Electroweak interaction beyond the Standard Model and Dark Matter in the Tangent Bundle Quantum Field Theory

A generalized theory of electroweak interaction is developed based on the underlying geometrical structure of the tangent bundle with symmetries arising from transformations of tangent vectors along the fiber axis at a fixed space-time point, leaving the scalar product invariant. Transformations with this property are given by the $SO(3,1)$ group with the little groups $SU(2),E^{c}(2)$ and $SU(1,1)$ where the group $E^{c}(2)$ is the central extended group of the Euclidian group $E(2).$ Electroweak interaction beyond the standard model (SM) is described by the transformation group $SU(2)\otimes E^{c}\mathbf{(}2)$ without a priori introduction of a phenomenologically determined gauge group. The Laplacian on this group yields the known internal quantum numbers of isospin and hypercharge, but in addition the extra $E^{c}$-charge $\varkappa$ and the family quantum number $n$ which explains the existence of families in the SM. The connection coefficients deliver the SM gauge potentials but also hypothetical gauge bosons and other hypothetical particles as well as candidate Dark Matter particles are predicted. It is shown that the interpretation of the $SO(3,1)$ connection coefficients as elctroweak gauge potentials is compatible with teleparallel gauge gravity theory based on the translational group.Comment: Improved versio

The p-Laplace equation in domains with multiple crack section via pencil operators

The p-Laplace equation \n \cdot (|\n u|^n \n u)=0 \whereA n>0, in a bounded domain \O \subset \re^2, with inhomogeneous Dirichlet conditions on the smooth boundary \p \O is considered. In addition, there is a finite collection of curves \Gamma = \Gamma_1\cup...\cup\Gamma_m \subset \O, \quad \{on which we assume homogeneous Dirichlet boundary conditions} \quad u=0, modeling a multiple crack formation, focusing at the origin 0 \in \O. This makes the above quasilinear elliptic problem overdetermined. Possible types of the behaviour of solution $u(x,y)$ at the tip 0 of such admissible multiple cracks, being a "singularity" point, are described, on the basis of blow-up scaling techniques and a "nonlinear eigenvalue problem". Typical types of admissible cracks are shown to be governed by nodal sets of a countable family of nonlinear eigenfunctions, which are obtained via branching from harmonic polynomials that occur for $n=0$. Using a combination of analytic and numerical methods, saddle-node bifurcations in $n$ are shown to occur for those nonlinear eigenvalues/eigenfunctions.Comment: arXiv admin note: substantial text overlap with arXiv:1310.065