Unification of twistors and Ramond vectors: transmutation of superconformal boosts into local supersymmetry

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1 Unification of twistors and Ramond vectors: transmutation of superconformal boosts into local supersymmetry arxiv: v1 [hep-th] 23 Jul 2007 A.A. Zheltukhin Kharkov Institute of Physics and Technology, Kharkov, Ukraine, Fysikum, AlbaNova, University of Stockholm, SE Stockholm, Sweden Abstract The relation between the scale and conformal symmetries in the presence of supersymmetry is studied and new supersymmetric twistors dual to the supertwistors are constructed. The dual twistor, called the θ-twistor, includes the composite Ramond vector [11] well known from the spinning string/particle dynamics. It is shown that the spin structure introduced by θ-twistor breaks superconformal symmetry but still preserves global supersymmetry and scale symmetry generalizing the Gross-Wess mechanism for bosonic scattering amplitudes. We establish that the half of the broken D = 4, N = 1 (super)conformal boosts is restored in the form of local supersymmetry. 1 Introduction Conformal invariance is one of the guiding principles both in theory of particles and condenced matter. The conception of scale invariance plays a great role in the description of classical and quantum phase transitions. In field theories scaling appears as a symmetry of Lagrangians of massles fields with dimensionless couplings or as an asymptotic highenergy symmetry of scattering amplitudes. The superconformal symmetry unifies strings in AdS 5 S 5 space and the supersymmetric Yang-Mills theory on its boundary [1]. Moreover, this symmetry is assumed to be a hidden symmetry of M theory [2]. One of widespread convictions is that the scale invariance implies the conformal invariance. However, the previous study of scale invariant amplitudes of scalar-scalar, scalarspinor, scalar-photon scatterings has shown that the latter are not automatically conformal invariant because of the additional constraints originating in spins [3]. Since the spin structure is built in the super Poincare group, considered as a fundamental symmetry of the space-time, it is important to study the relationship between superconformal and scale symmetries encoded in the geometry of supersymmetric spaces. The twistor spaces [4] and their supersymmetric generalizations [5], [6] are important superspaces connected with superstring and super Yang-Mills theories [7], [8], [9], [10], [11], [12], [13], [14], [15],[16], [17], [18]. Relationship between the scale and conformal symmetries in the presence of supersymmetry is studied here. We reveal a dual symmetry of the fundamental quadratic form [5] defining the supertwistor space which takes into account a fine spin structure ot the chiral superspace. This dual symmetry transforms the scalar grassmannian constituent of the 1

2 supertwistor into the composite Ramond vector [11], which appears as the solution of the Dirac constraint for the original Ramond vector [19] in theories of massless spinning particles and strings [20],[21],[22],[23]. As a result, we obtain a new D = 4, N = 1 scale invariant supersymmetric extension of the twistor space, called the θ-twistor space [24], which is not invariant under superconformal boosts in contrast to the supertwistor space [5]. The breaking of the superconformal symmetry is a consequence of the change of the grassmannian scalar by the grassmannian vector or spinor. This mechanism of the (super)conformal symmetry breaking in the superspace correlates with the Gross-Wess mechanism of the conformal symmetry breaking [3] just triggered by the substitution of vector or spinor particles for scalar particle in their scattering amplitudes. Moreover, a new observation complementary the Gross-Wess effect and connected with the supersymmetry presence was found. It is a transmutation of half of the broken D = 4, N = 1 global superconformal boosts into the supercharges of a local supersymmetry described by complex scalar grassmannian parameter. The corresponding part of the broken global conformal boosts restores itself as the local Lorentz and scaling transformations of the composite Ramond vector. Because of the dual symmetry between the θ-twistor and supertwistor invariant Cartan forms in the supertwistor and θ-twistor spaces are also dual as well as the invariant actions constructed from them. So, we arrive to the dual Wess-Zumino terms and actions connected by the dual symmetry. We suppose that the local supersymmetry of the θ-twistor space will appear as κ-symmetry of supersymmetric particle/string actions formulated in terms of the θ-twistor components. 2 The supertwistor A commuting Weyl spinor ν α belonging to the Penrose spinor doublet (ν α, ν β x β α ) is inert under the transformations of D = 4, N = 1 supersymmetry δθ α = ε α, δx α α = 2i(ε α θ α θ α ε α ), δν α = 0. (1) To introduce the supertwistor [5] we consider the complex superspace (y α α, θ α, θ α ) and the supersymmetric Cartan-Volkov differential form ω α α associated with the superspace y α α x α α 2iθ α θ α, ω α α = dy α α + 4idθ α θ α. (2) The invariant scalar form (νω ν) constructed from ω α α (2) and ν α, ν α may be presented as a supersymmetric differential form formed by the triplet Z A and its complex conjugate Z A (νω ν) = s(z, d Z) = iz A d Z A. (3) The triplets unify the spinors (ν α, ν α ) with the composite coordinates q α, q α, η, η Z A ( iq α, ν α, 2 η), ZA (ν α, i q α, 2η), η ν α θ α, q α = (q α ) ν α y α α = ν α x α α 2iη θ α. (4) The triplet components form a linear representation of the supersymmetry δ q α = 4iη ε α, δη = ν α ε α, δν α = 0. (5) 2

3 The complex pair (Z A, Z A ) defines the supertwistor introduced in [5] as a supersymmetric generalization of the projective Penrose twistor. The supertwistor space may be equivalently defined as a complex projective superspace equipped with the invariant bilinear null form s(z, Z ) s(z, Z ) iz A Z A = q α ν α + ν α q α 4i ηη = 0, (6) where the complex conjugate triplet Z A is given by (4) with ν substituted for ν Z A (ν α, i q α, 2η ), q α = ν α y α α, η = ν α θ α. (7) The quadratic form (6) is invariant under the global superconformal symmetry as it was shown in [5]. The fermionic sector of the supertwistor (4) contains only the scalar projection η of θ α. It sets the question: whether it is possible to preserve the superconformal symmetry without the θ component reduction while the twistor supersymmetrization? 3 The θ-twistor: a unification of Penrose twistor and Ramond vector To answer this question we proposed an alternative supersymmetric generalization of the Z- triplet that is formed by three spinors [24]. The spinor triplet preserves θ α, but introduces a new composite spinor l α produced by the right multiplication of the chiral coordinate y α α (2) by ν α (in contrast to the left multiplication (4) generating q α (4) l α y α α ν α = x α α ν α 2iθ α η, l α = q α 4iθ α η. (8) The transformations of l α (8) under the supersymmetry (1) are nonlinear δl α = 4iθ α ( ν β ε β), δθ α = ε α, δ ν α = 0 (9) and yield a new supersymmetry representation formed by the complex spinor triplet Ξ A Ξ A ( il α, ν α, θ α ), ΞA (Ξ A ) = (ν α, i l α, θ α ). (10) The quadratic form (6) expressed in terms of Ξ A and Ξ A (10) becomes a nonlinear form s(z, Z ) iz A Z A = s(ξ, Ξ ) l α ν α + ν α l α 4i(ν α ν α )θ α θ α = 0 (11) in the Ξ-triplet space, where the supersymmetry generators take the form Q α = θ α + 4iν α ( θ β l β ), Q α (Q α ) = θ α + 4i ν α (θ β l β ) (12) with their anticommutator closed by the vector generator P βα = ( ν β l α + ν α l β ) {Q α, Q β} = 4iP βα, [Q γ, P βα ] = [ Q γ, P βα ] = {Q γ, Q β } = { Q γ, Q β} = 0. (13) We reveal here that the nonlinear term 4i(ν α ν α )θ α θ α in (11) is presented in the form 4i(ν α ν α)θ α θ α 4i ηη 2i( ν σ m θ)(ν σ m θ) 8i ηm η m, (14) 3

4 where η m and η m are the composite Ramond grassmannian vectors previously introduced in [11] (see details in [25]) to prove the equivalence between superparticles and spinning particles η m 1 (νσ 2 m θ), η m = (η m ) = 1 ( ν σ 2 mθ), (15) ν α θ α η α α = (σ m ) α α η m, η m η n + η n η m = 0. As a result, the quadratic form (11) defining the supertwistor space becomes the quadratic form s = s(z, Z ) = s(ξ, Ξ ) iξ A Ξ A = l α ν α + ν α l α 8i η m η m = 0 (16) in the complex projective space of the Ξ-triplets including the composite Ramond vector Ξ A ( il α, ν α, 2 2 η m ), Ξ A (Ξ A ) = (ν α, i l α, 2 2η m ). (17) This quadratic form is invariant under the linear supersymmetry transformations δl α = 4i(σ m ε) α η m, δ η m = 1 2 (εσ m ν), δ ν α = 0. (18) and their c.c. which follow from the nonlinear transformation (9) after the substitution of η m for θ α. So, the supertwistor Z-triplet (4) transforms into the Ξ- triplet (17) under the substitution: (q α, η) (l α, 2 η m ) resulting in equivalent representations of the same quadratic form (6) either in the terms of the Z-triplet or the Ξ-triplet. Thus these two triplets occur to be dual giving a reason to call the new Ξ-triplet (or the Ξ-triplet (10)) the θ-twistor. The supertwistor and θ-twistor are the general solutions of different supersymmetric constraints. The supertwistor solves the generalized chiral constraint in the superspace (y α α, θ α ) complemented by the left Weyl spinor ν α and the new scalar operator ν α D α, D α F(x, θ, θ) = 0 F = F(y, θ), ν α D α F(y, θ, ν) = 0 F = F( Z A ). (19) In contrast, the θ-twistor solves the supersymmetric constraints in the chiral space complemented by the right Weyl spinor ν α and the new spinor operator ν α x α α, D α F(x, θ, θ) = 0 F = F(y, θ), ν α x α α F(y, θ, ν) = 0 F = F(Ξ A ). (20) It is easy to see that the second constraint (20) is equivalent to the Dirac constraint selecting the Ξ-triplet as it follows from its contraction with θ α θ α ν α x α α F(y, θ, ν) η α α x α α F(y, θ, ν) η m m F(y, θ, ν) = 0. (21) Thus, the θ-twistor is a new triplet restoring all spin degrees of freedom of θ and dual to the supertwistor. Our question is: whether the θ-twistor forms a representation of the superconformal symmetry? 4

5 4 Θ-twistor and superconformal symmetry breaking It is easy to see the invariace of nonlinear form (11) (or equivalently quadratic form (16)) under the scaling and phase symmetries given by the transformations [24] l β = eϕ l β, l β = e ϕ l β, ν β = e ϕ ν β, ν β = e ϕ ν β, θ β = eϕ θ β, θ β = e ϕ θ β, (22) described by the complex parameter ϕ = ϕ R + iϕ I, as well as under the γ 5 rotations θ β = eiλ θ β, θ β = e iλ θ β. (23) But, the θ-twistor space is not closed under the the superconformal boosts S α, S α [5], [26] of the support chiral superspace (y β β, θ β ) δy α α = 4iθ α (ξ β y β α ), δθ α = y α β ξ β + 4iθ α (ξ β θ β ) (24) as it was shown in [24], because the transformations of θ α and δl α = y α βδ ν β + 4iθ α (ξ β l β ) following from (24) include the chiral coordinate y α β. To remove the y α β dependence in δl α we choose δ ν β to be proportional ν β δl α = l α δ ϕ + 4iθ α (ξ β l β ), δ ν β = ν β δϕ. (25) However, the partial fixing of the ν α transformation (25) can not cancel the dependence of δθ α (24) on y α β. For the supertwistor case such a dependence has been removed by projecting δθ α (24) on the left Weyl spinor ν α. But, now we have only the right Weyl spinor ν α as a component of the Ξ-triple. Thus, the difference in the chiralities of ν and the θ- coordinate in the Ξ-triple obstructs to the realization of the superconformal boosts. We find the symmetry properties of supertwistor and θ-twistor to be different in spite of their dual character. As a result, the θ-twistor forms a representation of only the maximal subgroup of the superconformal group [24]. This breakdown is a consequence of the different chiral structures associatred with these twistor objects and it is in correspondence with the Gross- Wess observation. Next we show the presence of a sudden effect partially compensating this superconformal symmetry breaking. 5 Transmutation of superconformal boosts into local supersymmetry The problem of the y α β dependence in the θ variation (24) is solved by the transition from the global parameter ξ α to the coordinate dependent ξ α = ξν α + λv α, ν α v α = 1, (26) assuming a localization (or nonlinear realization) of the superconformal boosts and permitting the expansion of ξ α along ν α and v α which form a dyad basis. The local parameters ξ, λ associate with a local supersymemtry originating from the superconformal boosts. If we fix ξ α = ν α ξ, i.e. λ (ν α ξ α ) = 0, the variations (24), (25) take the form δθ α = l α ξ 4iθα (ν β θ β )ξ, δl α = l α δ ϕ + 4iθ α (ν β l β )ξ, δ ν α = ν α δϕ. (27) 5

6 The fixing δ ϕ = 4i(ν β θ β )ξ results in the Ramond vector transformation: δ η m = 1 2 (lσ m ν) ξ. Then the transformations (27) of the holomorphic triple Ξ A ( il α, ν α, θ α ) present as δl α = 4i[ (ν β θ β )l α + (ν β l β )θ α ]ξ, δ ν α = 4i ν α (ν β θ β )ξ, δθ α = l α ξ 4iθα (ν β θ β )ξ. (28) and their generators S (Ξ), S (Ξ) are given by S (Ξ) = 4iν α S α (Ξ), S α (Ξ) = (θ α l. + l α θ. ) l. θ α (θ. θ. ) + θ α ( ν. ν. ), S (Ξ) = (l β θ β ) (l. θ. ), {S (Ξ), S (Ξ) } = { S (Ξ), S (Ξ) } = 0. (29) The anticommutator of the local supercharges S (Ξ), S (Ξ) (29) of the holomorphic sector Ξ A of the θ-twistor space closes by the generator K (Ξ) originating from the conformal boosts {S (Ξ), S (Ξ) } = 32 K (Ξ) ), [S (Ξ), K (Ξ) ] = [ S (Ξ), K (Ξ) ] = [K (Ξ), K (Ξ) ] = 0. (30) K (Ξ) is projection of a holomorphic operator K (Ξ) α on the antiholomorphic coordinate ν α K (Ξ) ν α K α (Ξ), K(Ξ) α = l α [(l. ) 1 l. 2 ( ν. )] + θ α (l. ). (31) ν. θ. and generates the following transformations of the Ξ-triple components and η m δθ α = l α (ν. θ. )κ, δl α = l α (ν. l. )κ, δ ν α = 1 2 ν α (ν. l. )κ, δ η m = η m (ν. l. )κ + (νσ mn l) η n κ. (32) The parameter κ originates from the conformal boost parameter κ α β after its localization: κ α β = κ(ν α ν β) or, equivalently after projection of the conformal boost generator K βα on the light-like vector ν β ν β composed from the holomorphic and antiholomorphic parts of ν. It results in from the definiton of K βα [24] in the support chiral superspace (y α α, θ α ) δy α α = (yκy) α α, δθ α = (yκθ) α (33) and proves that the generator (31) in fact originates from the global conformal boosts. So, the global conformal boost (32) transmute into the local Lorentz transformations and dilatations of the Ramond vector η m. The generators of the antiholomorphic sector Ξ A = (ν α, i l α, θ α ) (10) S ( Ξ) = 4i ν α S( Ξ) α, S( Ξ) α = ( θ α l. + l α θ. ) l. θ α ( θ. θ. ) + θ α (ν. ν. ), S ( Ξ) = ( l. θ. ), K( Ξ) = ν α K( Ξ) α, K( Ξ) α = l α [( l. l. ) 1(ν 2. ν. )] + θ α ( l. θ. ), contain the coordinate ν α of the holomorphic sector in S ( Ξ) and K ( Ξ). Thus, the generators of the holomorphic sector S (Ξ) (29) and K (Ξ) (31) have nonzero (anti)commutators with the generators of the antiholomorphic sector S ( Ξ) and T ( Ξ) (34) {S (Ξ), S ( Ξ) } = 4i [ η S (Ξ) + η S ( Ξ) ], [ K (Ξ), K ( Ξ) ] = 1 2 [ (ν. l. ) K ( Ξ) ( ν. l. )K (Ξ) ] [ S (Ξ), K ( Ξ) ] = 1 2 ( ν. l. )S (Ξ) 4iη K ( Ξ), [ S ( Ξ), K (Ξ) ] = 1 2 (ν. l. ) S ( Ξ) 4i ηk (Ξ), where the fermionic constituent η = (ν. θ. ) of the supertwistor (4) together with the bosonic scalar (ν. l. ) play the role of structure functions. The obtained superalgebra is the open superalgebra of the local supersymmetry originated from the broken (super)conformal boosts and mixing the holomorphic and antiholomorphic sectors of the θ-twistor space. 6 (34) (35)

7 6 Dual Wess-Zumino terms and dual actions It is known that the open (super)algebras appear in gauge or locally diffeomorphic theories similar to (super)gravity. It was found in [27] that the κ-symmetry generators of the Green-Schwartz superstring in twistor formulation are projections of the infinitely reducible fermionic constraints on the Lorentz harmonics. As a result, the Lorentz covariant generators of κ-symmetry and Virasoro generators have formed an open superalgebra. The open subalgebra (35) has a similar structure and we assume that the associated local supersymmetry could be realizable as κ-symmetry of supersymmetric actions of particles, strings/branes reformulated in the θ-twistor space. The actions may be constructed using supersymmetric differentiall forms in the θ-twistor space. It can be illustrated by a simple example. Because of the dual symmetry connecting the θ-twistor and supertwistor the supersymmetric differential one-form (3) has two equivalent dual representations (νω ν) = (Z, d Z) = iz A d Z A = q α dν α + ν α d q α 4i ηdη = s(ξ, d Ξ) = iξ A d Ξ A = l α dν α + ν α d l α 8i η m dη m, (36) It means that an action firstly formulated in the supertwistor representation has to have the dual θ-twistor representation. For example, two dual representations of the one-form (36) iz A d Z A = [ ν α dq α + ν α d q α 2i( ηdη d ηη) ] d(νx ν) = iξ A d Ξ A = [ ν α dl α + ν α d l α 4i( η m dη m d η m η m ) ] d(νx ν). (37) result in two dual actions. To find the actions we observe that the first terms in r.h.s of (37) are Wess-Zumino terms because they are invariant under the supersymmeties (5) and respectively (18) up to the total differential of the variation of the scalar (νx ν) which absorbs the space-time coordinate x m. 1 As a result, the first terms in (37) produce two dual representations for the Wess-Zumino term and the latter yields two supersymmetric dual actions for particles. In the θ-twistor representation the action is as follows S = dτ{[ ν α lα + ν α l α 4i( η m η m η m η m ) ] + λ s(ξ, Ξ) }, (38) where λ is the Lagrange multiplier fixing the constraint s(ξ, Ξ) = 0 (17). 7 Conclusion Stimulated by the Gross-Wess observation [3] about the spin structures in scattering amplitudes of massless particles as obstructions preventing the scale symmetry extension up to the conformal symmetry we have addressed the same question to the superspaces having inherent chiral spin structures. On this way the supersymmetric twistors called the θ-twistors and dual to the well known supertwistors [5], [6] were revealed. The fermionic constituent of the θ-twistor is presented by the composite grassmannian Ramond vector [11] or by the chiral superspace coordinate θ α [24] contrarily to the scalar grassmannian constituent of the supertwistor. The chiral structure of the θ-twistor space is not invariant under the (super)conformal boosts, but it is invariant under the maximal subgroup of the superconformal 1 The Wess-Zumino terms linear in derivatives were previously considered in [29]. 7

8 group. However, we found here that the half of broken (super)conformal boosts restores in the form of the local supersymmetry generators forming the open superalgebra typical for theories invariant under (super)diffeomorphisms. We assume that the local supersymmetry realizes as the κ-symmetry of supersymmetric actions of particles/strings represented in the terms of the θ-twistor space coordinates. This conjecture correlates with the recent observations [2], [30] on the connection between diffeomorphisms, κ-transformations and non-linear realizations. The latter were previously considered in [31], where strings embedded in D- dimensional space-time have been described as exactly solvable sector either of the non-linear σ-models, associated with the cosets of the Lorentz group SO(1, D 1), or equivalently of the two dimensional SO(1, 1) SO(D 2) gauge theory. Taking into account the property of the θ-twistor space to be closed under the local supersymmetry mixing the holomorphic and antiholomorphic sectors of the chiral superspaces extended by the Penrose spinor ν it is interesting to understand their possible role in the description of perturbative scattering amplitudes in Yang-Mills theory other than the MHV amplitudes. The θ-twistor construction automatically generalizies to the case of extended supersymmetries similarly the supertwistor construction [5] and it is interesting to investigate the geometrical properties of the corresponding supermanifolds along the line developed in [32]. Also, because the dual symmetry naturally introduces the composite Ramond vector as the θ-twistor constituent it seems that it would shed a new light on the mystery of the GSO projection [33]. 2 We hope to study these issues elsewhere. 8 Acknowledgements The author is grateful to Fysikum at the Stockholm University for kind hospitality and Ingemar Bengtsson and Steven Giddings for useful discussions. The work was partially supported by the grant of the Royal Swedish Academy of Sciences. References [1] J. Maldacena, Adv. Theor. Math. Phys. 2 (1980) 231. [2] P. West, JHEP 08 (2004) 007. [3] D. Gross and J. Wess, Phys. Rev. D2 (1970) 753. [4] R. Penrose and W. Rindler, Spinor and twistor methods in space-time geometry. Cambridge University Press, [5] A. Ferber, Nucl. Phys. B132 (1978) 55. [6] E. Witten, Phys. Lett. B77 (1978) 215. [7] E. Witten, Commun. Math. Phys. 252 (2004) 189; Adv.Theor.Math.Phys. 8 (2004) Taking into account of the spontaneous vacuum transitions [34] in the Veneziano and Neveu-Schwarz dual models has given an alternative mechanism of the tachyon elimination and reveals the existence of the broken symmetry group with an infinite number of generator containing the group SU(2) SU(2) U(1) U(1)... U(1)... as a subgroup, in the Neveu-Schwarz model. 8

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