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1138 Brazilian Journal of Physics, vol. 35, no. 4B, December, 2005 The Super-Poincare´ Algebra Via Pure Spinors and the Interaction Principle in 3D Euclidean Space R. da Rocha IFGW, Universidade Estadual de Campinas, CP 6165, 13083-970 Campinas, SP, Brazil (Received on 12 September, 2005) The Poincare´ superalgebra is introduced from a generalization of the Cartan’s triality principle based on the extension of Chevalley product, between semispinor spaces and even subspaces of the extended exterior algebra over Euclidean space R3. The pure spinor formalism and the framework of Clifford algebras are used, in order to provide the necessary tools to introduce the Poincare´ superalgebra where all the operators in space and superspace are constructed via pure spinors in R3 and the interaction principle, that generalizes the SO(8) triality principle. I. INTRODUCTION A totally symmetric trilinear tensor is defined as φ φ φ The seminal paper of Wess and Zumino [1] treating the 4- T( 1; 2; 3) = h(u1;x2v3) + h(u1;x3v2) + h(u2;x3v1) dimensional super-Poincare´ algebra, has boosted new investi- + h(u2;x1v3) + h(u3;x2v1) + h(u3;x1v2): gations on supersymmetry in Lagrangian field theory and in particle physics. On the other hand, Cartan’s triality principle, It is possible to endow the vector space E with an asso- based on the group SO(8) and its double covering Spin(8), ciative and non-commutative product M: E £ E ! E, called has been applied in recent physical theories, like supergravity the Chevalley product, that is implicitly defined by [2, 4] and superstrings. Only in dimension 8, bosons and fermions T(φ1;φ2;φ3) = B˘(φ1 M φ2;φ3). It is immediate to see that have the same number of degrees of freedom. This general- x M u = xu and x M v = xv [2]. ized supersymmetry comes from the triality principle, which I Proposition 1: The inclusions V M S+ ⊆ S¡; S+ M asserts that bosons and fermions are equivalent under the trial- S¡ ⊆ V; S¡ M V ⊆ S+ hold. J ity map [2–4], based on the Chevalley product [4]. This paper Hereon x;u and v are respectively denoted elements of is presented as follows: in Section 2 the main results involv- V, S+ and S¡. The spinor representation ρ of the Clifford ing the structure of the Chevalley product, giving rise to the product in S§ and the vector (adjoint) representation χ in V triality principle, are presented [2], and we briefly revisit the naturally induce a irreducible representation Y in E. Given pure spinor formalism. In Section 3 the Poincare´ superalgebra s 2 Γ+, that denotes the Clifford group of elements of unit is introduced in R3, via orthosymplectic graded Lie algebras norm, such representation is defined by Y(s)(x + u + v) = and via the interaction principle, which generalizes the cyclic χ(s)x + ρ(s)u + ρ(s)v. property of the Chevalley product. This work was supported The representation Y(s) is shown to be orthogonal in rela- by CAPES. tion to the metric on B˘ [2]. In addition, it can be shown that, if 2 2 x0 2 V is such that x0 = 1, then Y (x0) = 1. Now consider a + spinor u0 2 S of unitary norm, i.e., h(u0;u0) = 1. A homo- + II. TRIALITY PRINCIPLE AND PURE SPINORS morphism τ : S ! V is defined is such a way that τ(u0)(x) = x M u0. The map τ(u0) is clearly orthogonal and is uniquely ¡ ¡ Consider a vector space V endowed with metric g. The extended to an order two automorphism in V ©S . If v 2 S is τ τ spinor space S can be written as the direct sum S = S+ © S¡, such that v = (u0)x for a unique x 2V, defining (u0)(v) = x, τ + where S§ denotes semispinor spaces that carry, each one, a and besides, the map (u0) is defined in S as a reflection, + τ non-equivalent irreducible representation of the even subalge- given a fixed spinor u0 2 S , as (u0)(u) = 2h(u;u0)u0 ¡ u. bra C`+(V;g). Define the space E = V © S+ © S¡. In order In order to define an order-three automorphism, that permutes + ¡ to find a vector space V such that the semispinor space S§ of cyclically the vector spaces S ;S and V, we can verify that τ τ τ V has the same dimension n of V, the condition n = 8 must (u0)Y(x0) (u0) = Y(x0) (u0)Y(x0), and then the triality op- Θ Θ τ hold. We consider hereon V ' R8;0 [5]. The space E is there- erator is defined as (x0;u0) = Y(x0) (u0), which is an Θ3 fore 24-dimensional and its elements are hereon denoted by order three automorphism, since (x0;u0) = 1. + ¡ φ = x + u + v, where x 2 V, u 2 S+ and v 2 S¡. The spinor I Proposition 2: The subspaces V, S and S of E are cycli- Θ Θ metric h : S§ £ S§ ! R is defined as cally permuted by (x0;u0), in such way that (x0;u0):V ½ + + ¡ ¡ S ; Θ(x0;u0):S ½ S ; Θ(x0;u0):S ½ V. J χ h(u;xv) = h(xu;v) = (xu) v (1) Now pure spinor formalism is briefly reviewed. For more details, see e.g. [2–4, 6–8]. Given the complex C2n vector where ( ¢ )χ is a dual operator from S§ to (S§)¤. A metric space and its corresponding Clifford algebra C`(2n;C) with B˘ : E £ E ! R is defined from the spinor metric h and the generators xa, a spinor u is a vector of the 2n-dimensional rep- metric g: resentation space of C`(2n;C) = End S, defined by the Car- tan’s equation zu = 0. For u 6= 0, z 2 C2n may only be null. B˘(φ1;φ2) = g(x1;x2) + h(u1;u2) + h(v1;v2): (2) Defining the volume element z, Weyl spinors u§ are defined R. da Rocha 1139 by z(1§x2n+1)u§ = 0, and are elements of the representation space of the even subalgebra C`+(2n;C). Each Weyl spinor n¡1 u§ spans a 2 -dimensional spinor space, and the expres- 2ξ0 = e1 + e˚3; 2ξ1 = e2 + e˚1; 2ξ2 = ie3 + e˚1; sion z(1 § x2n+1)u§ = 0 defines d-dimensional totally null ξ ξ ξ § 2 3 = e1 ¡ e˚3; 2 4 = e2 ¡ e˚1; 2 5 = ie3 ¡ e˚1 (4) planes Td(u§). It may be shown that every u§ 2 S , where C`+(2n;C) = End S§ define such planes. For d = n, that is, where 2g(ξ ;ξ ) = 1 = ξ ξ + ξ ξ ; 0 · i · 2 [6]. for maximal dimension of the totally null planes T (u ) := i 3+i i 3+i 3+i i n § Choosing the subspace of the exterior algebra, the underlying M(u ), the corresponding Weyl spinor u is named simple or § § vector space of the Clifford algebra, as pure and, as shown by Cartan, M(u§) and §u§ are isomor- phic (mod Z2). All Weyl spinors are simple or pure for n · 3. Λ2 3 3 Λ6 3 3 A = C(R © R ) © C(R © R ); (5) III. POINCARE´ SUPERALGEBRA IN R3 in particular the following products can be obtained: In four dimensions the (translational) super-Poincare´ alge- ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ bra [1] is given by 3 4 5 M 1 4 = 2 5 3; 3 4 5 M 2 3 5 = ¡2 5 3; ξ ξ ξ M ξ ξ ξ = 2ξ ξ ; ξ ξ ξ M ξ ξ = 2ξ ξ : (6) µ 1 2 3 1 4 5 1 3 1 2 3 3 4 2 3 fQ ;Q g = 2σ P ; [P ;Pν] = 0; a b ab˙ µ µ From the properties fvξ ξ ξ ;wξ ξ ξ g = ¡(vξ ξ ξ ) M fQa;Qbg = fQa˙;Qb˙ g = [Pµ;Qa] = [Pν;Qb˙ ] = 0: (3) 3 4 5 3 4 5 3 4 5 (wξ3ξ4ξ5); fvξ3ξ4ξ5;wξ3ξ4ξ5g = fvξ3ξ4ξ5;wξ3ξ4ξ5g = § The operators Pµ;Qa can be written, in terms of the superspace 0; [ψ;φ] = 0; where v;w 2 A , ψ;φ = ξAei or ξAe˚i, and using coordinates, as Pµ = i∂µ, and the results in [6], if we define µ b˙ b µ Q = ∂θa ¡ iσ θ ∂ ; Q = ¡∂θa˙ + iθ σ ∂ a ab˙ µ a˙ ba˙ µ Qµ Qµ˙ Pµ µ = 0 ξ ξ ξ + ξ ξ ξ ξ ξ ξ ξ + ξ ξ ξ ξ ξ ξ + ξ ξ M ξ ξ We now extend the results in [6], where the Poincare´ super- 1 4 3 0 3 4 0 2 4 5 3 4 5 2 4 5 0 1 2 3 µ = 1 ξ2ξ4ξ5 ¡ ξ1ξ3ξ5 ¡ξ2ξ3ξ4 ¡ ξ1ξ2ξ5 ξ1ξ0ξ5 + ξ3ξ4 M ξ1ξ2ξ5 algebra is extended by generalizing Prop.1. Instead of con- µ = 2 ξ3ξ4ξ5 + ξ1ξ3ξ4 ¡ξ3ξ4ξ5 + ξ1ξ2 ξ3ξ4 M ξ1ξ3ξ4ξ5 ¡ ξ4 sidering a vector space V, it is possible to consider any sub- µ = 3 ξ0ξ4ξ5 ¡ ξ3ξ4ξ5 ξ3ξ4ξ5 + ξ0ξ2ξ5 ξ2ξ3ξ5 M ξ5 ¡ ξ0ξ3ξ5 space A of the Clifford algebra C`(V;g). Denoting A§ ⊆ C`§(V;g), where C`¡(V;g) denotes the subspace of multivec- tors of C`(V;g) written as the Clifford product of an odd num- ber of vectors in V, the proposed generalization in [6] allows it can be shown that such operators satisfy eqs.(3), where the two possibilities: a) If dim(V) = n is such that n=2 is odd, we super-Poincare´ algebra is derived from a generalization of the have: A+ M S+ ⊆ S+ S+ M S¡ ⊆ A+; S¡ M A+ ⊆ S¡; b) Chevalley product, between even subspaces of a Clifford alge- If dim(V) = n is such that n=2 is even, it follows that bra, pure spinor formalism.
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  • Generalized Reduction and Pure Spinors

    Generalized Reduction and Pure Spinors

    Instituto Nacional de Matem´aticaPura e Aplicada Generalized reduction and pure spinors. Author: Thiago L. Drummond Rio de Janeiro 2010 2 Agradecimentos. Agrade¸co`aminha fam´ıliao apoio incondicional `aminha longa jornada longe de Bras´ılia,sem eles nada seria poss´ıvel. A` Francyne, por estar ao meu lado nos ´ultimos3 anos, per´ıodo mais cr´ıtico no doutorado. Ela foi meu porto seguro nas muitas idas e vindas ao mundo das id´eias. Aos meus amigos novos e antigos, sempre me proporcionando momentos de alegria. Principalmente, `aJu, Renatinha, Caio, Andr´e,Arthur, Marcus Vin´ıcius, Alvarenga e Bruno. Aos meus colegas de faculdade, Santiago, Lu- cas e Mauro. Eles tiveram uma influ^enciaenorme na minha forma¸c~aocomo matem´atico.Ao Professor Celius Magalh~aes,t~aodecisivo na minha entrada no mundo da pesquisa como tutor do PET. Aos meus amigos do IMPA, Thiago Fassarela, Luiz Gustavo, Rener, Andr´e Contiero. Aos professores do IMPA, Carlos Isnard, Marcelo Viana, Jorge Vit´orio. Agrade¸comuito aos meus colegas Cristi`anOrtiz, Alejandro Cabrera e Fer- nando del Carpio, pelas muitas conversas, palestras e troca de id´eiasque me ajudaram enormemente `aconcluir essa tese. A` meu orientador Henrique Bursztyn por sua confian¸cae enorme generosi- dade. Seu constante apoio e disponibilidade para conversar sobre minhas id´eias foram fundamentais para a conclus~aoda tese. Finalmente, ao CNPq e ao MCT pelo aux´ıliofinanceiro 1 2 Contents 1 Introduction. 5 1.1 Generalized geometry. 5 1.2 Generalized reduction. 8 1.3 The pure spinor point of view. 9 1.4 Contents. 11 2 Reduction on the linear algebra setting 15 2.1 Extensions of finite-dimensional vector spaces.