Narrow Results

We discuss ${𝒩}=1$ Klein and Klein-conformal superspaces in $D=(2,2)$ space-time dimensions, realizing them in terms of their functor of points over the split composition algebra ${ℂ}_s$. We exploit the observation that certain split forms of orthogonal groups can be realized in terms of matrix groups over split composition algebras. This leads to a natural interpretation of the sections of the spinor bundle in the critical split dimensions $D=4,6$ and $10$ as ${ℂ}_s^2$, ${ℍ}_s^2$ and ${𝕆}_s^2$, respectively. Within this approach, we also analyze the non-trivial spinor orbit stratification that is relevant in our construction since it affects the Klein-conformal superspace structure.

The classification of the octonionic realizations of the one-dimensional extended supersymmetries is here furnished. These are non-associative realizations which, albeit inequivalent, are put in correspondence with a subclass of the already classified associative representations for 1D extended supersymmetries. Examples of dynamical systems invariant under octonionic realizations of the extended supersymmetries are given. We cite among the others the octonionic spinning particles, the N = 8 KdV, etc. Possible applications to supersymmetric systems arising from dimensional reduction of the octonionic superstring and M-theory are mentioned.

Representations of SO(4, 2) are constructed using 4×4 and 2×2 matrices with elements in ℍ' ⊗ ℂ and the known isomorphism between the conformal group and SO(4, 2) is written explicitly in terms of the 4×4 representation. The Clifford algebra structure of SO(4, 2) is briefly discussed in this language, as is its relationship to other groups of physical interest.

It is shown that besides the standard real algebraic framework for M-theory a consistent octonionic realization can be introduced. The octonionic M-superalgebra and superconformal M-algebra are derived. The first one involves 52 real bosonic generators and presents a novel and surprising feature, its octonionic M5 (super-5-brane) sector coincides with the M1 and M2 sectors. The octonionic superconformal M-algebra is given by OS_p(1,8∣O) and admits 239 bosonic and 64 fermionic generators.

The Composition Algebra-based Methodology (CAM) [B. Wolk, Pap. Phys.9, 090002 (2017); Phys. Scr.94, 025301 (2019); Adv. Appl. Clifford Algebras27, 3225 (2017); J. Appl. Math. Phys.6, 1537 (2018); Phys. Scr.94, 105301 (2019), Adv. Appl. Clifford Algebras30, 4 (2020)], which provides a new model for generating the interactions of the Standard Model, is geometrically modeled for the electromagnetic and weak interactions on the parallelizable sphere operator fiber bundle$B_n=(T𝕄,S^n\to {\mathcal{𝒮}}^n,SO(n+1),\pi )$ consisting of base space, the tangent bundle $T𝕄$ of space–time $𝕄$, projection operator $\pi$, the parallelizable spheres $S^n=\{S^1,S^3\}$ conceived as operator fibers $S^n\to {\mathcal{𝒮}}^n$ attaching to and operating on $T_p𝕄$$\forall p\in 𝕄$ as $p$ varies over $𝕄$, and as structure group, the norm-preserving symmetry group $SO(n+1)$ for each of the division algebras which is simultaneously the isometry group of the associated unit sphere. The massless electroweak $SU{(2)}_L\otimes U{(1)}_Y$ Lagrangian is shown to arise from $B_{3\otimes 1}$’s generation of a local coupling operation on sections of Dirac spinor and Clifford algebra bundles over $𝕄$. Importantly, CAM is shown to be a new genre of gauge theory which subsumes Yang–Mills Standard Model gauge theory. Local gauge symmetry is shown to be at its core a geometric phenomenon inherent to CAM gauge theory. Lastly, the higher-dimensional, topological architecture which generates CAM from within a unified eleven $(1,10)$-dimensional geometro-topological structure is introduced.

We provide an analogue of Wedderburn’s factorization method for central polynomials with coefficients in an octonion division algebra, and present an algorithm for fully factoring polynomials of degree $n$ with $n$ conjugacy classes of roots, counting multiplicities.

There must exist a reformulation of quantum field theory which does not refer to classical time. We propose a pre-quantum, pre-spacetime theory, which is a matrix-valued Lagrangian dynamics for gravity, Yang–Mills fields, and fermions. The definition of spin in this theory leads us to an eight-dimensional octonionic spacetime. The algebra of the octonions reveals the standard model; model parameters are determined by roots of the cubic characteristic equation of the exceptional Jordan algebra. We derive the asymptotic low-energy value 1/137 of the fine structure constant, and predict the existence of universally interacting spin one Lorentz bosons, which replace the hypothesised graviton. Gravity is not to be quantized, but is an emergent four-dimensional classical phenomenon, precipitated by the spontaneous localisation of highly entangled fermions.

In this paper, we present a complete method for finding the roots of all polynomials of the form $\varphi (z)=c_n{z}^n+c_{n-1}{z}^{n-1}+\cdots +c_{1}z+c_0$ over a given octonion division algebra. When $\varphi (z)$ is monic, we also consider the companion matrix and its left and right eigenvalues and study their relations to the roots of $\varphi (z)$, showing that the right eigenvalues form the conjugacy classes of the roots of $\varphi (z)$ and the left eigenvalues form a larger set than the roots of $\varphi (z)$.

The derivation of the standard model from a higher-dimensional action suggests a further study of the fiber bundle formulation of gauge theories to determine the variations in the choice of structure group that are allowed in this geometrical setting. The action of transformations on the projection of fibers to their submanifolds are characteristic of theories with fewer gauge vector bosons, and specific examples are given, which may have phenomenological relevance. The spinor space for the three generations of fermions in the standard model is described algebraically.

It is shown that the algebra $J_3[C\otimes O]\otimes Cl(4,C)$ based on the complexified Exceptional Jordan, and the complex Clifford algebra in 4D, is rich enough to describe all the spinorial degrees of freedom of three generations of fermions in 4D, and include additional fermionic dark matter candidates. Furthermore, the model described in this paper can account also for the Standard Model gauge symmetries. We extend these results to the Magic Star algebras of Exceptional Periodicity developed by Marrani–Rios–Truini and based on the Vinberg cubic $T$ algebras which are generalizations of exceptional Jordan algebras. It is found that there is a one-to-one correspondence among the real spinorial degrees of freedom of four generations of fermions in 4D with the off-diagonal entries of the spinorial elements of the $pair$$T_3^{8,n},({\bar{T}}_3^{8,n})$ of Vinberg matrices at level $n=2$. These results can be generalized to higher levels $n>2$ leading to a higher number of generations beyond 4. Three $pairs$ of $T$ algebras and their conjugates $\bar{T}$ were essential in the Magic Star construction of Exceptional Periodicity that extends the $e_8$ algebra to $e_8^{(n)}$ with $n$ integer.

We discuss basic properties (uniqueness and regularity) of viscosity solutions to fully nonlinear elliptic equations of the form F(x, D²u) = 0, which includes also linear elliptic equations of nondivergent form. In the linear case we consider equations with discontinuous coefficients.