where in the first line we have explicitly indicated the sum over the

spinorial indices, which are suppressed for simplicity in the second line.

They cover basic

spinorial material, lowest eigenvalues of the Dirac operator on closed spin manifolds, special spinor field and geometries, and Dirac spectra of model spaces.

In a more recent work [2], we studied with Bayard and Lawn the

spinorial characterization of surfaces into 4-dimensional space forms.

The even sector is expressed by the 2x2 diagonal operators [J.sup.[mu]] and R ("R" stands for the R-symmetry u(1)-generator) which can be recovered from the saturated (due to the presence of both

spinorial [alpha], [beta] and internal A, B indices) generalized supersymmetry (see [26])

As we see from (2) in addition to Poincare generators [P.sup.[mu]], [M.sup.[mu]v] there are more than one anticommuting Majorana

spinorial charges (supersymmetry generators [P.sup.[mu]], [M.sup.[mu]v] (the indices A = 1, ..., N label the representation of the internal symmetry group to which belongs).

As an example in non-SUSY SO(10) [117, 118], at first a precision unification frame has been achieved with the modification of the TeV scale spectrum of the minimal SUSY GUT by taking out the full scalar super partner content of the

spinorial super field representation 16 [subset] SO(10).

The duality operates under the exchange of the total number of

spinorial 16 [direct sum] [bar.16] and vectorial 10 representations of SO(10).

Besides this, the field [PSI] has not a

spinorial character.

Using (8), the

spinorial affine connections are derived as follows [23]:

The relevant physical feature of this measure factor V[([D.sup.5]).sup.1/4] is that it encodes the

spinorial degrees of freedom of the electron, like the factor of 8[pi] encodes the two-helicity states of the massless photon.

The investigations on the interaction between a torsion and a Dirac

spinorial field have attracted attention for a long time [39-48].

Here first of all we demonstrate how to incorporate spin as well as internal symmetries for both fermions and gauge bosons for general SO(n) gauge groups (see [31,32] for related work) and we specialize to the case of S0(10) for the chiral fermions of the SM, for the 16 fermions of the SM in a single generation transform as a

spinorial representation of S0(10), upon including a right-handed antineutrino (see [33, 34] for related work).