International Journal of Geometric Methods in Modern Physics

A two-dimensional (2D) surface can be considered as three-dimensional (3D) shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of 2D sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [x_i, p_j] = iℏ(δ_ij - x_ix_j/r²) rather than [x_i, p_j] = iℏδ_ij that does not hold true anymore. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.

Seiberg–Witten equations on 8-manifolds with Spin(7)-holonomy are introduced in [A. H. Bilge, T. Dereli and S. Koçak, Monopole equations on 8-manifolds with Spin(7) holonomy, Commun. Math. Phys.203(1) (1999) 21–30] and it is given a local solution for these equations. In this work, we write down similar Seiberg–Witten equations on 8-dimensional manifolds with SU(4)-structure and give a global solution for these equations.

We will establish some extension theorems for functions belonging to anisotropic Besov–Morrey type spaces and defined on bounded domains in ℝⁿ which satisfying certain geometric properties.

Let M be a complex manifold and let PM ≔ C^∞([0, 1], M) be space of smooth paths over M. We prove that the induced almost complex structure on PM is weak integrable by extending the result of Indranil Biswas and Saikat Chatterjee of [Geometric structures on path spaces, Int. J. Geom. Meth. Mod. Phys.8(7) (2011) 1553–1569]. Further we prove that if M is smooth manifold with corner and N is any complex manifold then induced almost complex structure 𝔍 on Fréchet manifold C^∞(M, N) is weak integrable.

In general, topological quantum field theory (TQFT) is studied in detail from the n-dimensional cobordism (nCob) to the Hilbert vector space. However, we study the TQFT in the different way in this paper, that is, the TQFT is studied from the Hilbert vector space to nCob. To do this, the theory called the ϕ-mapping topological current theory is used. The relation between the objects and zero points of the Hilbert states in the Hilbert vector space is studied in this frame. The relation between the morphism and topological current is revealed too.

The aim of this paper is to improve Hamiltonian energy equations for the complex jet bundles using Kähler manifolds. The coordinates on the bundle structure of Kähler manifolds have been given for real and imaginary dimensions. For given bundle structures, all fundamental geometrical properties have been investigated in Hamiltonian energy equations and applications to complex bundle structures. The improved Hamiltonian energy equations have been applied to the presented example in order to test its performance. Moreover, we have presented a new concept of velocity and time dimensions for energy movement equations. Results showed that imaginary Hamiltonian values are opposed to real Hamiltonian values in 2n dimensions. Finally, this study showed a physical application and interpretation of velocity and time dimensions in Hamiltonian energy equations for given examples.

In this paper, the natural foliations in cotangent bundle T*M of Cartan space (M, K) is studied. It is shown that geometry of these foliations are closely related to the geometry of the Cartan space (M, K) itself. This approach is used to obtain new characterizations of Cartan spaces with negative constant curvature.

We discuss the mass gap in quantum Hall solitons embedded in superstring theory. In particular, we give two holographic models which are obtained from D-brane configurations in type IIB superstring compactifications. The first one deals with the monolayered system in the D3/D7 brane set up. The second model corresponds to a multilayered system which is described by intersecting D5-branes wrapping a particular set of 3-cycles. In both models, we have shown that the mass gap is related to the filling factor.

This paper shows that the Ricci flow on Finsler manifolds with Berwald metrics cannot possibly be strictly parabolic. Then, we will define a modified flow which is strictly parabolic and by using it, we will prove the existence and uniqueness for solution of Ricci flow on Finsler manifolds.

This paper is a natural continuation of our previous paper: "Teleparallel Lagrange geometry and a unified field theory, Class. Quantum Grav.27 (2010) 045005 (29 pp)". In this paper, we apply a linearization scheme on the field equations obtained in the above-mentioned paper. Three important results under the linearization assumption are accomplished. First, the vertical fundamental geometric objects of the EAP-space lose their dependence on the positional argument x. Secondly, our linearized theory in the Cartan-type case coincides with the GFT in the first-order of approximation. Finally, an approximate solution of the vertical field equations is obtained.

In this paper, the jump problem and the auxiliary Cauchy problem for Helmholtz equation have been studied. The diffraction problems for the junctions of open waveguide in the classes of distributions are investigated. Furthermore some results and theorems concerning the existence and uniqueness for the solutions of the suggested problems are given. The methods adapted for the investigation are the integral Fourier transformation.

On the basis of the theory of algebraic curves, the continuous flow related to the discrete mKdV hierarchy is straightened using the Abel–Jacobi coordinates. The meromorphic function and the Baker–Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions of the discrete mKdV hierarchy are constructed with the help of the asymptotic properties and the algebro-geometric characters of the meromorphic function and the hyperelliptic curve.

The purpose of this study is to investigate the new dark matter candidates and their production mechanisms. In this study, we concentrated on the dark matter production by analyzing the topology change. Our results revealed the variety of the dark matter productions.

A geometrical framework for the de Broglie–Bohm quantum theory is presented, in which the trajectories of an N-particle system are interpretable as the integral curves of a particular vector field defined on a 3N-dimensional manifold constructed from physical space M. It is mathematically valid even when M is curved. If M is flat, the usual theory is recovered and automatically expressed in whatever curvilinear coordinates one may wish to choose. The general construction is illustrated by the case of a free particle moving on the surface of a sphere. (A modified Bohr quantization condition for angular momentum is obtained, with a first correction proportional to the curvature.) The Zeeman effect and some bound states on the sphere are also considered.