### Bianchi identities, Yang-Mills and Higgs field produced on ${\tilde{S}}^{\left(2\right)}M$-deformed bundle.

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It is a common belief among theoretical physicists that the charge conjugation of the Dirac equation has an analogy in higher dimensional space-times so that in an 8-dimensional space-time there would also be Maiorana spinors as eigenspinors of a charge conjugation, which would swap the sign of the electric charge of the Dirac equation. This article shows that this mistaken belief is based on inadequate distinction between two kinds of charge conjugation: the electric conjugation swapping the sign...

On a pseudo-Riemannian manifold $\mathbb{M}$ we introduce a system of partial differential Killing type equations for spinor-valued differential forms, and study their basic properties. We discuss the relationship between solutions of Killing equations on $\mathbb{M}$ and parallel fields on the metric cone over $\mathbb{M}$ for spinor-valued forms.

Inspired by the results on symmetries of the symplectic Dirac operator, we realize symplectic spinor fields and the symplectic Dirac operator in the framework of (the double cover of) homogeneous projective structure in two real dimensions. The symmetry group of the homogeneous model of the double cover of projective geometry in two real dimensions is $\tilde{}(3,)$.

It is well known that starting with real structure, the Cayley-Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 27. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of...

We introduce the symplectic twistor operator ${T}_{s}$ in symplectic spin geometry of real dimension two, as a symplectic analogue of the Dolbeault operator in complex spin geometry of complex dimension 1. Based on the techniques of the metaplectic Howe duality and algebraic Weyl algebra, we compute the space of its solutions on ${\mathbb{R}}^{2}$.

Using the Clifford bundle formalism we show that Frenet equations of classical differential geometry or its spinor version are the appropriate equations of motion for a classical spinning particle. We show that particular values of the curvatures appearing in Darboux bivector of the spinor form of Frenet equations produce a "classical" Dirac-Hestenes equation. Using the concept of multivector Lagrangians and Hamiltonians we provide a Lagrangian and Hamiltonian approach for our theory which then...