Abstract:Quaternionic and octonionic realizations of Clifford algebras and spinors are classified and explicitly constructed in terms of recursive formulas. The most general free dynamics in arbitrary signature space-times for both quaternionic and octonionic spinors is presented. In the octonionic case we further provide a systematic list of results and tables expressing, e.g., the relations of the octonionic Clifford algebras with the G 2 cosets over the Lorentz algebras, the identities satisfied by the higher-rank a… Show more
“…The supersymmetry transformations are here explicitly obtained by dressing the N = 8 root supermultiplet expressed in terms of the octonionic structure constants C ijk , see [34,4] …”
Section: Connectivity Symbol Of the N = 4 Fully Reducible Representatmentioning
We construct the non-minimal linear representations of the N = 4 Extended Supersymmetry in one-dimension. They act on 8 bosonic and 8 fermionic fields. Inequivalent representations are specified by the mass-dimension of the fields and the connectivity of the associated graphs. The oxidation to minimal N = 5 linear representations is given. Two types of N = 4 σ-models based on non-minimal representations are obtained: the resulting off-shell actions are either manifestly invariant or depend on a constrained prepotential. The connectivity properties of the graphs play a decisive role in discriminating inequivalent actions. These results find application in partial breaking of supersymmetric theories.
“…The supersymmetry transformations are here explicitly obtained by dressing the N = 8 root supermultiplet expressed in terms of the octonionic structure constants C ijk , see [34,4] …”
Section: Connectivity Symbol Of the N = 4 Fully Reducible Representatmentioning
We construct the non-minimal linear representations of the N = 4 Extended Supersymmetry in one-dimension. They act on 8 bosonic and 8 fermionic fields. Inequivalent representations are specified by the mass-dimension of the fields and the connectivity of the associated graphs. The oxidation to minimal N = 5 linear representations is given. Two types of N = 4 σ-models based on non-minimal representations are obtained: the resulting off-shell actions are either manifestly invariant or depend on a constrained prepotential. The connectivity properties of the graphs play a decisive role in discriminating inequivalent actions. These results find application in partial breaking of supersymmetric theories.
“…We can write down a possible set of (2 5 × 2 5 ) gamma-matrices, in the Weyl representation [8], satisfying the Clifford algebra given in Eq. (2.1):…”
Section: Building Up the Space-time: D = (5 + 5) Dimensionsmentioning
We show that Majorana-Weyl spinors of a given chirality in dimensions higher than four can be interpreted, in (1+3)-dimensions, as four Majorana particles with different masses in general. This mechanism borrows a peculiar connection with the See-Saw scheme associated to neutrinos with Majorana-type masses.
“…On the other hand, it is well-known [15], that the seven 8 × 8 antisymmetric gamma matrices of Cl(0, 7) can be recovered by the left-action of the imaginary octonions on the octonionic space. As a result, the entries of the seven antisymmetric gamma-matrices of Cl(0, 7) can be expressed in terms of the totally antisymmetric octonionic structure constants C i jk 's.…”
The complete classification of the irreducible representations of the N-extended one-dimensional supersymmetry algebra linearly realized on a finite number of fields is presented. Off-shell invariant actions of one-dimensional supersymmetric sigma models are constructed. The role of both Clifford algebras and the Cayley-Dickson's doublings of algebras in association with the N-extended supersymmetries is discussed. We prove in specific examples that the octonionic structure constants enter the N = 8 invariant actions as coupling constants. We further explain how to relate one-dimensional supersymmetric quantum mechanical systems to the dimensional reduction of higher-dimensional supersymmetric theories.
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