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

TL;DR

This paper classifies generalized supersymmetries in various space-time dimensions using division algebra constraints, revealing new hermitian and holomorphic structures in complex and quaternionic settings relevant to M-theory.

Contribution

It provides a comprehensive classification of maximal generalized supersymmetries compatible with division algebra constraints, including new hermitian and holomorphic supersymmetries in complex and quaternionic cases.

Findings

Classified maximal generalized supersymmetries in different space-time signatures.

Identified the existence of quaternionic holomorphic supersymmetry in specific dimensions.

Showed that the algebra admits a 12-dimensional Euclidean F-algebra presentation.

Abstract

Relying upon the division-algebra classification of Clifford algebras and spinors, a classification of generalized supersymmetries (or, with a slight abuse of language,"generalized supertranslations") is provided. In each given space-time the maximal, saturated, generalized supersymmetry, compatible with the division-algebra constraint that can be consistently imposed on spinors and on superalgebra generators, is furnished. Constraining the superalgebra generators in both the complex and the quaternionic cases gives rise to the two classes of constrained hermitian and holomorphic generalized supersymmetries. In the complex case these two classes of generalized supersymmetries can be regarded as complementary. The quaternionic holomorphic supersymmetry only exists in certain space-time dimensions and can admit at most a single bosonic scalar central charge. The results here presented pave the way for a better understanding of the various $M$ algebra-type of structures which can be introduced in different space-time signatures and in association with different division algebras, as well as their mutual relations. In a previous work, e.g., the introduction of a complex holomorphic generalized supersymmetry was shown to be necessary in order to perform the analytic continuation of the standard $M$-theory to the 11-dimensional Euclidean space. As an application of the present results, it is shown that the above algebra also admits a 12-dimensional, Euclidean, $F$-algebra presentation.