Spinor Algebras and Extended Superconformal Algebras (original) (raw)
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Considerations on Super Poincaré Algebras and Their Extensions to Simple Superalgebras
Reviews in Mathematical Physics, 2002
We consider simple superalgebras which are a supersymmetric extension of the spin algebra in the cases where the number of odd generators does not exceed 64. All of them contain a super Poincaré algebra as a contraction and another as a subalgebra. Because of the contraction property, some of these algebras can be interpreted as de Sitter or anti de Sitter superalgebras. However, the number of odd generators present in the contraction is not always minimal due to the different splitting properties of the spinor representations under a subalgebra. We consider the general case, with arbitrary dimension and signature, and examine in detail particular examples with physical implications in dimensions d = 10 and d = 4.
On the Embedding of Spacetime Symmetries into Simple Superalgebras
Letters in Mathematical Physics - LETT MATH PHYS, 2001
We explore the embedding of Spin groups of arbitrary dimension and signature into simple superalgebras in the case of extended supersymmetry. The R-symmetry, which generically is not compact, can be chosen compact for all the cases that are congruent mod 8 to the physical conformal algebra so(D - 2,2), D = 3. An so(1,1) grading of the superalgebra is found in all cases. Central extensions of super translation algebras are studied within this framework.
A N-extended version of superalgebras in
1997
A set of generalized superalgebras containing arbitrary tensor p-form operators is considered in dimensions D = 2n + 1 for n = 1, 4 mod 4 and the general conditions for its existence expressed in the form of generalized Jacobi identities is established. These are then solved in a univoque way and some lowest dimensional cases D = 3, 9, 11 of possible interest are made explicit. Introduction Supersymmetry (SUSY) was a deeply studied subject during the last two decades since the no-go theorem of Coleman and Mandula about all the possible symmetries of the S-matrix in D = 4 was by-passed by the discovery of Haag, Lopuzsanski and Sohnius (HLS) on possible fermionic extensions of the corresponding symmetry algebra [8]. While I do not know the relevance of the mentioned theorem in dimensions greater than four and outside the domain of local quantum field theory (and the present paper do not intend to address this question) it is certainly true that SUSY in higher dimensions becamed more a...
Communications in Mathematical Physics, 1997
We classify extended Poincari Lie super algebras and Lie algebras of any signature (p, q\ that is Lie super algebras (resp. Z2-graded Lie algebras) fl = flo + fli, where Jlo = 50(V)+V is the (generalized) Poincare Lie algebra of the pseudo-Euclidean vector space V = R p ' g of signature (p, q) and fli = 5 is the spinor S0(F)-module extended to a go-module with kernel V. The remaining super commutators {flii,fli} (respectively, commutators [fl i, 01 ]) are defined by an $0(V)-equivariant linear mapping V 2 fli-* V (respectively, A 2 fli->• V). Denote by V + (n, s) (respectively, V~ (n, s)) the vector space of all such Lie super algebras (respectively, Lie algebras), where n = p + q = dim V and s = p-q is the classical signature. The description of V^in, s) reduces to the construction of all S0(\^)-invariant bilinear forms on 5 and to the calculation of three Z2-valued invariants for some of them. This calculation is based on a simple explicit model of an irreducible Clifford module S for the Clifford algebra Cl P)q of arbitrary signature (p, q). As a result of the classification, we obtain the numbers L ± (n, s) = dimV ± (n 1 s) of independent Lie super algebras and algebras, which take values 0,1,2,3,4 or 6. Due to Bott periodicity, L ± (n, s) may be considered as periodic functions with period 8 in each argument. They are invariant under the group F generated by the four reflections with respect to the axes n =-2, n = 2, s-1 =-2 and 5-1 = 2. Moreover, the reflection (n, s)-> (-n, s
A N-extended version of superalgebras in D=3,9 mod 8
1997
A set of generalized superalgebras containing arbitrary tensor p-form operators is considered in dimensions D=2n+1D=2n+1D=2n+1 for n=1,4mod4n=1,4 mod 4n=1,4mod4 and the general conditions for its existence expressed in the form of generalized Jacobi identities is established. These are then solved in a univoque way and some lowest dimensional cases $ D= 3, 9, 11 $ of possible interest are made explicit.
2-Spinors, Twistors and Supersymmetry in the Spacetime Algebras
Spinors, Twistors, Clifford Algebras and Quantum Deformations, 1993
We present a new treament of 2-spinors and twistors, using the spacetime algebra. The key rôle of bilinear covariants is emphasized. As a by-product, an explicit representation is found, composed entirely of real spacetime vectors, for the Grassmann entities of supersymmetric field theory.
Seven-sphere and the exceptional N = 7 and N = 8 superconformal algebras
Nuclear Physics B, 1996
We study realizations of the exceptional non-linear (quadratically generated, or Wtype) N = 8 and N = 7 superconformal algebras with Spin(7) and G 2 affine symmetry currents, respectively. Both the N = 8 and N = 7 algebras admit unitary highestweight representations in terms of a single boson and free fermions in 8 of Spin(7) and 7 of G 2 , with the central charges c 8 = 26/5 and c 7 = 5, respectively. Furthermore, we show that the general coset Ansätze for the N = 8 and N = 7 algebras naturally lead to the coset spaces SO(8) × U(1)/SO(7) and SO(7) × U(1)/G 2 , respectively, as the additional consistent solutions for certain values of the central charge. The coset space SO(8)/SO is the seven-sphere S 7 , whereas the space SO(7)/G 2 represents the sevensphere with torsion, S 7 T . The division algebra of octonions and the associated triality properties of SO(8) play an essential role in all these realizations. We also comment on some possible applications of our results to string theory.
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.
a New N=4 Superconformal Algebra
Modern Physics Letters A, 1993
It is shown that the previously known N=3N=3N=3 and N=4N=4N=4 superconformal algebras can be contracted consistently by singular scaling of some of the generators. For the later case, by a contraction which depends on the central term, we obtain a new N=4N=4N=4 superconformal algebra which contains an SU(2)timesU(1)4SU(2)\times {U(1)}^4SU(2)timesU(1)4 Kac-Moody subalgebra and has nonzero central extension.