Hermitian versus holomorphic complex and quaternionic generalized supersymmetries of the M-theory. A classification

Relying upon the division-algebra classification of Clifford algebras andspinors, a classification of generalized supersymmetries (or, with a slightabuse of language,"generalized supertranslations") is provided. In each givenspace-time the maximal, saturated, generalized supersymmetry, compatible withthe division-algebra constraint that can be consistently imposed on spinors andon superalgebra generators, is furnished. Constraining the superalgebragenerators in both the complex and the quaternionic cases gives rise to the twoclasses of constrained hermitian and holomorphic generalized supersymmetries.In the complex case these two classes of generalized supersymmetries can beregarded as complementary. The quaternionic holomorphic supersymmetry onlyexists in certain space-time dimensions and can admit at most a single bosonicscalar central charge. The results here presented pave the way for a better understanding of thevarious $M$ algebra-type of structures which can be introduced in differentspace-time signatures and in association with different division algebras, aswell as their mutual relations. In a previous work, e.g., the introduction of acomplex holomorphic generalized supersymmetry was shown to be necessary inorder to perform the analytic continuation of the standard $M$-theory to the11-dimensional Euclidean space. As an application of the present results, it isshown that the above algebra also admits a 12-dimensional, Euclidean,$F$-algebra presentation.Comment: 25 pages, LaTe