International Journal of Geometric Methods in Modern Physics

In this paper we give a geometric parametrization to the Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix and the Jarlskog invariant, which is based on two flag manifolds SU(3)/U(1)².

To treat a fourth generation of quarks on CP violation we generalize the parametrization to one based on two flag manifolds SU(4)/U(1)³.

This paper studies spinor two-point functions for spin-1/2 and spin-3/2 fields in maximally symmetric spaces such as de Sitter (dS) space–time, by using intrinsic geometric objects. The Feynman, positive- and negative-frequency Green functions are then obtained for these cases, from which we eventually display the supercommutator and the Peierls bracket under such a setting in two-component-spinor language.

A novel supergeometric generalization of common n-sphere, called ν-sphere, is introduced. It is shown that a canonical volume form exists on a ν-sphere. Then a concept of degree is developed for the maps between them. Finally, a semigroup action on ν-spheres is discussed.

In this paper, we show that para-Kahler–Norden (or paraholomorphic) metrics do not exist on warped manifolds.

Here we define the concept of L-regularity for coherent sheaves on the Grassmannian G(1,4) as a generalization of Castelnuovo–Mumford regularity on Pⁿ. In this setting we prove analogs of some classical properties. We use our notion of L-regularity in order to prove a splitting criterion for rank 2 vector bundles with only a finite number of vanishing conditions. In the second part, we give the classification of rank 2 and rank 3 vector bundles without "inner" cohomology (i.e. for any i = 2,3,4) on G(1,4) by studying the associated monads.

A description of how a theory of gravity can be considered as a gauge theory (in the sense of Trautman) of the Poincaré group is given. As a result, it is shown that a gauge theory of this kind is consistent with the Equivalence Principle only if the Lagrangian and the constraints are preserved not only by the gauge transformations but also by an additional family of transformations, called pseudo-translations. Explicit expressions of pseudo-translations and of their action on gravitational gauge fields are given. They are expected to be useful for geometric interpretations of their analogues in supergravity theories.

We investigate pseudo-symmetric Lorentzian three-manifolds for the different possible Segre types of the Ricci operator. After determining all three-dimensional pseudo-symmetric Lorentzian algebraic curvature tensors, we classify pseudo-symmetric Lorentzian three-spaces which are either homogeneous, curvature homogeneous up to order 1 or curvature homogeneous, and we also provide some examples which are not curvature homogeneous.

Within bicovariant differential calculi framework, the BRST operator Ω is constructed. We showed that Ω is nil-potent (Ω²=0).

The aim of the paper is to establish a natural affine frame for affine Lagrangians and Hamiltonians, generalizing the well-known classical field theory. Scalar and volume-valued Lagrangians and Hamiltonians can be lifted to the new classes. Using the Hamilton–Jacobi principle, we analyze variational problems corresponding to actions defined by the affine Lagrangians and Hamiltonians. The extremals verify generalizations of the Euler–Lagrange and De Donder–Weyl PDEs. They improve the information about the dynamical solutions of the classical variational problems and refresh the Lagrange–Hamilton theories.

The main purpose of this paper is to investigate the fractional differential systems associated to dynamics of the rigid body on the Lie group SO(2,1) in terms of fractional Caputo derivatives.

We calculate the relative conserved currents, superpotentials and conserved quantities between two homogeneous and isotropic universes. In particular, we prove that their relative "energy" (defined as the conserved quantity associated to cosmic time coordinate translations for a comoving observer) is vanishing and so are the other conserved quantities related to a Lie subalgebra of vector fields isomorphic to the Poincaré algebra. These quantities are also conserved in time. We also find a relative conserved quantity for such a kind of solution which is conserved in time though non-vanishing. This example provides at least two insights in the theory of conserved quantities in General Relativity. First, the contribution of the cosmological matter fluid to the conserved quantities is carefully studied and proved to be vanishing. Second, we explicitly show that our superpotential (that happens to coincide with the so-called KBL potential although it is generated differently) provides strong conservation laws under much weaker hypotheses than the ones usually required. In particular, the symmetry generator is not needed to be Killing (nor Killing of the background, nor asymptotically Killing), the prescription is quasi-local and it works fine in a finite region too and no matching condition on the boundary is required.

We discuss local F-theory geometries and their gauge theory dualities in terms of intersecting D7-branes wrapped four-cycles in Type IIB superstring. The manifolds are built as elliptic K3 surface fibrations over intersecting F₀ = CP¹ × CP¹ base geometry according to ADE Dynkin Diagrams. The base is obtained by blowing up the extended ADE hyper-Kähler singularities of eight-dimensional manifolds considered as sigma model target spaces with eight supercharges. The resulting gauge theory of such local F-theory models are given in terms of Type IIB D7-branes wrapped intersecting F₀. The four-dimensional N = 1 anomaly cancelation requirement translates into a condition on the associated affine Lie algebras.

A geometric method is described to characterize the different kinds of extremals in optimal control theory. This comes from the use of a presymplectic constraint algorithm starting from the necessary conditions given by Pontryagin's Maximum Principle. The algorithm must be run twice so as to obtain suitable sets that once projected must be compared. Apart from the design of this general algorithm useful for any optimal control problem, it is shown how to classify the set of extremals and, in particular, how to characterize the strict abnormality. An example of strict abnormal extremal for a particular control-affine system is also given.