Hermitian versus holomorphic complex and quaternionic generalized supersymmetries of the M-theory. A classification (original) (raw)
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Generalized space–time supersymmetries, division algebras and octonionic M-theory
Physics Letters B, 2002
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 U α U(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.
On a Division Algebra Classification of Constrained Generalized Supersymmetries
2005
In this talk we present a division-algebra classification of the generalized supersymmetries admitting bosonic tensorial central charges. We show that for complex and quaternionic supersymmetries a whole class of compatible division-algebra constraints can be imposed. Possible applications to M-theory related dynamical systems are briefly mentioned.
Constrained generalized supersymmetries and their classification
2005
Abstract. Generalized superymmetries going beyond the HLS scheme and admitting thepresence of bosonic tensorial central charges are constructed and classified in terms ofthe division algebras R , C , H and O . The eleven-dimensional M-algebra falls into thisclass of supersymmetries. Division-algebra compatible constraints can be introduced andfully classified. They can be used to construct and analyze various dynamical systems, thesimplest examples being the superparticles with tensorial central charges which generalizethe Rudychev-Sezgin and the Bandos-Lukierski models. Key words: Supersymmetry, M-Theory, Tensorial Central Charges 1. INTRODUCTIONThegeneralized supersymmetriesgoing beyond the Haag, Lopusza´nskiandSohniusclassi-fication[1]werefirstintroducedbyD’AuriaandFr´e in1982 [2]. Thefermionicsupersymmetrygenerators are, essentially, square roots operators. Their anticommutators produce a r.h.s.which is totally saturated and has to be expanded in terms of higher-rank bosonic tensors...
Spinor Algebras and Extended Superconformal Algebras
Quantum Theory and Symmetries, 2002
We consider supersymmetry algebras in arbitrary spacetime dimension and signature. Minimal and maximal superalgebras are given for single and extended supersymmetry. It is seen that the supersymmetric extensions are uniquely determined by the properties of the spinor representation, which depend on the dimension D mod 8 and the signature |ρ| mod 8 of spacetime.
Division algebras, extended supersymmetries and applications
2001
I present here some new results which make explicit the role of the division algebras R, C, H, O in the construction and classification of, respectively, N = 1, 2, 4, 8 global supersymmetric quantum mechanical and classical dynamical systems. In particular an N = 8 Malcev superaffine algebra is introduced and its relation to the non-associative N = 8 SCA is discussed. A list of present and possible future applications is given.
The symmetry algebras of Euclidean M-theory
Physics Letters B, 2004
We study the Euclidean supersymmetric D = 11 M -algebras. We consider two such D = 11 superalgebras: the first one is N = (1, 1) self-conjugate complex-Hermitean, with 32 complex supercharges and 1024 real bosonic charges, the second is N = (1, 0) complex-holomorphic, with 32 complex supercharges and 528 bosonic charges, which can be obtained by analytic continuation of known Minkowski M -algebra. Due to the Bott's periodicity, we study at first the generic D = 3 Euclidean supersymmetry case. The role of complex and quaternionic structures for D = 3 and D = 11 Euclidean supersymmetry is elucidated. We show that the additional 1024−528 = 496 Euclidean tensorial central charges are related with the quaternionic structure of Euclidean D = 11 supercharges, which in complex notation satisfy SU (2) pseudo-Majorana condition. We consider also the corresponding Osterwalder-Schrader conjugations as implying for N = (1, 0) case the reality of Euclidean bosonic charges. Finally, we outline some consequences of our results, in particular for D = 11 Euclidean supergravity. *
On the Octonionic M-superalgebra
The generalized supersymmetries admitting abelian bosonic tensorial central charges are classified in accordance with their division algebra structure (over R, C, H or O). It is shown in particular that in D = 11 dimensions, the M-superalgebra admits a consistent octonionic formulation, involving 52 real bosonic generators (in place of the 528 of the standard M-superalgebra). The octonionic M 5 (super-5-brane) sector coincides with the octonionic M 1 and M 2 sectors, while in the standard formulation these sectors are all independent. The octonionic conformal and superconformal M-algebras are explicitly constructed. They are respectively given by the Sp(8|O) (OSp(1, 8|O)) (super)algebra of octonionic-valued (super)matrices, whose bosonic subalgebra consists of 232 (and respectively 239) generators. * Speaker. † A large part of the results here reported is a fruit of a collaboration with J. Lukierski. PrHEP unesp2002 Workshop on Integrable Theories, Solitons and Duality Francesco Toppan length in the following. Perhaps the most remarkable and the most unexpected of such features consists in the fact that the different bosonic sectors expressed by the tensorial abelian central charges are no longer independent, as for the standard generalized supersymmetries admitting associative realizations, but they are all interrelated. This phenomenon is a peculiar characteristic of the octonionic construction. It is worth noticing that the Minkowskian 11-dimensional spacetime supports two inequivalent structures, the real structure and the octonionic one. Therefore, besides the standard M-algebra leading to the OSp(1|32) superalgebra [9] (and its OSp(1|64) superconformal algebra), a different M-algebra can be introduced in terms of the octonionic structure and consistently defined as a closed algebra. This is the octonionic M-algebra (it will also sometimes be referred to as M-superalgebra) which will be discussed in this talk. Of course, it is too early to say whether this octonionic M-algebra can be of any relevance for physics. On the other hand, the mere fact that it exists, side by side with the standard M-algebra (not to mention its puzzling features) justifies a thorough investigation of this and its related mathematical structures. The plan of this talk is as follows. In the next section the classification of Clifford algebras and spinors (i.e. the necessary ingredients to introduce supersymmetry) is recalled. Later, in section 3, the connection of division algebras with the classification of Clifford algebras will be elucidated. In particular the octonionic-valued realizations (which are usually disregarded in the literature) of the Clifford algebras and their corresponding spinors will be introduced. This paves the way for the construction, in Section 4, of the generalized supersymmetries based on the division algebras and, in Section 5, of the octonionic M-algebra. A detailed discussion of its properties will also be given. In particular a table, based on the octonionic structure constants, expressing the equivalence of the different brane sectors in the octonionic description, will be furnished. In Section 6 the octonionic superconformal M-algebra will be introduced. Finally, in the Conclusions, the relation of the octonionic M-algebra with other algebraic structures such as Jordan algebras will be elucidated. Some possible geometrical interpretations underlining the octonionic description will be pointed out and the outline for further future investigations will be given.
Starting with a review of the Extended Relativity Theory in Clifford-Spaces, and the physical motivation behind this novel theory, we provide the generalization of the nonrelativistic Supersymmetric pointparticle action in Clifford-space backgrounds. The relativistic Supersymmetric Clifford particle action is constructed that is invariant under generalized supersymmetric transformations of the Clifford-space background's polyvector-valued coordinates. To finalize, the Polyvector Super-Poincare and M, F theory superalgebras, in D = 11, 12 dimensions, respectively, are discussed followed by our final analysis of the novel Clifford-Superspace realizations of generalized Supersymmetries in Clifford spaces.
A Clifford algebra realization of Supersymmetry and its Polyvector extension in Clifford Spaces
It is shown explicitly how to construct a novel (to our knowledge) realization of the Poincare superalgebra in 2D. These results can be extended to other dimensions and to (extended) superconformal and (anti) de Sitter superalgebras. There is a fundamental difference between the findings of this work with the other approaches to Supersymmetry (over the past four decades) using Grassmannian calculus and which is based on anti-commuting numbers. We provide an algebraic realization of the anticommutators and commutators of the 2D super-Poincare algebra in terms of the generators of the tensor product Cl1,1(R) ⊗ A of a two-dim Clifford algebra and an internal algebra A whose generators can be represented in terms of powers of a 3 × 3 matrix Q, such that Q 3 = 0. Our realization dif f ers from the standard realization of superalgebras in terms of dif f erential operators in Superspace involving Grassmannian (anticommuting) coordinates θ α and bosonic coordinates x µ . We conclude in the final section with an analysis of how to construct Polyvector-valued extensions of supersymmetry in Clifford Spaces involving spinor-tensorial supercharge generators Q µ 1 µ 2 .....µn α and momentum polyvectors Pµ 1 µ 2 ....µn . Clifford-Superspace is an extension of Clifford-space and whose symmetry transformations are generalized polyvector-valued supersymmetries.