curved" NC tori, where the role of the usual Laplacian is played by Laplace-Beltrami operators associated with arbitrary Riemannian metrics. The Cwikel-type estimates of our previous paper are extended to pseudodifferential operators and powers of Laplace-Beltrami operators. There are several applications of these estimates. First, we get L p -versions of the usual formula for the trace of ΨDOs on NC tori, i.e., for combinations of ΨDOs with L p -position operators. Next, we get L p -versions of the analogues for NC tori Connes' trace theorem and Connes' integration formula. They give formulas for the NC integrals (a.k.a. Dixmier traces) of products of L p -position operators with ΨDOs or powers of the Laplace-Beltrami operators. Moreover, by combining our Cwikel-type estimates with suitable versions of the Birman-Schwinger principle we get versions of the CLR and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators associated with powers of Laplace-Beltrami operators and L p -potentials. As in the original Euclidean case the Lieb-Thirring inequalities imply a dual Sobolev inequality for orthonormal families. Finally, we discuss spectral asymptotics and semiclassical Weyl's laws for our classes of operators on curved NC tori. This supersedes a previous conjecture of [63]. [ABSTRACT FROM AUTHOR]

Discretized Kaluza-Klein theory in spacetime can be constructed based on the concepts of noncommutative geometry. In this paper, we show that it is possible to incorporate the non-Abelian gauge fields in this framework. The generalized Hilbert-Einstein action is gauge invariant only in two cases. In the first case, the gauge group must be Abelian on one sheet of spacetime and non-Abelian on the other one. In the second case, the gauge group must be the same on two sheets of spacetime. Actually, the theories of electroweak and strong interactions can fit into these two cases. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of static spherically symmetric wormhole solutions along with noncommutative geometry in the background of F( T, T) gravity. We assume a nonzero redshift function as well as two well-known models of this gravity and discuss the behavior of null/weak energy conditions graphically. We conclude that there does not exist any physically acceptable wormhole solution for the first model, but there is a chance to develop physically acceptable wormhole solution in a particular region for the second model. [ABSTRACT FROM AUTHOR]