Supertwistors, massive superparticles and k-symmetry
aa r X i v : . [ h e p - t h ] J a n Supertwistors, massive superparticlesand κ -symmetry J.A. de Azc´arraga,Departamento de F´ısica Te´orica and IFIC (CSIC-UVEG),46100-Burjassot (Valencia), SpainJ. M Izquierdo,Departamento de F´ısica Te´orica, Universidad de Valladolid,47011-Valladolid, Spain,andJ. LukierskiInstitute of Theoretical Physics, Wroc law University,50-204 Wroc law, Poland
August 15, 2008; revised November 28
Abstract
We consider a D = 4 two-twistor lagrangian for a massive particlethat incorporates the mass-shell condition in an algebraic way, andextend it to a two-supertwistor model with N = 2 supersymmetryand central charge identified with the mass. In the purely super-twistorial picture the two D = 4 supertwistors are coupled through aWess-Zumino term in their fermionic sector. We demonstrate how the κ -gauge symmetry appears in the purely supertwistorial formulationand reduces by half the fermionic degrees of freedom of the two super-twistors; a formulation of the model in terms of κ -invariant degrees offreedom is also obtained. We show that the κ -invariant supertwistorcoordinates can be obtained by dimensional ( D =6 → D =4) reductionfrom a D = 6 supertwistor. We derive as well by 6 → N = 2, D =4 massive superparticle model with Wess-Zumino termintroduced in 1982. Finally, we comment on general superparticlemodels constructed with more than two supertwistors. Introduction
The conformal structure of twistor theory (see e.g. [1]-[4]) implies that rel-ativistic particles described by single twistors are massless [1, 3, 5, 6]. Todescribe massive particles at least two twistors are needed [7, 8, 3],[9]-[13]( cf. [14]). Indeed, with two D = 4 twistors Z Ai = ( λ αi , ¯ ω ˙ αi ) , i = 1 , , α, ˙ α = 1 , , (1.1)the standard formula for the composite fourmomentum p α ˙ β = λ αi ¯ λ ˙ βi , p µ = √ p α ˙ β ( σ µ ) ˙ βα , (1.2)implies the algebraic relation p µ p µ = p α ˙ β p ˙ βα = | M | , (1.3)where the Lorentz-invariant bilinear M = √ λ αi λ αi ( ¯ M = √ ¯ λ ˙ αi ¯ λ ˙ αi ) (1.4)gives the composite complexified mass and breaks the conformal invariancedown to the Poincar´e one. To obtain a real mass m it suffices to consider atwistor theory invariant under the phase transformations λ ′ αi = e iϕ λ αi , ¯ λ ′ ˙ αi = e − iϕ ¯ λ ˙ αi ; (1.5)by a suitable gauge fixing, a real M → | M | = m is obtained.Twistorial particle models constructed from several twistors are known[3, 7, 8, 9]; in particular a two-twistor model was proposed recently to de-scribe free relativistic particles with mass and spin [11]-[13]. These consider-ations confirm that the introduction of a non-vanishing Pauli-Luba´nski vec-tor, describing the relativistic spin fourvector in terms of twistor coordinates,requires at least two twistors [9]. In this paper we extend the two-twistorparticle dynamics by considering two N = 2 supertwistors to describe the de-grees of freedom of our elementary system. The supersymmetric two-twistorgeometry will be arranged in a way that leads algebraically to the conditionΞ = M , (1.6)where Ξ is the complex central charge of the D =4, N =2 superPoincar´e alge-bra. We shall use the phase symmetry (1.5) so that the value M of the central λ αi = ǫ ij λ αj ; ǫ ij = (cid:18) − (cid:19) , ǫ ij ǫ jk = δ kj . In general, a α ˙ β ≡ √ σ µα ˙ β a µ , σ α ˙ β =( σ , σ i ) , σ ˙ βα = ( σ , − σ i ) , i = 1 , ,
3. Then, a ν = √ a α ˙ β σ ν ˙ βα , a α ˙ β a ˙ βγ = δ αγ a µ a µ and a α ˙ β b ˙ βα = a · b , a = ( a ) − ¯ a . Complex conjugated spinors are denoted by ( λ αi ) ∗ ≡ ¯ λ ˙ αi ,etc. In [11, 12] the variable f = λ α λ α = √ M was used in place of M . κ -gauge transformations ina purely twistorial formulation of the massive superparticle model. Further,it may be shown that the presence of the mass in the twistorial frameworkreduces the conformal supersymmetry realized in terms of the two N =2 su-pertwistors to the N =2 superPoincar´e symmetry with a composite centralcharge given by (1.4).The plan of the paper is the following: In Sec. 2 we introduce a massivetwo-twistor particle model without supersymmetry using a hybrid spinor/spacetimeformulation. We also present there two other equivalent spacetime and purelytwistorial geometry formulations and exhibit their relation with the associ-ated six-dimensional massless particle model. In Sec. 3 we introduce the N = 2 supersymmetric extension of the bosonic massive model, with degreesof freedom described by two ( i = 1 , N = 2 ( r = 1 ,
2) supertwistors [15] Z Ri = ( Z Ai , η ri ) = ( λ αi , ¯ ω ˙ α i , η ri ) , (1.7)where Z Ai and η ri are, respectively, Grassmann even and odd. This supersym-metric model will be formulated first in a hybrid spinor/superspace geometricframework [5]. We discuss ( cf. [16, 17]) the constraints and, in particular,the fermionic first class constraints that generate the κ -transformations withodd gauge parameters [18, 19].Further, present in Sec. 3 a purely supertwistorial (two-supertwistor)formulation of our model. Its lagrangian will be the sum of two terms,i) a first one describing the free two-supertwistor lagrangian, andii) an additional bilinear term in the fermionic sector which couples the twosupertwistors; this will turn out to be a supertwistorial Wess-Zumino (WZ)term.The model is completed by adding the Lagrange multipliers that describeits algebraic constraints.Sec. 4 is devoted to quantizing the fermionic sector of the two super-twistor model, to uncover the quaternionic structure behind it and to calcu-lating the fermionic gauge κ -transformations. These κ -symmetries reduce byhalf (from four to two) the number of complex Grassmannian supertwistorialdegrees of freedom. We find new features of the two-supertwistor formula-tion with respect to single supertwistor models: the appearance of a WZ termand the presence of the fermionic κ -gauge transformations associated withthe non-physical fermionic degrees of freedom in the multi-supertwistorialformulation. If we impose the quaternionic SU (2) Majorana condition in thefermionic sector, the redundant degrees of freedom of the two supertwistorcoordinates described in the fermionic sector by the κ -transformations canbe eliminated. Subsequently, we show also in Sec. 4 that it is possible tointroduce κ -invariant fermionic variables which describe the fermionic sec-tor of our model in terms of two (rather than four) complex odd degrees offreedom. It is then seen that these κ -invariant fermionic variables can be3nterpreted as the dimensionally reduced ( D = 6 → D = 4) odd compo-nents of a six-dimensional complex supertwistor, satisfying SU (2)-Majoranaconditions.Sec. 5 exploits the above six-dimensional interpretation of the two-supertwistor model, using a pair of N = 1, D = 4 (super)twistors to describea single D = 6 (super)twistor. We perform the D = 6 → D = 4 dimensionalreduction by taking the six-momentum coordinates p , p as constants. As aresult, we recover the D =4 N =2 massive superparticle model formulated bytwo of the present authors a quarter century ago, the first one with mass inthe fermionic sector introduced by means of a WZ term [18]. The last sec-tion provides an outlook. In particular, it is argued there that using N D =4supertwistors we can extend our construction and obtain a N -supertwistorialmodel with N supersymmetries as well as with N ( N − complex compositemass-like parameters playing the role of central charges. In particular, the D = 4 , N = 4 model would be especially interesting, since its κ -invariant for-mulation could be given by the coordinates of a single D = 10 supertwistorwith octonionic structure. D = 6 and its D = 4 twistorial picture It is known that D = 4 massive free particle models can be obtained bydimensional reduction from massless particle ones in D >
4. Because thetwistorial mass (1.4) is complex, we shall take D = 6 and denote M = p − ip ,¯ M = p + ip (( p µ , p , p ) are the six-momentum coordinates) and, similarly, z = x + ix , ¯ z = x − ix . In the first order formalism, the Lagrangian of a D =6 massless particle can be written as L B = p µ ˙ x µ + 12 ( M ˙ z + ¯ M ˙¯ z ) + e p − | M | ) , µ = 0 , , , . (2.1)Using eqs. (1.2) and (1.4) the ‘mixed’ or spinor/spacetime formulation of(2.1) is obtained, in which L B = ¯ λ ˙ βi ˙ x ˙ βα λ αi + 12 √ (cid:0) λ αi λ αi ˙ z + ¯ λ ˙ αi ¯ λ ˙ αi ˙¯ z (cid:1) , (2.2)where the spacetime vector is expressed as a second rank spinor, x µ = √ ( σ µ ) ˙ βα x α ˙ β (see the first footnote) and the D =6 zero mass shell condi-tion, p k p k = p µ p µ − | M | = 0 ( k = 0 , , . . . , λ αi , ¯ λ ˙ αi . Clearly, the lagrangian (2.2)is invariant under the rigid phase transformations (1.5) if the z ’s have a ( cf. (1.5)) double U (1) charge, z ′ = e − iϕ z , ¯ z ′ = e iϕ ¯ z . (2.3)4o obtain a purely twistorial description of the lagrangian (2.2) we nowintroduce a new Weyl spinor and its conjugate ( ω αi , ¯ ω ˙ αi ) and postulate for thecomponents of the twistors Z Ai = ( λ αi , ¯ ω ˙ αi ) the following incidence relations: ω αi = i (¯ λ ˙ βi x ˙ βα + √ λ αi z ) , ¯ ω ˙ αi = − i ( x ˙ αβ λ βi + √ ¯ λ ˙ αi ¯ z ) , (2.4)( x ˙ αβ ) † = x ˙ βα ( x µ real). Eqs. (2.4) generalize the well known Penrose formula[1]-[4]. Using them, the lagrangian (2.2) can be written as L B = − i (cid:16) ˙ ω αi λ αi + ˙¯ λ ˙ αi ¯ ω ˙ αi (cid:17) + i (cid:16) ω αi ˙ λ αi + ¯ λ ˙ αi ˙¯ ω ˙ αi (cid:17) = i (cid:16) ¯ Z Ai ˙ Z Ai − ˙¯ Z Ai Z Ai (cid:17) , (2.5)where the scalar product of two twistors is given by¯ Z Ai Z Ak = ω αi λ αk + ¯ λ ˙ αi ¯ ω ˙ αk , (2.6)with ¯ Z Ai = ( ω αi , ¯ λ ˙ αi ).One can check, using eqs. (2.4) and the relations λ αi λ αj = − √ ǫ ij M , ¯ λ ˙ αi ¯ λ ˙ αj = − √ ǫ ij ¯ M , (2.7)that ¯ Z Ai Z Aj = i (cid:0) M z − ¯ M ¯ z (cid:1) δ ij . (2.8)Since ¯ Z Ak Z Ak = i (cid:0) M z − ¯ M ¯ z (cid:1) , it follows that the z, ¯ z coordinates may beremoved by the relations (2.8), which can be rewritten as (see also [6])¯ Z Ai Z Aj − δ ij ¯ Z Ak Z Ak = 0 , (2.9)an expression that does not fix the conformal norm of the twistors Z Ak butstates that the two norms are equal.Summarizing, L B in eq. (2.5) appears as the sum of two free twistorlagrangians, associated with two non-null, orthogonal twistors having thesame non-vanishing length for M = 0. To specify that the purely twistoriallagrangian (2.5) describes a massive particle one has to incorporate relations(2.9) and (1.4) by means of suitable Lagrange multipliers. D = 4 , N = 2 supersymmetry and mass a) Supersymmetrization of the model (2.2) and κ -transformations. dx α ˙ β , dz and d ¯ z by the cor-responding supertranslation-invariant ones on D = 4, N = 2 superspace ex-tenden by a central charge and parametrized by ( x α ˙ β , z, ¯ z ; θ αr , ¯ θ ˙ αr ), r = 1 , ˙ x ˙ βα → ω ˙ βα = ˙ x ˙ βα − i √ (cid:16) ˙ θ αr ¯ θ ˙ βr − θ αr ˙¯ θ ˙ βr (cid:17) , (3.1a)˙ z → ω = ˙ z + 2 i θ αr ˙ θ αr , ˙¯ z → ¯ ω = ˙¯ z + 2 i ¯ θ ˙ αr ˙¯ θ ˙ αr , (3.1b)where a r b r ≡ a r ǫ rs b s and z, ¯ z play the rˆole of the D =4, N =2 superspacecomplex central charge coordinates. These replacements in (2.2) give thesupersymmetric lagrangian L SUSY = ¯ λ ˙ βi ω ˙ βα λ αi + 12 √ (cid:0) λ αi λ αi ω + ¯ λ ˙ αi ¯ λ ˙ αi ¯ ω (cid:1) . (3.2)The action obtained from (3.2) is invariant under the supertranslationsof the N = 2 superPoincar´e group extended by a complex central charge, x ′ ˙ βα = x ˙ βα − i √ ǫ αr ¯ θ ˙ βr − θ αr ¯ ǫ ˙ βr ) ,θ ′ αr = θ αr + ǫ αr , ¯ θ ′ ˙ αr = ¯ θ ˙ αr + ¯ ǫ ˙ αr ,z ′ = z + 2 iθ αr ǫ αr , ¯ z ′ = ¯ z + 2 i ¯ θ ˙ αr ¯ ǫ ˙ αr ,λ ′ αi = λ αi , ¯ λ ′ ˙ αi = ¯ λ ˙ αi , (3.3)which leave λ αi invariant. Expression (3.1a) is also invariant under the U (2)internal transformation of the odd superspace coordinates θ αr , θ ˙ αr ; this sym-metry is broken by the ω ’s of eq.(3.1b) down to the U (2) ∩ Sp (2 , C ) = U Sp (2) ≈ SU (2) internal symmetry.The lagrangian (3.2) describes a superparticle in a mixed spinorial/superspaceconfiguration space M (6;8 | parametrized by M (6;8 | = { q M } = ( x ˙ αβ , z, ¯ z ; λ αi , ¯ λ ˙ αi | θ αr , ¯ θ ˙ αr ) , (3.4)with 4+2+8 =14 real bosonic and 8 real fermionic coordinates. The canonicalmomenta ( π αr = ∂L/∂ ˙ θ αr , etc.) P M = ∂L∂ ˙ q M ≡ ( p ˙ αβ , q, ¯ q ; ρ αi , ¯ ρ ˙ αi | π αr , ¯ π ˙ αr ) , M = 1 . . . , (3.5)define the following set of primary constraints: R α ˙ β := p α ˙ β − λ αi ¯ λ ˙ βi = 0 , The √ ω µ ( τ ) = √ σ µα ˙ β ω ˙ βα ( τ ) = ˙ x µ − i ( ˙ θ αr σ µα ˙ β ¯ θ ˙ βr − θ αr σ µα ˙ β ˙¯ θ ˙ βr ) is the pull-back to to the worldline of the particle of the super-space Maurer-Cartan one-form Π µ = dx µ − i ( dθ αr σ µα ˙ β ¯ θ ˙ βr − θ αr σ µα ˙ β d ¯ θ ˙ βr ), which is invariantunder the D = 4 , N = 2 superPoincar´e transformations in eqs. (3.3). := q − √ λ αi λ αi = q − M = 0 , ¯ R := ¯ q − √ ¯ λ αi ¯ λ αi = q − ¯ M = 0 ,R αi := ρ αi = 0 , R ˙ αi := ¯ ρ ˙ αi = 0 , (3.6)and ( ¯ G ˙ αr = − ( G αr ) + , ¯ π ˙ αr = − ( π αr ) + ) G αr := π αr + i √ p α ˙ β ¯ θ ˙ βr − iM ǫ rs θ αs = 0 , (3.7a)¯ G ˙ αr := ¯ π ˙ αr + i √ θ βr p β ˙ α − i ¯ M ǫ rs ¯ θ ˙ αs = 0 . (3.7b)Let us restrict ourselves to the set (3.7a), (3.7b) of fermionic constraints,which determine the elements of the Poisson brackets (PB) matrix C AB = (cid:18) { G αr , G βs } , { G αr , ¯ G ˙ βs }{ ¯ G ˙ αr , G βs } , { ¯ G ˙ αr , ¯ G ˙ βs } (cid:19) . (3.8)Using the canonical PB { θ αr , π βs } = ǫ αβ δ rs , { ¯ θ ˙ αr , ¯ π ˙ βs } = ǫ ˙ ǫ ˙ β δ rs , (3.9)it follows that the four 4 × C AB matrix are given by ( p ˙ βα =( p α ˙ β ) T ) C AB = 2 i (cid:18) − ǫ αβ ǫ rs M √ δ rs p α ˙ β √ δ rs p β ˙ α − ǫ ˙ α ˙ β ǫ rs ¯ M (cid:19) . (3.10)Using the formula for the determinant of a 2 × C = det A · det( D − CA − B ) , C = (cid:18) A BC D (cid:19) , (3.11)one finds that det C = 2 ( p − | M | ) . (3.12)The first constraint in (3.6) reproduces eq. (1.2), implying p − | M | = 0 (3.13)(eq. (1.3)), and thus we conclude from (3.12) that the 8 × p α ˙ γ and (3.7b) by ǫ rs M . This gives¯ C ˙ βr = π αr p α ˙ β + M √ ǫ rs ¯ π ˙ βs − i √ θ ˙ βr ( p − | M | ) = 0 . (3.14)7quation (3.14) determines in principle four complex constraints, but theircomplex-conjugate ones are equivalent to them. Indeed, if we multiply theconstraints ( C + βr = − ( ¯ C ˙ βr )) C βr = p β ˙ α ¯ π ˙ αr + ¯ M √ ǫ rs π βs − i √ θ βr ( p − | M | ) = 0 , (3.15)by p ˙ γβ we get back the constraints (3.14), plus terms that contain the factor( p − | M | ). Therefore our model has effectively only four real first class con-straints, which in the 8-dimensional real Grassmann odd sector of the config-uration space (3.4) generate four real odd gauge transformations. These arethe κ -symmetries of the model (3.2) that eliminate the unphysical fermionicgauge degrees of freedom i.e. , half of the odd N =2 superspace coordinates.The explicit expression of these κ -transformations, parametrized by a pair ofanticommuting Weyl spinors κ αr and their complex conjugates ¯ κ ˙ αr , is givenby the graded Poisson brackets δ κ θ αr := { κ βs C βs , θ αr } = κ βs { C βs , θ αr } = − ǫ rs ¯ M √ κ αs ,δ ¯ κ θ αr := { ¯ κ ˙ βs ¯ C ˙ βs , θ αr } = ¯ κ ˙ βs { ¯ C ˙ βs , θ αr } = − p α ˙ β ¯ κ ˙ βr ,δ κ ¯ θ ˙ αr := { κ βs C βs , ¯ θ ˙ αr } = κ βs { C βs , ¯ θ ˙ αr } = p ˙ αβ κ βr ,δ ¯ κ ¯ θ ˙ αr := { ¯ κ ˙ βs ¯ C ˙ βs , ¯ θ ˙ αr } = ¯ κ ˙ βs { ¯ C ˙ βs , ¯ θ ˙ αr } = − ǫ rs M √ κ ˙ αs . (3.16)If δ κ now denotes the variation under both κ and ¯ κ , the variation of thefour-dimensional spinor ( θ αr , ¯ θ ˙ αr ) is written as δ κ (cid:18) θ αr ¯ θ ˙ αr (cid:19) = − ǫ rs δ βα ¯ M √ − δ rs p α ˙ β δ rs p ˙ αβ − ǫ rs δ ˙ α ˙ β M √ ! (cid:18) κ βs ¯ κ ˙ βs (cid:19) . (3.17)The above matrix can be rewritten as the product − ǫ rt δ γα ¯ M √ − ǫ rt δ ˙ α ˙ γ M √ ! δ ts δ βγ − ǫ ts √ M p γ ˙ β ǫ ts √ M p ˙ γβ δ ts δ ˙ γ ˙ β ! . (3.18)The first matrix just produces a scaling of the κ -transformations, and thesecond matrix is the sum of the four-dimensional unit matrix plus one withonly non-zero 2 × i.e. , it is a projec-tion operator. Thus, δ κ θ r in eq. (3.17) has the standard projector structureeffectively halving the parameters of the κ -transformations.The behaviour of the remaining configuration space variables (3.4) under δ κ is given by δ κ x ˙ βα = i √ δ κ θ αr ¯ θ ˙ βr − θ αr δ κ ¯ θ ˙ βr ) , κ z = − iθ αr δ κ θ αr ,δ κ ¯ z = − i ¯ θ ˙ αr δ κ ¯ θ ˙ αr ,δ κ λ αi = δ κ ¯ λ ˙ αi = 0 . (3.19)These relations differ from eqs. (3.3) by the replacement ǫ → − δ κ θ ; the rela-tive minus sign characterizes κ -symmetry as a ‘right’ (local) supersymmetry(see e.g. [20]). The κ -transformations (3.16), (3.19), may be used to checkexplicitly the κ -invariance of the action based on (3.2). b) From the hybrid (spinor/spacetime) formulation to the purely super twisto-rial one. To introduce a purely supertwistorial formulation of the model (3.2), eqs.(2.4) are further extended in the two supertwistors case by ω αi = i ¯ λ ˙ βi ( x ˙ βα − i √ θ αr ¯ θ ˙ βr ) + i √ ǫ ij ( λ αj z + 2 iλ βj θ αr θ βr ) , ¯ ω ˙ αi = − i ( x ˙ αβ + i √ θ βr ¯ θ ˙ αr ) λ βi − i √ ǫ ij (¯ λ ˙ αj ¯ z − i ¯ λ ˙ βj ¯ θ ˙ αr ¯ θ ˙ βr ) , (3.20)which are the generalized incidence relations for the bosonic components ofthe Z Ai part of Z Ri (eq. (1.7)). These relations, which involve the fermionicsuperspace coordinates besides the real spacetime x ˙ αβ and complex z vari-ables ( cf. eq. (2.4)), have to be complemented by those affecting the oddcomposite variables η ri , ¯ η ri that make up [15] the coordinates triple of thetwo D =4 N = 2 supertwistors,¯ Z Rri = ( ω αi , ¯ λ ˙ αi , ¯ η ri ) , Z R ri = ( λ αi , ¯ ω ˙ αi , η ri ) , i = 1 , , (3.21)where r = 1 , N =2 supertwistor index. Eqs. (3.20) are accordinglysupplemented by [15] η ri = √ θ αr λ αi , ¯ η ri = √ θ ˙ αr ¯ λ ˙ αi . (3.22)Using (1.4), the above expressions can be inverted with the result θ αr = λ αj η rj M , ¯ θ ˙ αr = ¯ λ ˙ αj ¯ η rj ¯ M . (3.23)It follows from the definition (3.22) and from eqs. (3.3) that under super-symmetry the odd supertwistor variables transform as δ ǫ η ri = √ ǫ αr λ αi , δ ¯ ǫ ¯ η ri = √ ǫ ˙ αr ¯ λ ˙ αi . (3.24)Finally we note that, in terms of the fermionic composite coordinates, the ω, ¯ ω components (3.20) of the two N = 2 supertwistors can be rewritten as ω αi = i (¯ λ ˙ βi x ˙ βα + 1 √ λ αi z ) + ( θ αr ¯ η ri − θ αr η ir ) , See [21] for similar relations in the framework of D =6 Lorentz harmonics. ω ˙ αi = − i ( x ˙ αβ λ βi + 1 √ λ ˙ αi ¯ z ) − (¯ θ ˙ αr η ri − ¯ θ ˙ αr ¯ η ir ) , (3.25)which are the supersymmetric generalizations of eqs. (2.4).The supersymmetric extension (3.2) of the bosonic model (2.2) can bewritten in the form L SUSY = L B − i (cid:16) λ αi ˙ θ αr ¯ η ri + ¯ λ ˙ αi ˙¯ θ ˙ αr η ri (cid:17) + i (cid:16) M θ βr ˙ θ βr + ¯ M ¯ θ ˙ βr ˙¯ θ βr (cid:17) . (3.26)A calculation now shows (modulo a total time derivative) that, after intro-ducing (3.23), the purely supertwistorial lagrangian for our model reads L SUSY = L SUSY + L SUSY , L SUSY ≡ i ( ¯ Z Ai ˙ Z Ai − ˙¯ Z Ai Z Ai ) , L SUSY ≡ i √ η ri ˙¯ η ri − ˙ η ri ¯ η ri ) − i √ (cid:0) ˙ η ri η ri + ˙¯ η ri ¯ η ri (cid:1) , (3.27)where the scalar product of the ¯ Z Ai = ( ω αi , ¯ λ ˙ αi ) and Z Ai = ( λ αi , ¯ ω ˙ αi ) twistors,in which ω αi , ¯ ω ˙ αi are those in (3.20), is given by eq. (2.6). Using eq. (3.23) itis seen that the L SUSY part of L SUSY depends only on η , ¯ η, λ , ¯ λ and on thetime derivatives of the λ ’s; all the dependence of L SUSY on the derivativesof the η, ¯ η variables is contained in L SUSY above.The SU (2 , | N =2 supertwistors (3.21) is definedby ¯ Z Ri Z Rj = ¯ Z Ai Z Aj + √ η ri η rj . (3.28)The bosonic subgroup of SU (2 , |
2) is SU (2 , × U (2) ≈ f SO (2 , × U (2),of which SU (2 ,
2) acts on the A indices and U (2) on the index r ; each factorpreserves the two terms in (3.28) independently. Using (3.28), the lagrangian(3.27) takes the form L SUSY = i (cid:16) ¯ Z Ri ˙ Z Ri − ˙¯ Z Ri Z Ri (cid:17) − i √ (cid:0) ˙ η ri η ri + ˙¯ η ri ¯ η ri (cid:1) . (3.29)The first term in (3.29) is the free lagrangian for two supertwistors, whichare coupled only through the second term. This last one is the pull-backto the worldline of the supertwistor space one-form dη η + d ¯ η ¯ η , a potentialone-form of the closed, supersymmetry-invariant two-form dη dη + d ¯ η d ¯ η and,thus, L SUSY is the supertwistorial WZ part of L SUSY (for the geometry ofWZ rerms, see [22]). In fact, using δ ǫ η , δ ¯ ǫ ¯ η in (3.24), it is seen that dηη + d ¯ η ¯ η is invariant modulo an exact term. The different components of the (super)twistors are not dimensionally homogeneous.In natural units, [ λ ] = L − , [¯ ω ] = L , [ η ] = L ; the scalar products of (super)twistors(eqs. (2.6), (3.28)) are, of course, dimensionless. Note also that (1.7) implies that thecomponents λ , ¯ ω and η of the (super)twistors transform in the same manner under asymmetry group acting on the index i that labels the two (super)twistors.
10e calculate now, using (3.20), (3.22) and (2.6) the value of the scalarproducts (3.28), and obtain¯ Z Ri Z Rj = i (cid:0) M z − ¯ M ¯ z (cid:1) δ ij − √ (cid:0) η ri η rj + ¯ η ri ¯ η rj (cid:1) , (3.30)( cf. (2.8)). Since η ri η ri ≡ η ’s odd Grassmann parity, oneobtains ¯ Z Rk Z Rk = i ( M z − ¯ M ¯ z ) (3.31)as in the non-supersymmetric case. Proceeding as in Sec. 2 and using (3.31),the constraints (3.30) may be rewritten just in terms of supertwistorial vari-ables as ¯ Z Ri Z Rj − δ ij ¯ Z Rk Z Rk + 1 √ (cid:0) η ri η rj + ¯ η ri ¯ η rj (cid:1) = 0 , (3.32)which extend those in (2.9) to the supertwistorial case. The constraints (3.32)and the relations (1.4) that characterize the model (3.29) can be incorporatedto it by means of suitable lagrange multipliers.We now turn to the fermionic gauge symmetries of our model. κ -symmetry and κ -invariant formulation ofthe fermionic sector Let us consider now the L part of the action (3.27) involving the derivativesof the fermionic variables. Introducing new complex Grassmann variables( η ri = ǫ rs ǫ ij η sj etc.) ξ ri ≡ η ri + ¯ η ri ( ¯ ξ ri ≡ ¯ η ri + η ri ) , (4.1)and using the Grassmann nature of η ri , ¯ η ri one gets, up to a total derivative, L SUSY = i √ ξ ri ˙ ξ ri ≡ i √ ǫ rs ǫ ij ξ sj ˙ ξ ri . (4.2)If we observe that the variables (4.1) satisfy the SU (2)-reality conditionrepresenting quaternionic structure ξ ri = ¯ ξ ri = ǫ rs ǫ ij ¯ ξ sj , (4.3)we see that the action (4.2) can be written in two different ways, L SUSY = i √ ξ ri ˙ ξ ri = √ ξ i ˙ ξ i . (4.4)where we have chosen ξ i ≡ ξ i and used (4.3).11hus, the fermionic action (4.4) effectively depends on two complex Grass-mann variables only. The κ -transformations with Grassmann parameters ρ ri that satisfy the condition ( cf. (4.3)) such that ¯ ρ ri = − ρ ri δη ri = ρ ri , δ ¯ η ri = − ρ ri , (4.5)leave invariant the variables ξ ri in (4.1) as well as the action (3.27). Using in(3.17) the spinor bilinears (1.2) for p α ˙ β , and comparing eq. (3.17) with (4.5)one obtains, using eqs. (1.2), (2.7) and (3.22), ρ ri = − ¯ M ǫ rs λ αi κ αs + ǫ is M ¯ λ ˙ βs ¯ κ ˙ βr = − ¯ ρ ri . (4.6)It is easy to deduce the reality condition (4.3) if we assume that thefermionic Grassmann sector of the D = 4 supertwistor degrees of freedomis described by a single quaternionic D = 6 supertwistor coordinate: itsGrassmann sector is given by the odd quaternionic variable ξ = ξ (0) + ξ ( r ) e r ,where ξ (0) , ξ ( r ) ( r = 1 , ,
3) are four real Grassmann variables and e r e s = − δ rs + ǫ rst e t . In the matrix representation obtained by replacing the quater-nionic units by the Pauli matrices,1 → σ , e r ↔ − iσ r , (4.7)the quaternionic variable ξ becomes the 2 × µ = 0 , r ) ξ = σ ξ (0) − iσ i ξ ( i ) = (cid:18) ξ (0) − iξ (3) − iξ (1) − ξ (2) − iξ (1) + ξ (2) ξ (0) + iξ (3) (cid:19) . (4.8)It is trivial to check that ξ ri as given by the elements of the matrix (4.8) satis-fies the subsidiary condition (4.3); clearly, the hermitian matrix ξ † describesthe conjugate quaternion ξ = ξ (0) − ξ ( r ) e r . We see therefore that our super-twistor model, described by the lagrangian (3.27), reflects the quaternionicstructure inherent to the D = 6 geometry. The κ -transformations in ourformulation with two independent complex D = 4 supertwistors representthe redundant degrees of freedom which disappear if we pass to the N = 1, D = 6 quaternionic supertwistor coordinates.The complex Grassmann coordinates ξ r (= ξ r , r = 1 ,
2) satisfy, when thecanonical quantization of the action (4.4) is performed, the relations ( ~ = 1) { ξ r , ¯ ξ s } = δ rs , { ξ r , ξ s } = { ¯ ξ r , ¯ ξ s } = 0 . (4.9)If we supplement the above anticommutators with the canonical twistorialequal-time commutators,[ λ αi , ω βj ] = iδ βα δ ij , [¯ λ ˙ αi , ¯ ω ˙ βj ] = iδ ˙ β ˙ α δ ij , (4.10)which follow from the symplectic twistorial two-form, we can postulate thefollowing formulae for the four complex supercharges describing the algebraicbasis of N = 2, D = 4 supersymmetry algebra Q (1) α = λ iα ξ i , ¯ Q (1)˙ α = ¯ λ i ˙ α ¯ ξ i ,Q (2) α = λ αi ¯ ξ i , ¯ Q (2)˙ α = ¯ λ ˙ αi ξ i . (4.11)12ndeed, using the canonical commutation relations (4.9), (4.10) one obtains n Q ( r ) α , ¯ Q ( s )˙ β o = δ rs λ αi ¯ λ ˙ βi = δ rs P α ˙ β , n Q ( r ) α , Q ( s ) β o = ǫ rs λ αi λ iβ = − √ ǫ rs ǫ αβ M , n ¯ Q ( r )˙ α , ¯ Q ( s )˙ β o = ǫ rs ¯ λ ˙ αi ¯ λ i ˙ β = − √ ǫ rs ǫ ˙ α ˙ β ¯ M , (4.12)which reproduces the fermionic sector of the N = 2, D = 4 supersymmetryalgebra with a composite central charge M . D = 4 , N = 2 superparticlemodel with WZ term from the D = 6 su-pertwistorial framework Our lagrangian (3.2) may be written in terms of D = 6 four-component Weylspinors. We introduceΛ A = (cid:18) λ α ¯ λ ˙ α (cid:19) , Λ A = (cid:18) − λ α ¯ λ ˙ α (cid:19) . (5.1)The complex spinors (5.1) satisfy the D =6 symplectic Majorana reality con-dition Λ Ar = ǫ rs C A ˙ A Λ ˙ As , (5.2)where Λ ˙ A ≡ (Λ A ) ∗ and C = (cid:18) ǫ αβ − ǫ ˙ α ˙ β (cid:19) , C T = − C , C = − , (5.3)is the charge conjugation matrix.The D = 6 generalization of Pauli matrices can be obtained by replacingin the expression of the D = 4 Pauli matrices the imaginary unit by the threequaternionic imaginary units e i ( i = 1 , ,
3) as follows σ k = (cid:18) , (cid:18) − e i e i (cid:19) , (cid:18) (cid:19) , (cid:18) − (cid:19)(cid:19) , (5.4) k = 0 , i, ,
5. Making a similarity transformation σ ′ k = Aσ k A − with A = √ ( σ + σ ) ( A = 1 , A = A − ) and using the realization (4.7) in the expres-sion of the σ ′ k , we obtain six complex 4 × A ˙ B in the form (cid:18) , (cid:18) − σ i σ i (cid:19) , − (cid:18) (cid:19) , i (cid:18) − (cid:19)(cid:19) . (5.5)13ince C Σ C − = Σ ∗ , the undotted A and the dotted ˙ A indices transformsimilarly under SO (1 ,
5) and, unlike in the D = 4 case, there is no metricallowing us to raise and lower the D =6 Weyl spinorial indices.Let Λ A ≡ Λ A . It follows that Λ A (Σ µ ) A ˙ B Λ ˙ B = λ αi ( σ µ ) α ˙ β ¯ λ ˙ βi = √ p µ , Λ A (Σ ) A ˙ B Λ ˙ B = − √ M + ¯ M ) = √ p , Λ A (Σ ) A ˙ B Λ ˙ B = − i √ M − ¯ M ) = √ p , (5.6)where we have used eqs. (1.2) ((1.4)) in the first (second and third) expressionabove and that M = p − ip , ¯ M = p + ip . This gives the algebraic D =6zero mass shell condition, p k p k = p µ p µ − p − p = 0 . (5.7)Then, the bosonic lagrangian (2.2) may be now written in a six-dimensionalform as the lagrangian for the D =6 massless particle in a hybrid spino-rial/spacetime formulation, L B = √ A ˙ x A ˙ B Λ ˙ B , (5.8)where ˙ x A ˙ B = 12 ˙ x k (Σ k ) A ˙ B , (5.9)and the D =6 zero mass condition (5.7) is built in algebraically.To supersymmetrize the bosonic lagrangian (5.8) we introduce D =6 su-perspace with Weyl-Grassmann spinors. The lagrangian (3.2) is then ob-tained by the replacement˙ x ˙ BA −→ ω ˙ BA = ˙ x ˙ BA − i ( ˙ θ A ¯ θ ˙ B − θ A ˙¯ θ ˙ B ) , (5.10)where we use the following four-component D =6 Grassmann spinors θ A = (cid:18) θ α ¯ θ ˙ α (cid:19) , θ ˙ A = (cid:18) ¯ θ ˙ α θ α (cid:19) . (5.11)The substitution (5.10) may be now written in terms of a pair of two-dimensional D = 4 Weyl spinors as˙ x µ −→ ω µ = ˙ x µ − i ( ˙ θ αr ( σ µ ) αβ ¯ θ ˙ βr − θ αr ( σ µ ) α ˙ β ˙¯ θ ˙ βr ) , Since the Σ’s are now four-dimensional and ˜Σ= Σ ˙ BA ≡ (Σ , − Σ s ), s = 1 , . . . , k e Σ l + Σ l e Σ k = 2 η lk ( η kl = diag(1 , − , . . . − a A ˙ B ≡ Σ kA ˙ B a k , b ˙ BA ≡ Σ ˙ BA k b k . As a result, we have a A ˙ B b ˙ BA = a · b as for D =4, but now a A ˙ B Σ ˙ BA k = a k , a ˙ BA Σ kA ˙ B = a k , k = µ, ,
5. The various √ D =4-adaptedfactors in eqs. (1.2), (1.4) (see footnote 1) within a D =6 context. z −→ ω = ˙ z + 2 i θ αr ˙ θ αr , ˙¯ z −→ ¯ ω = ˙¯ z + 2 i ¯ θ ˙ αr ˙¯ θ ˙ αr . (5.12)It may be checked that the supersymmetric lagrangian (3.2) in D = 6notation can be written as follows L SUSY = √ A ω A ˙ B Λ ˙ B = 1 √ ω k Λ A Σ kA ˙ B Λ ˙ B , (5.13)where k = 0 , , . . . , ω k = ( ω µ , (¯ ω + ω ) , i (¯ ω − ω ) ) and ω and ¯ ω aregiven in eq. (3.1b).In our model (see (3.2) or (5.13)) the central charge coordinates z , ¯ z , aswell as the dual central charges M , ¯ M , are dynamical variables; however thecentral charges are constants on-shell (the field equations are ˙ M = ˙¯ M = 0).The static approximation M = ¯ M = c onst. can be achieved consistently inour first order formulation by the D = 6 → D = 4 reduction procedure intarget space. We set in eq. (5.13)) p = p = const. p + p = m = const. (5.14)Putting p = m sin ϕ , p = m cos ϕ , M = me − iϕ , (5.15)where ϕ is a constant phase, the dimensional reduction L SUSY → L
SUSYD =4 gives L SUSYD =4 = p µ ω µ + im (cid:16) e − iϕ θ αr ˙ θ αr + e iϕ ¯ θ αr ˙¯ θ αr (cid:17) + e p − m ) , (5.16)where the D =4 mass shell condition, eq. (5.7) after using (5.15), is imposedby a Lagrange multiplier. We further observe that we can set ϕ = 0 because ω µ is invariant under the constant phase transformations θ ′ αr = e i ϕ θ αr , ¯ θ ′ ˙ αr = e − i ϕ ¯ θ ˙ αr . (5.17)Subsequently, we obtain the first order formulation of the D = 4, N = 2superparticle model with WZ term introduced by the two of present authors[18]. Indeed, after eliminating p µ and e from (5.16) by the algebraic fieldequations we obtain L SUSYD =4 = m p ω µ ω µ + i m ( θ αr ˙ θ αr + ¯ θ αr ˙¯ θ αr ) . (5.18)The model (5.18) corresponds to the case where the central charge is rep-resented by a constant real mass parameter. It is worth stressing here that See [23, 24]; for the application of the dimensional reduction procedure to superpar-ticles see [25]-[27]. There, one performs the dimensional reduction procedure in targetspace, in consistency with the on-shell values of the reduced solutions. Note that thereis no need of restricting x , x because in the first order formalism the only term in thelagrangian depending on these variables becomes a total derivative for constant p , p . m in front of the first (Nambu-Goto-like) and sec-ond (WZ) term in the Lagrangian (5.18) corresponds in our two-supertwistormodel to the equality of the numerical coefficients in front of the two terms in(3.27), necessary for the invariance under the local κ -transformations (4.5) intwo-supertwistor space. In the N = 2 superspace formulation, the equalityof the ‘bosonic’ and ‘fermionic’ masses in the two terms of the lagrangian(5.18) allows as well for the invariance under the κ -gauge transformations(3.16), (3.19), which are necessary to balance the number of fermionic andbosonic degrees of freedom in the p =0 super- p -brane model, as it is the casefor extended objects in general [28]. We have considered here the supertwistorial formulation of D =4 superpar-ticles with mass and N = 2 supersymmetry. Our interest in the massivecase is due to the fact that it is the massive superparticle model with WZterm [18], rather than the massless one, which is the pointlike p =0 analogueof the extended p > p -branes. By constructing a model with two N = 2, D = 4 supertwistors we have been able to study the appearance ofboth the WZ term and the fermionic κ -transformations in a (super)twistorialframework.It is known that massless superparticles with N -extended supersymmetrycan be described by a single N -extended supertwistor Z = ( λ α , ¯ ω ˙ α , η r ) with N complex Grassmann coordinates η r , r = 1 . . . N . The degrees of freedomof one superstwistor are invariant under κ -transformations i.e. , for masslesssuperparticles the coordinates of the single supertwistor already describe the‘physical’ degrees of freedom. Thus, since the N -extended D = 4 superspacecontains 2 N complex Grassmann coordinates θ αr ( α = 1 , r = 1 , . . . N ), theequivalence between the supertwistorial and the superspace formulations ofthe massless superparticle requires the removal of half of the odd superspacedegrees of freedom by means of the fermionic gauge κ -transformations [18,19].However, if we wish to describe a D =4 massive superparticle in a super-twistorial approach, we necessarily need at least two supertwistors to allowfor a non-vanishing mass [7, 8, 3],[9]-[13]. In this paper we have consid-ered the N = 2 supersymmetry case using for two D = 4 supertwistors,which give rise to a supertwistorial WZ term and to the κ -gauge transfor-mations. Indeed, it turns out that the number of Grassmannian degrees offreedom of our supertwistorial model is the same as in N =2 superspace (seeeqs. (3.23)), and thus the familiar local fermionic transformations of thesuperspace framework must appear as well in the purely two-supertwistorialdescription.In order to obtain the κ -invariant formulation of our model we observethat two N = 2, D = 4 supertwistors can be obtained from a single N = 1,16 = 6 supertwistor provided that the odd D = 4 supertwistor coordinatessatisfy the SU (2)-Majorana reality condition (eq. (4.3)). One can concludetherefore that our κ -transformations account for the degrees of freedom thatdisappear when we use such a pair of constrained D = 4 complex super-twistors, equivalent to the single D = 6 supertwistor with associated quater-nionic geometry.For the D =4, N -extended supersymmetry case one can introduce N ( N − mass-like parameters corresponding to as many complex central charges ( i =1 . . . N >
2) [29] n Q iα , ¯ Q j ˙ β o = δ ij P α ˙ β , (cid:8) Q iα , Q jβ (cid:9) = ǫ αβ Ξ ij , n ¯ Q i ˙ α , ¯ Q j ˙ β o = ǫ ˙ α ˙ β ¯Ξ ij , (6.1)where Ξ ij = − Ξ ji is the complex N × N skewsymmetric matrix of generatorsof the central charges . To introduce in a supertwistorial formalism all possi-ble massive parameters as independent spinorial bilinears, one can generalizethe relation Ξ = M (eq. (1.6)) to allow for antisymmetric charges as followsΞ ij = − Ξ ji ∝ λ iα λ αj , i, j = 1 , . . . N (6.2)( cf. (1.4)). For such a purpose N independent copies of N -extended super-twistors (1.7) are needed, with N complex fermionic degrees of freedom .Superparticle models characterized by having several mass-like parameterscorresponding to the central charges (6.2) are not known, but by generalizingof our two-supertwistor framework one may guess how to construct a super-twistor lagrangian in terms of N > N -extended supertwistors.One of the primary tasks in building such a model in D =4 would be to de-scribe the corresponding generalized κ -transformations which would requirethe maximal number N ( N −
1) of odd complex parameters.The most interesting case one could study is that of a D = 4, N = 4model with 12 κ -gauge odd parameters. If we could introduce four D = 4supertwistors with suitable constraints to describe the degrees of freedom ofan octonionic N = 1, D = 10 supertwistor [6], the κ -invariant formulationwould determine the corresponding N = 4 , D = 4 supertwistor dynamicswith octonionic structure .We conclude by mentioning that, recently, there has been a renewed in-terest in twistor theory and in the general Penrose programme as a result ofthe applications of twistors and supertwistors in various modern contexts as e.g. , in the analysis and computation of N =4 Yang-Mills amplitudes [30, 31],in various (super)string models [30, 32, 33, 34] or in connection with an al-gebraic description of the BPS states in M-theory [35]. One may assume, Since for D =4 there are two linearly independent constant spinors, the N >
D > [ D ] ). In such a formalism a single octonion coordinates spanning R would be described byfour complex split octonionic units (see e.g. [36]). It is unclear, however, whether theproblems associated with non-associativity can be avoided. Acknowledgments
The authors would like to thank Dima Sorokin for valuable discussions. Thiswork has been partially supported by research grants from the Spanish Min-istry of Science and Innovation (FIS2008-01980, FIS2005-03989) and EUFEDER funds, the Junta de Castilla y Le´on (VA013C05), the Polish Ministryof Science and Higher Education (J.L., NN202318534) and the EU ‘ForcesUniverse’ network (MRTN-CT-2004-005104).
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