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Clifford algebras and the minimal representations of the 1D N-Extended supersymmetry algebraToppan, FrancescoCentro Brasileiro de Pesquisas Físicas (CBPF)2008The Atiyah-Bott-Shapiro classification of the irreducible Clifford algebra is used to derive general properties of the minimal representations of the 1D N-Extended Supersymmetry algebra (the $Z_2$-graded symmetry algebra of the Supersymmetric Quantum Mechanics) linearly realized on a finite number of fields depending on a real parameter t, the time.
Minimal representations of supersymmetry and 1D N-extended sigma-modelsToppan, FrancescoCentro Brasileiro de Pesquisas Físicas (CBPF)2008The Atiyah-Bott-Shapiro classification of the irreducible Clifford algebra is used to derive general properties of the minimal representations of the 1D N-Extended Supersymmetry algebra (the $Z_2$-graded symmetry algebra of the Supersymmetric Quantum Mechanics) linearly realized on a finite number of fields depending on a real parameter t, the time.
On Alphabetic Presentations of Clifford Algebras and Their Possible Applications [Impresso] / Francesco Toppan and Piet W. Verbeek.Toppan, Francesco-2009In this paper we address the problem of constructing a class of representations of Clifford algebras that can be named 'alphabetic (re)presentations'. The Clifford algebras generators are expressed as $m$-letter words written with a $3$-character or a $4$-character alphabet. We formulate the problem of the alphabetic presentations, deriving the main properties and some general results. At the end we briefly discuss the motivations of this work and outline some possible applications.
One-dimensional sigma-models with N = 5, 6, 7, 8 off-shell supersymmetriesGonzales, M. (Marcelo); Rojas, M. (Moises); Toppan, Francesco-2008We computed the actions for the $1D$ $N=5$ $\sigma$-models with respect to the two inequivalent $(2,8,6)$ multiplets. $4$ supersymmetry generators are manifest, while the constraint originated by imposing the $5$-th supersymmetry automatically induces a full $N=8$ off-shell invariance. The resulting action coincides in the two cases and corresponds to a conformally flat $2D$ target satisfying a special geometry of rigid type. \par To obtain these results we developed a computational method (for {\em Maple 11}) which does not require the notion of superfields and is instead based on the nowadays available list of the inequivalent representations of the $1D$ $N$-extended supersymmetry. Its application to systematically analyze the $\sigma$-models off-shell invariant actions for the remaining $N=5,6,7,8$ $(k,8,8-k)$ multiplets, as well as for the $N>8$ representations, only requires more cumbersome computations.