International Journal of Geometric Methods in Modern Physics

An octonionic ternary gauge field theory is explicitly constructed based on a ternary-bracket defined earlier by Yamazaki. The ternary infinitesimal gauge transformations do obey the key closure relations [δ₁, δ₂] = δ₃. An invariant action for the octonionic-valued gauge fields is displayed after solving the previous problems in formulating a nonassociative octonionic ternary gauge field theory. These octonionic ternary gauge field theories constructed here deserve further investigation. In particular, to study their relation to Yang–Mills theories based on the G₂ group which is the automorphism group of the octonions and their relevance to noncommutative and nonassociative geometry.

The main goal of this paper is to study singularities of lightlike surfaces and focal surfaces of spacelike curves in Hyperbolic space times sphere. To do that, we construct a de Sitter height function and a Lightcone height function, and then show the relation between singularities of the lightlike surfaces (respectively, the focal surfaces) and that of the de Sitter height functions (respectively, the Lightcone height functions). In addition, some geometry properties of the spacelike curves are studied from geometric point of view.

Ebadian et al. [Stability of bi-θ-derivations on JB*-triples, Int. J. Geom. Meth. Mod. Phys.9(7) (2012) 1250051, 12 pp.] proved the Hyers–Ulam stability of bi-θ-derivations on JB*-triples. Under the conditions of the definition of bi-θ-derivation given in the above paper, the bi-θ-derivation must be zero. So the results given in the above paper must be trivial. Under the conditions in the main theorem, we can show that the related mapping must be zero. In this paper, we correct the statements of the results and prove the corrected theorems and corollaries.

In this paper, we put forward a systematic and unifying approach to construct gauge invariant composite fields out of connections. It relies on the existence in the theory of a group-valued field with a prescribed gauge transformation. As an illustration, we detail some examples. Two of them are based on known results: the first one provides a reinterpretation of the symmetry breaking mechanism of the electroweak part of the Standard Model of particle physics; the second one is an application to Einstein's theory of gravity described as a gauge theory in terms of Cartan connections. The last example depicts a new situation: starting with a gauge field theory on Atiyah Lie algebroids, the gauge invariant composite fields describe massive vector fields. Some mathematical and physical discussions illustrate and highlight the relevance and the generality of this approach.

It is shown that for a mechanical system with a finite number of degrees of freedom, subject to nonholonomic constraints, there exists an infinite number of Hamiltonians and symplectic structures such that the equations of motion can be written as the Hamilton equations, with the original constraints incorporated in the Hamiltonian structure.

The quantization of Yang–Mills theories relies on the gauge-fixing procedure. However, in the non-Abelian case this procedure leads to the well-known Gribov ambiguity. In order to solve the ambiguity a modification of the functional integral formula must be introduced. As a consequence of this, the Green functions get deep modifications in the infrared. We consider, in particular, the SU(N) case and show that in the pure gauge case the ghost propagator is enhanced, while the gluon propagator is suppressed in this limit, therefore the study of the Gribov ambiguity may shed some light on the mass gap problem and on color confinement. We discuss some recent developments on the subject in the case of a curved background. We argue that the concurrent presence of a spacetime curvature and the Gribov ambiguity may introduce further modifications to the Green functions in the infrared.

In this paper, we consider an axial torsion to build metric-compatible connections in conformal gravity, with gauge potentials; the geometric background is filled with Dirac spinors: scalar fields with suitable potentials are added eventually. The system of field equations is worked out to have torsional effects converted into spinorial self-interactions: the massless spinors display self-interactions of a specific form that gives them the features they have in the non-conformal theory but with the additional character of renormalizability, and the mechanisms of generation of mass and cosmological constants become dynamical. As a final step we will address the cosmological constant problem and the coincidence issue.

In this paper, we define and investigate a notion of parallel transport on finite projective modules over finite matrix algebras. Given a derivation-based differential calculus on the algebra and a connection on the module, we construct for every derivation X a module parallel transport, which is a lift to the module of the one-parameter group of algebra automorphisms generated by X. This parallel transport morphism is determined uniquely by an ordinary differential equation depending on the covariant derivative along X. Based on these parallel transport morphisms, we define a basic set of gauge invariant observables, i.e. functions from the space of connections to the complex numbers. For modules equipped with a Hermitian structure, we prove that this set of observables is separating on the space of gauge equivalence classes of Hermitian connections. This solves the gauge copy problem for fuzzy gauge theories.

We study the twistor spaces of oriented Riemannian 4-manifolds as a source of almost Hermitian 6-manifolds of constant or strictly positive holomorphic, Hermitian and orthogonal bisectional curvatures. In particular, we obtain explicit formulas for these curvatures in the case when the base manifold is Einstein and self-dual, and observe that the "squashed" metric on ℂℙ³ is a non-Kähler Hermitian–Einstein metric of positive holomorphic bisectional curvature. This shows that a recent result of Kalafat and Koca [M. Kalafat and C. Koca, Einstein–Hermitian 4-manifolds of positive bisectional curvature, preprint (2012), arXiv: 1206.3941v1 [math.DG]] in dimension four cannot be extended to higher dimensions. We prove that the Hermitian bisectional curvature of a non-Kähler Hermitian manifold is never a nonzero constant which gives a partial negative answer to a question of Balas and Gauduchon [A. Balas and P. Gauduchon, Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z.190 (1985) 39–43]. Finally, motivated by an integrability result of Vezzoni [L. Vezzoni, On the Hermitian curvature of symplectic manifolds, Adv. Geom.7 (2007) 207–214] for almost Kähler manifolds, we study the problem when the holomorphic and the Hermitian bisectional curvatures of an almost Hermitian manifold coincide. We extend the result of Vezzoni to a more general class of almost Hermitian manifolds and describe the twistor spaces having this curvature property.

Mechanical systems, called by use, mechanical (ρ, η)-systems, Lagrange mechanical (ρ, η)-systems or Finsler mechanical (ρ, η)-systems, are presented. The canonical (ρ, η)-semi(spray) associated to a mechanical (ρ, η)-system is obtained. New and important results are obtained in the particular case of Lie algebroids. The Lagrange mechanical (ρ, η)-systems are the spaces necessary to develop a new Lagrangian formalism. We obtain the (ρ, η)-semispray associated to a regular Lagrangian L and external force F_e and we derive the equations of Euler–Lagrange type. So, a new solution for the Weinstein's problem in the general framework of generalized Lie algebroids is presented.

The world lines of null particles admit arbitrary parametrizations. In the presence of a family of observers one may introduce along a null world line an extension of the so-called Cattaneo's relative standard time parameter (valid for massive particles) which plays a special role. Another possibility is to use the coordinate time itself as a parameter. The relation between relative standard time and coordinate time allows for the introduction of an observer-dependent optical path and associated refraction index. Both these quantities are studied here working out explicit examples concerning familiar null orbits and observers in black hole spacetimes.

The main aim of this paper is to investigate the geometric structures admitting by the Gödel spacetime which produces a new class of semi-Riemannian manifolds. We also consider some extension of Gödel metric.