International Journal of Geometric Methods in Modern Physics

We give an elegant formulation of the structure equations (of Cartan) and the Bianchi identities in terms of exterior calculus without reference to a particular basis and without the exterior covariant derivative. This approach allows both structure equations and the Bianchi identities to be expressed in terms of forms of arbitrary degree. We demonstrate the relationship with both the conventional vector version of the Bianchi identities and to the exterior covariant derivative approach. Contact manifolds, codimension one foliations and the Cartan form of classical mechanics are studied as examples of its flexibility and utility.

We consider the supercircle $S^{1|2}$ equipped with the standard contact structure. The conformal Lie superalgebra ${𝒦}(2)$ acts on $S^{1|2}$ as the Lie superalgebra of contact vector fields; it contains the M$\ddot{\text{o}}$bius superalgebra $𝔬𝔰𝔭(2|2)$. We study the space of linear differential operators on weighted densities as a module over $𝔬𝔰𝔭(2|2)$. We introduce the canonical isomorphism between this space and the corresponding space of symbols. This result allows us to give, in contrast to the classical setting, a classification of the ${𝒦}(2)$-modules ${𝔇}_{\lambda ,\mu }^k$ of linear differential operators of order $k$ acting on the superspaces of weighted densities. This work is the simplest superization of a result by Gargoubi and Ovsienko [Modules of differential operators on the real line, Funct. Anal. Appl. 35(1) (2001) 13–18.]

The object of the present paper is to study a spacetime admitting conharmonic curvature tensor and some geometric properties related to this spacetime. It is shown that in a conharmonically flat spacetime with cyclic parallel Ricci tensor, the energy–momentum tensor is cyclic parallel and conversely. Finally, we prove that for a radiative perfect fluid spacetime if the energy–momentum tensor satisfying the Einstein’s equations without cosmological constant is generalized recurrent, then the fluid has vanishing vorticity and the integral curves of the vector field $U$ are geodesics.

The two-dimensional Dirac equation with spin and pseudo-spin symmetries is investigated in the presence of the maximally superintegrable potentials. The integrals of motion and the quadratic algebras of the superintegrable quantum $E{3}^{\prime }$, anisotropic oscillator and the Holt potentials are studied. The corresponding Casimir operators and the structure functions of the mentioned superintegrable systems are found. Also, we obtain the relativistic energy spectra of the corresponding superintegrable systems. Finally, the relativistic energy eigenvalues of the generalized Yang–Coulomb monopole (YCM) superintegrable system (a $SU(2)$ non-Abelian monopole) are calculated by the energy spectrum of the eight-dimensional oscillator which is dual to the former system by Hurwitz transformation.

In this work, according to the electromagnetic field tensor in the framework of generalized uncertainty principle (GUP), we obtain the Lorentz force and Faraday’s law of induction in the presence of a minimal length. Also, the ponderomotive force and ponderomotive pressure in the presence of a measurable minimal length are found. It is shown that in the limit $\beta \to 0$, the generalized Lorentz force and ponderomotive force become the usual forms. The upper bound on the isotropic minimal length is estimated.

In the bottom-up approach of emergent gravity, we attempt to find symplectic gauge fields emerging from Euclidean Schwarzschild instanton, which is studied as electromagnetism defined on the symplectic space $(M,\omega )$. Geometrical engineering with the emergent metric sets up the Seiberg–Witten map between commutative and non-commutative gauge fields, preparing the ground for the evaluation of topological invariants in terms of the underlying gauge theory quantities.

We describe the definition of Jacobi (generalized)–Lie bialgebras $((g,{\varphi }_0),(g^{\ast },X_0))$ in terms of structure constants of the Lie algebras $g$ and $g^{\ast }$ and components of their 1-cocycles $X_0\in g$ and ${\varphi }_0\in g^{\ast }$ in the basis of the Lie algebras. Then, using adjoint representations and automorphism Lie groups of Lie algebras, we give a method for classification of real low-dimensional Jacobi–Lie bialgebras. In this way, we obtain and classify real two- and three-dimensional Jacobi–Lie bialgebras.

In this paper, we study conformal deformations and $C$-conformal deformations of Ricci-directional and second type scalar curvatures on Finsler manifolds. Then we introduce the best equation to study the Yamabe problem on Finsler manifolds. Finally, we restrict conformal deformations of metrics to $C$-conformal deformations and derive the Yamabe functional and the Yamabe flow in Finsler geometry.

In a attempt to treat a supergravity as a tensor representation, the four-dimensional $N$-extended quaternionic superspaces are constructed from the (diffeomorphyc) graded extension of the ordinary Penrose-twistor formulation, performed in a previous work of the authors [D. J. Cirilo-Lombardo and V. N. Pervushin, Int. J. Geom. Methods Mod. Phys., doi: http://dx.doi.org/10.1142/S0219887816501139.], with $N=p+k.$ These quaternionic superspaces have $4+k(N-k)$ even-quaternionic coordinates and $4N$ odd-quaternionic coordinates, where each coordinate is a quaternion composed by four $ℂ$-fields (bosons and fermions respectively). The fields content as the dimensionality (even and odd sectors) of these superspaces are given and exemplified by selected physical cases. In this case, the number of fields of the supergravity is determined by the number of components of the tensor representation of the four-dimensional $N$-extended quaternionic superspaces. The role of tensorial central charges for any $N$ even $USp(N)=Sp(N,{ℍ}_{ℂ})\cap U(N,{ℍ}_{ℂ})$ is elucidated from this theoretical context.

In classical mechanics, a light particle bound by a strong elastic force just oscillates at high frequency in the region allowed by its initial position and velocity. In quantum mechanics, instead, the ground state of the particle becomes completely de-localized in the limit $m\to 0$. The harmonic oscillator thus ceases to be a useful microscopic physical model in the limit $m\to 0$, but its Feynman path integral has interesting singularities which make it a prototype of other systems exhibiting a “quantum runaway” from the classical configurations near the minimum of the action. The probability density of the coherent runaway modes can be obtained as the solution of a Fokker–Planck equation associated to the condition $S=S_{\mathrm{\text{min}}}$. This technique can be applied also to other systems, notably to a dimensional reduction of the Einstein–Hilbert action.

We study the bound state problem for $N$ attractive point Dirac $\delta$-interactions in two- and three-dimensional Riemannian manifolds. We give a sufficient condition for the Hamiltonian to have $N$ bound states and give an explicit criterion for it in hyperbolic manifolds ${ℍ}^2$ and ${ℍ}^3$. Furthermore, we study the same spectral problem for a relativistic extension of the model on ${ℝ}^2$ and ${ℍ}^2$.

We demonstrate that all gauge instantons in a $d=3+1$ Yang–Mills theory, with generic topological vacuum charge $K$, correspond to soliton solutions and kink scalar fields in $d=4+1$ space-time.

By means of two different approaches, we construct the (2+1)-dimensional supersymmetric integrable equations based on the super Lie algebra osp(3/2). We relax the constraint condition of homogenous space of super Lie algebra osp(3/2) in the first approach. In another one, the technique of extending the dimension of the systems is used. Furthermore for the $(2+1)$-dimensional supersymmetric integrable equations, we also derive their Bäcklund transformations.

The Dirac equation in four time and four space dimensions (or ($4+4$)-dimensions) is considered. Step by step, we show that such an equation admits Majorana and Weyl solutions. In order to obtain the Majorana or Weyl spinors, we used a method based on the construction of Clifford algebra in terms of $2\times 2$-matrices. We argue that our approach can be useful in supergravity, superstrings and qubit theory.

We consider the $𝔰𝔩(2,ℝ)$-module structure on the spaces of $3$-ary differential operators acting on the spaces of weighted densities. We classify $𝔰𝔩(2,ℝ)$-invariant $3$-ary differential operators. This result allows us to compute the first differential cohomology of the Lie algebra $𝔰𝔩(2,ℝ)$ with coefficients in the space of trilinear differential operators.

We compute the first differential cohomology of the orthosymplectic Lie superalgebra $𝔬𝔰𝔭(n|2)$ with coefficients in the superspace of weighted densities ${𝔉}_{\lambda }^n$ on the (1, 2)-dimensional real superspace. We explicitly give 1-cocycles spanning these cohomologies. This work is the simplest generalization of a result by Basdouri and Essayari [On cohomology of the orthosymplectic superalgebra, Acta Math. Hungar. 130(1–2) (2011) 155–166].