Narrow Results

Following the formalism derived from one method of constructing common projective spaces along with using a special kind of odd module homomorphisms, denoted by ν, a novel supergeometric generalization of projective spaces is constructed. Existence of canonical line bundles over these spaces and their Chern classes are discussed.

Superspace is considered as space of parameters of the supercoherent states defining the basis for oscillator-like unitary irreducible representations of the generalized superconformal group $SU(2m,2n|2N)$ in the field of quaternions $ℍ$. The specific construction contains naturally the supertwistor one of the previous work by Litov and Pervushin [1] and it is shown that in the case of extended supersymmetry such an approach leads to the separation of a class of superspaces and its groups of motion. We briefly discuss this particular extension to the domain of quaternionic superspaces as nonlinear realization of some kind of the affine and the superconformal groups with the final end to include also the gravitational field [6] (this last possibility to include gravitation, can be realized on the basis of Ref. 12 where the coset $\frac{Sp(8)}{SL(4R)}\sim$$\frac{SU(2,2)}{SL(2C)}$ was used in the non supersymmetric case). It is shown that this quaternionic construction avoid some unconsistencies appearing at the level of the generators of the superalgebras (for specific values of $p$ and $q$; $p+q=N$) in the twistor one.