International Journal of Geometric Methods in Modern Physics

In this paper, we present a quantization of the functions of spacetime, i.e. a map, analog to Weyl map, which reproduces the $\kappa$-Minkowski commutation relations, and it has the desirable properties of mapping square integrable functions into Hilbert–Schmidt operators, as well as real functions into symmetric operators. The map is based on Mellin transform on radial and time coordinates. The map also defines a deformed ∗ product which we discuss with examples.

We study a set of cohomology classes which emerge in the cohomological formulations of the calculus of variations as obstructions to the existence of (global) solutions of the Euler–Lagrange equations of Chern–Simons gauge theories in higher dimensions $2p+1>3$.

In this paper, we study the M-string realization of chiral $(2,0)$-super-conformal field theory in 6 dimensions and its orbifold compactification down to three-dimensional (3D). We analyze its fractionally charged BPS particle spectrum in connection with effective 3D Chern–Simons gauge theory and the supersymmetric fractional quantum Hall effect in $2+1$ dimensions. We construct the set of underlying fractionally charged BPS particles in the ground state of the compactified M string and find that it contains 144 BPS states that are generated by four basic quasi-particles (two bosonic-like and two fermionic like) and their CPT conjugate. Two representations of the gauge bosons and the gauginos as condensates of the basic quasiparticles are found and explicit realizations are also given. Other features concerning generalizations are also discussed.

We investigate the cosmological and thermodynamic aspects of Weyl tensor corrected $f(R)$ gravity. For this purpose, we assume some well-known cosmological bouncing scenarios such as symmetric bounce cosmology, oscillatory cosmology, matter bounce cosmology, little rip cosmology, superbounce cosmology and develop some cosmological parameters. For instance, the equation of state parameter ${\omega }_R$ describes the quintessence phase for symmetric bounce cosmology, vacuum phase for oscillatory, little rip and matter bounce cosmology while it gives both quintessence and vacuum phases for matter bounce cosmology. It is also observed that the squared speed of sound $C_s^2$ gives positive behavior for all models resulting in that the models assumed are stable. We evaluate generalized second law of thermodynamics which remains valid for all cosmological models except symmetric bounce cosmology. Moreover, we also investigate the thermal equilibrium condition ${\ddot{S}}_{Atot}$ and found its validity for all models except symmetric bounce cosmological model.

In this paper, the charged thin-shell wormholes have been constructed by using cut-and-paste approach in the framework of $f(R)$ theory of gravity. The stability analysis is performed in $f(R)=R+\alpha R^n$ gravity formalism, where $\alpha$ and $n$ are nonzero constants, with a linear equation of state. The stable and unstable regions have been examined for different values of the parameters involved in the model. The effect of charge and mass on the throat radius is analyzed and stability of thin shell is obtained.

Nonlocal gravity models are constructed to explain the current acceleration of the universe. These models are inspired by the infrared correction appearing in Einstein–Hilbert action. Here, we develop the Hamiltonian formalism of a nonlocal model by considering only terms to quadratic order in Riemann tensor, Ricci tensor and Ricci scalar. We show how to count degrees of freedom using Hamiltonian formalism including Ricci tensor and Ricci scalar terms. In this model, we have also worked out with a choice of a nonlocal action which has only two degrees of freedom equivalent to GR. Finally, we find the existence of additional constraints in Hamiltonian required to remove the ghosts in our full action. We also compare our results with that of obtained using Lagrangian formalism.

The main intention of this paper is to analyze integrability for the derivative formulas of the rotation minimizing frame in the Lorentz–Minkowski 3-space. As far as we know, no one has yet given a method to study their integrability in the Lorentz–Minkowski 3-space. So, we introduce the coordinate system in order to provide a tool for studying the integrability. As an application, the position vectors of some special curves having an important place in mathematical and physical research are obtained in the natural representation form. Finally, we support our work with examples.

In this paper, we present a spatially homogeneous and anisotropic Bianchi type-I cosmological model with a viscous bulk fluid in $f(R,T)$ gravity where $R$ and $T$ are the Ricci scalar and trace of the energy-momentum tensor, respectively. The field equations are solved explicitly using the hybrid law of the scale factor, which is related to the average Hubble parameter and gives a time-varying deceleration parameter (DP). We found the deceleration parameter describing two phases in the universe, the early deceleration phase $(q>0)$ and the current acceleration phase $(q<0)$. We have calculated some physical and geometric properties and their graphs, whether in terms of time or redshift. Note that for our model, the bulk viscous pressure $(\overline{p})$ is negative and the energy density $(\rho )$ is positive. The energy conditions and the $\mathrm{\text{Om}}(z)$ analysis for our spatially homogeneous and anisotropic Bianchi type-I model are also discussed.

Using Perelman’s approach for geometrical flows in terms of an entropy functional, the Higgs mechanism is studied dynamically along flows defined in the space of parameters and in fields space. The model corresponds to two-dimensional gravity that incorporates torsion as the gradient of a Higgs field, and with the reflection symmetry to be spontaneously broken. The results show a discrete mass spectrum and the existence of a mass gap between the Unbroken Exact Symmetry and the Spontaneously Broken Symmetry scenarios. In the latter scenario, the geometries at the degenerate vacua correspond to conformally flat manifolds without torsion; twisted two-dimensional geometries are obtained by building perturbation theory around a ground state; the tunneling quantum probability between vacua is determined along the flows.

In this paper, we investigate a four-component Toda lattice (TL), which may be used to model the wave propagation in lattices just like the famous TL. By means of the Lax pair and gauge transformation, we construct the $N$-fold Darboux transformation (DT), which enables us to obtain multi-soliton or multi-solitary wave solution without complex iterative process. Through the obtained DT, $N$-fold explicit exact solutions of the system and their figures with proper parameters are presented from which we find the $1$-fold solution shows two-solitary wave structure, the amplitude and shape of the wave change with time. Finally, we derive an infinite number of conservation laws formulaically to illustrate the integrability of the system.

In this paper, we propose a scheme for implementing a Toffoli gate in a system with an atom that has six levels in a lambda configuration, interacting with a high-Q cavity containing four modes. Here, we reduce a six-level system into an effective three-level behavior by applying the adiabatic elimination method. Next, we calculate the probabilities of the states of the interest as well as the fidelity. We also study the effects of photonic and atomic decay rates on the evolution of the system which is reasonably less sensitive to decoherence.

Rényi Holographic Dark Energy (RHDE) Models have been studied in different dimensions considering a flat FRW metric and subsequently exploring the nature of evolution of different cosmological parameters. It is seen that the number of dimensions affect the evolution of the cosmological parameters and the transition of the universe from decelerating to accelerating phase. The statefinder diagnostics have been performed and it is seen that the model corresponds to the $\mathrm{\Lambda }\mathrm{\text{CDM}}$ model for the present and future universe.

In this paper, we obtain lower bounds for the normalized scalar curvature on statistical submersion with the normalized $\delta$-vertical Casorati curvatures. Also, we discuss the conditions for which the equality cases hold. Beside this, we determine the statistical solitons on statistical submersion from statistical manifolds and illustrate an example of statistical submersions from statistical manifolds.

After summarizing the basic concepts for the exterior algebra, we first discuss the gauge structure of the bundle over base manifold for deciding the form of the gravitational sector of the total Lagrangian in any dimensions. Then we couple minimally a Dirac spinor field to our gravitational Lagrangian 2-form which is quadratic in the nonmetricity and both linear and quadratic in the curvature in two dimensions. Subsequently, we obtain field equations by varying the total Lagrangian with respect to the independent variables. Finally, we find some classes of solutions of the vacuum theory and then a solution of the Dirac equation in a specific background and analyze them.

Bianchi type-III, V, and VI₀ perfect fluid anisotropic cosmological models have been discussed in the theory of gravity suggested by [T. Harko, F. S. N. Lobo, S. Nojiri and S. D. Odintsov, f(R, T) gravity, Phys. Rev. D 84 (2011) 024020] with two well-known functional forms, namely, $f(R,T)=R+kln(\lambda R)+\gamma T$ and $f(R,T)=R+\mu R^2-\frac{{\nu }^4}{R}+\gamma T$, here $k$, $\lambda$, $\gamma$, $\mu$, and $\nu$ are constants. To acquire the solutions of the field equations of the three models, the average scale factor and the proportionality of expansion scalar with shear scalar are used. The behavior of all the physical and geometrical properties of the three models is explained with their plots. For all the three models, the NEC is positive, which implies that the EoS parameter is in the quintessence region. For the Bianchi type-III model, the SEC is violated, which recommends that the present universe is accelerating and expanding. Also, we have observed that, if $k=0$, $\mu =0$, and $\nu =0$, then the two functional forms lead to the linear form and for $\nu =0$, the second form leads to the quadratic form, these two are interesting points in this paper. It can be observed that the stability analysis of the obtained models background solutions is discussed through perturbation techniques and observed that the background solutions are stable. The values of all the parameters in the three models match with the recent cosmological observations.

One of the foremost goals of theoretical modern physics is to obtain a reliable theory of quantum gravity. In so doing, new fundamental scales are being proposed. For example, in the Snyder–de Sitter (SdS) model, two scales manifest which relate measurement limits of the position and momentum. In this work, we study the thermodynamic functions of the Schwarzschild black hole in the SdS model within the presence of position independent and dependent gravity’s rainbow. Heuristically, we prove the presence of a nonzero lower bound value of the horizon, mass and temperature of the black hole. Then, we present the stability and remnant conditions of the black hole according to the entropy and heat capacity functions.