Generalized space–time supersymmetries, division algebras and octonionic M-theory

We describe the set of generalized Poincaré and conformal superalgebras in D=4,5 and 7 dimensions as two sequences of superalgebraic structures, taking values in the division algebras R,C and H. The generalized conformal superalgebras are described for D=4 by OSp(1;8|R), for D=5 by SU(4,4;1) and for D=7 by UαU(8;1|H). The relation with other schemes, in particular the framework of conformal spin (super)algebras and Jordan (super)algebras is discussed. By extending the division-algebra-valued superalgebras to octonions we get in D=11 an octonionic generalized Poincaré superalgebra, which we call octonionic M-algebra, describing the octonionic M-theory. It contains 32 real supercharges but, due to the octonionic structure, only 52 real bosonic generators remain independent in place of the 528 bosonic charges of standard M-algebra. In octonionic M-theory there is a sort of equivalence between the octonionic M2 (supermembrane) and the octonionic M5 (super-5-brane) sectors. We also define the octonionic generalized conformal M-superalgebra, with 239 bosonic generators.