aa r X i v : . [ h e p - t h ] J un TWO–TWISTOR DESCRIPTION OF MEMBRANE
Sergey Fedoruk, Jerzy Lukierski ∗ Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Region, Russia Institute for Theoretical Physics, University of Wroc law, pl. Maxa Borna 9, 50-204 Wroc law, Poland
We describe D = 4 twistorial membrane in terms of two twistorial three–dimensional world–volumefields. We start with the D –dimensional p –brane generalizations of two phase space string for-mulations: the one with p + 1 vectorial fourmomenta, and the second with tensorial momenta of( p + 1)-th rank. Further we consider tensionful membrane case in D = 4. By using the membranegeneralization of Cartan–Penrose formula we express the fourmomenta by spinorial fields and obtainthe intermediate spinor–space-time formulation. Further by expressing the world–volume dreibeinand the membrane space-time coordinate fields in terms of two twistor fields one obtains the purelytwistorial formulation. It appears that the action is generated by a geometric three–form on two–twistor space. Finally we comment on higher–dimensional ( D >
4) twistorial p –brane models andtheir superextensions.PACS numbers: 11.25.-w, 11.10.Ef, 11.30.Pb I. INTRODUCTION
Since long time the idea of twistor space (see e. g. [1])as describing the basic geometric arena for the physicalphenomena is tested in various ways. In particular it isknown that massless relativistic particles can be equiva-lently described by free one–twistor particle model (seee. g. [2, 3, 4, 5]); further it can be shown that massiverelativistic particles with spin require in twistorial ap-proach the two–twistor space [6, 7, 8, 9, 10, 11] . Fur-ther recently there was derived the twistor space actionfor general Yang–Mills (Euclidean) gauge theory [13, 14]providing a breakthrough in extending the twistor con-struction to non–self–dual field theories. Important con-tribution to the twistor programme is also provided bythe proof that large class of perturbative amplitudes in N = 4 D = 4 supersymmetric YM theory and con-formal supergravity can be derived by using tensionlesssuperstring moving in supertwistor space. In such ap-proach (see e. g. [15, 16]) the twistorial classical stringis described by two–dimensional CP (3 | σ –model with(twisted) N = 2 world sheet SUSY. The correspondencewith the space–time picture was derived [15] on the levelof quantized supertwistor string, providing twistor su-perstring field theory. By assuming the topological CSaction for the (Euclidean) twistor superstring field onecan reduce all the superstring excitations to the masslessspectrum describing SUSY YM and supergravity theo-ries. On the other hand recently in D = 4 the closelink between the tensionful string model (Nambu-Gotoaction) and twistor geometry has been proposed on the ∗ Supported by KBN grant 1 P03B 01828 One should add that unconventional application of one–twistorgeometry to massive particles has been proposed in [12]. classical level, with target space–time string coordinatescomposite in terms of twistor string fields [17, 18]. It ap-pears that in such a model the twistorial target space inbosonic D = 4 string model consists of a pair of twistorstring coordinates.Recently in M –theory the (super)strings have lost theirprivileged role as the candidates for the Theory of Ev-erything, and higher–dimensional p –branes ( p > p –branedescription in the framework of spinorial harmonics ([19];see also [20, 21]). It should be added that the presence ofcosmological term in Polyakov type action for membranepermits to obtain the purely twistorial model without theuse of gauge fixing procedure, which was a necessary stepin string case [17].For simplicity we shall describe our composite mem-brane models in detail in D = 4, and without supersym-metric extension. Taking however into consideration thatthe multidimensional and supersymmetric extensions oftwistors applied to twistorial string formulations has beenalready considered (see for example [15, 16, 22, 23]), webelieve that this paper can be also useful e. g. for theconsideration of the composite “ M –theoretic” D = 11supermembrane.In order to achieve our goal firstly in Sect. 2 we shallshow how to extend from string to p –brane the two knownphase space formulations of bosonic string theory: theone using vectorial momenta fields [24, 25] and the secondusing tensorial momenta fields (see e. g. [26, 27, 28, 29]).These two formulations are based on the following twoLiouville ( p +1)–forms which define the p –brane momentafields: a) Vectorial momenta model:Θ (1) = p µ dX µ → Θ ( p +1) = P µ ∧ dX µ (1)where P µ is the following p –form P µ = P mµ ǫ mn ...n p dξ n . . . dξ n p . (2)It is easy to see that if p = 1 we obtain the Siegelformula [24] for the string momenta one form P µ = P mµ dξ n ǫ mn . (3) b) Tensorial momenta model:Θ (1) = p µ dX µ → ˜Θ ( p +1) = P µ ... µ p +1 dX µ ∧ . . . ∧ dX µ p +1 . (4)If p = 1 one obtains the tensorial string momenta P µν = − P νµ ([26, 28]; for arbitrary p see [29]).In Sect. 2 we shall consider p –branes with arbitrary p and in arbitrary space–time dimension D . We shall showthat both phase space formulations using the momentafields (2) or (4) are equivalent to the p –brane Dirac–Nambu–Goto action [30, 31] as well as to the σ –modelaction ( p –brane extension of Howe–Tucker–Polyakov ac-tion for strings and membranes [32, 33, 34]). Further, inSect. 3 we consider for the case p = 2 and D = 4 (four-dimensional membrane) the intermediate spinor–space-time models for both phase space formulations. It ap-pears that in such a model the spinors are constrained. In Sect. 4 we shall consider purely twistorial action for D = 4 membrane. We show that in this action the La-grangian density is described by a canonical twistorialthree–form. In Sect. 5 we present an outlook: we com-ment on the description of twistorial membranes in higherdimensions, their supersymmetric extensions and presentthe remarks about the purely twistorial p –branes ( p > II. TWO PHASE SPACE FORMULATIONS OFTHE TENSIONFUL p –BRANE IN D –DIMENSIONAL SPACE–TIME The tensionful p –brane propagating in flat Minkowskispace is described by the nonlinear Dirac–Nambu–Gotoaction S = − T Z d p +1 ξ √− g (5) We propose alternative way of generating constraints in compar-ison with the framework of Lorentz harmonic approach [19, 20,21]. The indices m, n = 0 , , . . . , p are vector world-sheet indices; µ, ν = 0 , , . . . , D − η ab = ( − , + , . . . , +), η µν = ( − , + , . . . , +). where ξ m = ( τ, σ , . . . , σ p ) are the world–volume coordi-nates, g ≡ det( g mn ) (6)and g mn = ∂ m X µ ∂ n X µ (7)is the induced metric on the ( p + 1)–dimensional p –branevolume, T is the p –brane tension. In the case p = 1the action (5) is the Nambu–Goto action for string [31]whereas if p = 2 the action (5) is the Dirac action forrelativistic membrane [30].Because the twistor coordinates replace the standardphase space variables, the transition to twistorial formu-lation should be imposed on the Hamiltonian–like formu-lations. In the case of tensionful p –branes (5) there areknown two Hamiltonian descriptions. A. Phase space formulation with vectorialmomenta
Let us use firstly the vectorial momenta defined by theformula (2). The corresponding action of the tensionful p –brane looks as follows [25] S = Z d p +1 ξ h P mµ ∂ m X µ + T ( − h ) − / h mn P mµ P µ n ++ T ( p − − h ) / i . (8)We note that last ‘cosmological’ term in the action isabsent if p = 1 (string case).Let us write down the equation of motion obtainedafter varying the world–volume metric h mn . Using δh = hh mn δh mn = − hh mn δh mn one gets P mµ P µ n − h mn h h kl P kµ P lµ + ( p − hT i = 0 . (9)Further expressing P mµ in the action (8) by its equationof motion ( h mk h kn = δ mn ) P mµ = − T ( − h ) / h mn ∂ n X µ (10)one obtains the σ –model action for the p –brane of theHowe–Tucker–Polyakov type S = − T Z d p +1 ξ ( − h ) / h h mn ∂ m X µ ∂ n X µ − ( p − i (11) The string actions as d = 2 world sheet gravity interacting withstring coordinate fields were originally proposed in [35], whereas well the description of spinning string using interacting d = 2supergravity is presented. where the variables h mn can be treated as independentones. The equations of motions for h mn give ∂ m X µ ∂ n X µ − h mn h h kl ∂ k X µ ∂ l X µ − ( p − i = 0and lead to (using h mn h mn = p + 1) h mn = g mn (12)where g mn is the induced metric (7) on the p –brane vol-ume. After substitution of (12) in (11) we obtain aftersimple algebraic calculation the action (5).It can be shown that from the action (8) one can de-rive the p + 1 Virasoro constraints, generating the world–volume diffeomorphisms. We shall divide world–volumeindices m, n = 0 , , . . . , p into one with zero value andremaining m, n = 1 , . . . , p , that is m = (0 , m ). Theequations (10) lead to expression of the ‘auxiliary space momenta’ P mµ = − h n h n m P µ − T ( − h ) / h m n ∂ n X µ (13)where P µ = P µ is true momentum and ( p × p ) matrix h m n = h m n − h m h n h (14)is the inverse matrix for ( p × p ) space part of the worldvolume metric ( h m n h n k = δ mk ). We note that1 h = h det( h m n ) , h m h = − h n h n m . (15)Inserting (13) in the the action (8) we obtain the firstorder Lagrangian The relation (12) is unique for p = 1; for p = 1 one can introducein (12) an arbitrary local scaling factor. L = P µ ˙ X µ − √− h T det( h m n ) (cid:16) P µ P µ (cid:17) − T √− h h m n (cid:16) ∂ m X µ ∂ n X µ (cid:17) − h n h n m (cid:16) P µ ∂ m X µ (cid:17) + T ( p − √− h . (16)Using the equality h m n = p − det( h m n ) ǫ m m ...m p ǫ n n ...n p h m n . . . h m p n p and the equations of motions (12) we obtain that third term in Lagrangian (16) takes the form − T √− h h m n (cid:16) ∂ m X µ ∂ n X µ (cid:17) = − T √− h p det( h m n ) det( g m n ) . (17)If the equations (12) are valid we get det( h m n ) det( g m n ) = 1 (18)and the last term in the Lagrangian (16) can be written as follows T ( p − √− h = T ( p − √− h det( h m n ) det( g m n ) . (19)Inserting the expressions (17), (19) in the Lagrangian (16) we obtain the following standard form of the Lagrangiandensity in first order formalizm L = P µ ˙ X µ − √− h det( h m n ) h T P µ P µ + T det( g m n ) i − h n h n m (cid:16) P µ ∂ m X µ (cid:17) . (20)We see that the formula (20) describes the set of p + 1Virasoro constraints H ≡ T P µ P µ + T det( g m n ) ≈ , (21) H m ≡ P µ ∂ m X µ ≈ h m n . Let us finally deduce from the relations (9) the p –branemass–shell condition. Multiplying (9) by h mn we get for p > h mn P mµ P µ n = − ( p + 1) h T (23)or T ( − h ) − / h mn P mµ P µ n = ( p + 1)( − h ) / T . (24)In string case ( p = 1) the contraction of l. h. s. of (9)with h mn is identically vanishing and in any space–timedimension the string mass condition (23) is absent. B. Phase space formulation with tensorialmomenta
Other phase space formulation of the p –brane (5) is themodel with tensorial momenta. It is obtained by the use of the Liouville ( p + 1)–form (4). Such a formulation isdirectly related with the interpretation of p –branes as de-scribing the dynamical ( p + 1)–dimensional world volumeelements described by the following ( p + 1)–forms Total antisymmetric tensors ǫ m ... m p +1 , ǫ m ... m p +1 and ǫ a ... a p +1 , ǫ a ... a p +1 have the components ǫ ... p = 1, ǫ ... p = − dS µ ... µ p +1 = dX µ ∧ . . . ∧ dX µ p +1 = ∂ m X µ . . . ∂ m p +1 X µ p +1 ǫ m ... m p +1 d p +1 ξ . (25)The p –brane action with tensorial momenta looks as follows [29] S = √ ( p +1)! Z d p +1 ξ " P µ ... µ p +1 Π µ ... µ p +1 − Λ (cid:16) P µ ... µ p +1 P µ ... µ p +1 + T (cid:17) (26)whereΠ µ ... µ p +1 ≡ ǫ m ... m p +1 ∂ m X µ . . . ∂ m p +1 X µ p +1 . (27)Expressing P µ ... µ p +1 by its equation of motion, we get P µ ... µ p +1 = Π µ ... µ p +1 . (28)After substituting (28) in the action (26) we obtain the2( p + 1)-th order action S = √ ( p +1)! Z d p +1 ξ (cid:2) Λ − Π µ ... µ p +1 Π µ ... µ p +1 − Λ T (cid:3) . (29)Eliminating the auxiliary field Λ we obtain S = − T √ ( p +1)! Z d p +1 ξ q − Π µ ... µ p +1 Π µ ... µ p +1 . (30)But the determinant of the matrix (7) is given by theformuladet( g mn ) = p +1)! ǫ m ... m p +1 ǫ n ... n p +1 g m n . . . g m p +1 n p +1 = p +1)! Π µ ... µ p +1 Π µ ... µ p +1 . (31)We see that the action (30) is classically equivalent tothe action (5).The formula (29) is very useful if we wish to considerfor any p the tensionless limit T →
0. We obtain theformula [29, 37, 38] S T =0 = √ ( p +1)! Z d p +1 ξ Π µ ... µ p +1 Π µ ... µ p +1 . (32)The formula (32) describes the p –brane counterpart ofBrink–Schwarz action for massless particle [39]. III. TENSIONFUL D = 4 MEMBRANE ( p = 2 ) ININTERMEDIATE SPINOR–SPACE-TIMEFORMULATIONSA. Formulation with vectorial momenta In order to obtain from the action (8) the intermediatespinor–space-time action we should eliminate the fourmo-menta P mµ by means of the the membrane generalizationof the Cartan–Penrose formula expressing fourmomentaas spinorial bilinears. On d = 3 curved world volume ithas the form P mα ˙ α = e ˜ λ ˙ α ρ m λ α = ee ma ˜ λ i ˙ α ( ρ a ) ij λ αj (33)where λ αi ( i = 1 ,
2) are two D = 4 commutingWeyl spinors, ( ρ a ) ij are 2 × { ρ a , ρ b } = 2 η ab ) and ρ m = e ma ρ a . After using (33) the second term in the ac-tion (8) takes the form T ( − h ) − / h mn P mµ P nµ = T e ( λ αi λ αi )(˜ λ j ˙ α ˜ λ ˙ αj ) (34) h mn = e am e na is a world–volume metric, e am is the dreibein, e am e mb = δ ab , e = det( e am ) = √− h . The indices a, b = 0 , , m, n = 0 , , d = 3 vector indices; the indices i, j = 1 , d = 3 Dirac spinor indices. We use bar for complex conjugatequantities, ¯ λ i ˙ α = ( λ αi ), and tilde for Dirac–conjugated d = 2spinors, ˜ λ i ˙ α = ¯ λ j ˙ α ( ρ ) ji . The matrix of charge conjugation is the 2 × ǫ ij . For definiteness, we take ǫ = 1. The rules of liftingand lowering the indices are following: a i = ǫ ij a j , a i = ǫ ij a j where ǫ ij ǫ jk = δ ik . Also, we use D = 4 Penrose spinor–vectorconventions [1] in which P µ P µ = P α ˙ α P ˙ αα , P µ X µ = P α ˙ α X ˙ αα ( P α ˙ α = √ P µ σ µα ˙ α etc.) where we used Tr( ρ m ρ n ) = 2 h mn .Let us recall the condition (24) which is valid for p > λA ≡ ( λλ )(˜ λ ˜ λ ) − T = 0 (35)(we use notations ( λλ ) ≡ ( λ αi λ αi ), (˜ λ ˜ λ ) ≡ (˜ λ i ˙ α ˜ λ ˙ αi ); notethat ˜ λ i ˙ α ˜ λ ˙ αi = ¯ λ i ˙ α ¯ λ ˙ αi ). Putting (33) and (34) in (8) andimposing via Lagrange multiplier the constraint (35) weobtain the action S = Z d ξ " e (cid:16) ˜ λ ˙ α ρ m λ α ∂ m X ˙ αα + 2 T (cid:17) + Λ A (36)which provides the intermediate spinor–space-time for-mulation of the membrane. Let us observe that the ac-tion (36) is invariant under the following Abelian localgauge transformation λ ′ αi = e iγ λ αi (37)with real local parameter γ ( ξ ). By fixing the gauge (37)we can replace one real constraint (35) by the followingpair of constraints ( λλ ) = (˜ λ ˜ λ ) = √ T . (38)
B. Formulation with tensorial momenta
Let us consider the general action which has the fol-lowing form S = Z d ξ (cid:16) ee ma Q am + 2 eT (cid:17) (39)where Q am = Q am ( X, λ ) do not depend on e am . Using therelations e = − ǫ mnk ǫ abc e am e bn e ck , ee ma = − ǫ mnk ǫ abc e bn e ck we obtain the following equation of motion for e ma e am = − T Q am . (40)Subsequently the action (39) takes the following classi-cally equivalent form S = − T Z d ξ ǫ abc ǫ mnk Q am Q bn Q ck . (41)Choosing in the action (39) Q am = (˜ λ ˙ α ρ a λ α ) ∂ m X ˙ αα (42) Compare with the string case considered in [17]. and after supplementing the constraint (35) one gets ourmembrane action (36). Inserting the formula (40) weobtain S = √ Z d ξ (cid:16) P α ˙ α,β ˙ β,γ ˙ γ ǫ mnk ∂ m X ˙ αα ∂ n X ˙ ββ ∂ k X ˙ γγ +Λ A (cid:17) (43)where tensorial momenta are composites in term of fun-damental spinors P µνλ = P α ˙ α,β ˙ β,γ ˙ γ (44)= − √ T ǫ abc (˜ λ ˙ α ρ a λ α )(˜ λ ˙ β ρ b λ β )(˜ λ ˙ γ ρ c λ γ ) . We see that the intermediate spinor–space-time ac-tion (43) with composite tensorial momenta is obtainedafter the elimination of dreibein variables e am . Takinginto account the relation(˜ λ ˙ α ρ a λ α )(˜ λ ˙ α ρ b λ α ) = δ ab T , (45)following from (35), and ǫ abc ǫ abc = −
3! we can showeasily that the tensor (44) satisfies the membrane massshell condition (compare with the constraint in the ac-tion (26)) P µνλ P µνλ = P α ˙ α,β ˙ β,γ ˙ γ P α ˙ α,β ˙ β,γ ˙ γ = − T . (46)We see that in the action (43) the condition (46) followsfrom the constraint (35). IV. PURELY TWISTORIAL FORMULATION OFTHE MEMBRANE ( p = 2 ) IN D = 4 SPACE–TIME
Further we introduce second half of twistor coordinates µ ˙ αi , ¯ µ αi by postulating the Penrose incidence relationsgeneralized for D = 4 membrane fields µ ˙ αi = X ˙ αα λ αi , ˜ µ αi = ˜ λ i ˙ α X ˙ αα . (47)We shall rewrite the action (36) by taking into accountthe relations (47). Using the relations (33), (47) we ob-tain P mα ˙ α ∂ m X ˙ αα = e ˜ λ ˙ α ρ m λ α ∂ m X ˙ αα (48)= e e ma (cid:16) ˜ λ ˙ α ρ a ∂ m µ ˙ α − ˜ µ α ρ a ∂ m λ α (cid:17) + c . c . If we introduce the four–component twistors ( A =1 , · · · , Z Ai = ( λ αi , µ ˙ αi ) , ˜ Z Ai = (˜ µ αi , − ˜ λ i ˙ α ) , (49) The Lagrange multiplier in (43) is obtained from the one in (36)by the rescaling Λ → √ Λ. The coefficient in (44) are chosen in consistency with the general p –brane formula (26). the relations (48) takes the form P mα ˙ α ∂ m X ˙ αα = e ˜ λ ˙ α ρ m λ α ∂ m X ˙ αα (50)= e e ma (cid:16) ∂ m ˜ Z A ρ a Z A − ˜ Z A ρ a ∂ m Z A (cid:17) . Incidence relations (47) with real space–time mem-brane position field X ˙ αα imply that the twistor field vari-ables satisfy the constraints V ij ≡ λ αi ˜ µ αj − µ ˙ αi ˜ λ j ˙ α ≈ V ij = Z Ai ˜ Z Aj ≈ . (52)We obtain the following membrane action (36) in twistorformulation with dreibein S = Z d ξ " e e ma (cid:16) ∂ m ˜ Z A ρ a Z A − ˜ Z A ρ a ∂ m Z A (cid:17) ++ 2 e T + Λ A + Λ ji V ij (53)where Λ and Λ ij are the Lagrange multipliers. If we define the asymptotic twistors [1, 6] I AB = (cid:18) ǫ αβ
00 0 (cid:19) , I AB = (cid:18) ǫ ˙ α ˙ β (cid:19) (54)one can introduce the following notation( λλ ) ≡ λ αi λ αi = ǫ ij I AB Z Ai Z Bj ≡ ( ZZ ) , (55)(˜ λ ˜ λ ) ≡ ˜ λ i ˙ α ˜ λ ˙ αi = ǫ ij I AB ˜ Z Ai ˜ Z Bj ≡ ( ˜ Z ˜ Z ) (56)and write down the fourlinear constraint (35) in the fol-lowing twistorial form: A ≡ ( ZZ )( ˜ Z ˜ Z ) − T = 0 . (57)We shall eliminate the dreibein e am by employing theformula (40). The action (53) correspond to the choice Q am = (cid:16) ∂ m ˜ Z A ρ a Z A − ˜ Z A ρ a ∂ m Z A (cid:17) . (58)One gets the final action depending only on two twistorialfields Z Ai ( τ, σ , σ ) and suitably rescaled (in comparisonwith (53)) the Lagrange multipliers Λ, Λ ji : S = − T Z d ξ " ǫ abc ǫ mnk (cid:16) ∂ m ˜ Z A ρ a Z A − ˜ Z A ρ a ∂ m Z A (cid:17)(cid:16) ∂ n ˜ Z B ρ b Z B − ˜ Z B ρ b ∂ n Z B (cid:17)(cid:16) ∂ k ˜ Z C ρ c Z C − ˜ Z C ρ c ∂ k Z C (cid:17) ++ Λ A + Λ ji V ij . (59)The model (59) describes the D = 4 membrane inpurely twistorial formulation. Introducing three one–forms with world-volume–vectorial indexΘ a (1) ≡ d ˜ Z A ρ a Z A − ˜ Z A ρ a dZ A (60)one can obtain the action (59) as induced on the mem-brane world volume by the following three–formΘ (3) = ǫ abc Θ a (1) ∧ Θ b (1) ∧ Θ c (1) . (61) V. OUTLOOK
In this paper we presented the new description of thetwistorial membrane in D = 4 space–time. We wouldlike now to comment on two generalizations: i) to p –branes with p > D –dimensional( D > p + 1) space–time ii) to super– p –branes in higher dimensions. The basic relation in the construction of twistorial for-mulation of the p –branes in dimension D is a suitablegeneralization of Cartan–Penrose formula (33). For arbi-trary p and arbitrary D such a formula looks as follows: P mµ = ee ma ˜ λ ˆ αi ( ρ a ) ij λ ˆ βj ( γ µ ) ˆ α ˆ β (62)where γ µ are the Dirac matrices in D –dimensional space–time and ρ a are Dirac matrices in d = p + 1 dimensionalspace (tangent space to curved world volume geometryof the p –brane). Thus the spinor λ ˆ αi has two indices:it is spinor in D dimensions with the components de-scribed by index ˆ α and as well spinor in d dimensionswith index i ; we employ also the spinor ˜ λ ˆ αi which isa Dirac–conjugated spinor with respect to both indices:˜ λ ˆ αi = ( λ ˆ βj ) + ( ρ ) ji ( γ ) ˆ β ˆ α . For general D and p we getthat ˆ α = 1 , . . . , [ D ] and i = 1 , . . . , [ p +12 ] . Thus in or-der to obtain the composite p –brane momenta we mustuse at least 2 [ p +12 ] twistors. For definite dimensions p and D we can further decrease the number of the ele-mentary spinor components λ ˆ αi by imposing consistentlyMajorana-, Weyl- or Majorana–Weyl conditions.Inserting (62) in the action (8) we shall obtain the in-termediate spinor–space-time formulation. Due to themass–shell for the vectorial momenta (see (23)) theelementary spinors λ ˆ αi will be constrained (comparewith (35) and (45)). In order to get the purely twisto-rial formulation of p –branes one has to introduce D –dimensional incidence relation (47) which provides thedoubling of spinor components and lifts the Lorentzspinors to twistors. It should be stress however that ingeneral case the D –dimensional incidence relation will in-troduce extended D –dimensional space–time. Only suit-able use of the additional spinor structures (e. g. quater-nionic in D = 6) and imposition of the algebraic con-straints in twistor space (e. g. selecting only null twistorslying on null hyperplanes) permits to obtain the inci-dence relations just with the Minkowski space–time co-ordinates.The extension of the twistorial formalism for bosonic p –branes to super– p –branes requires the introduction of p –brane supertwistors. The techniques of supersym-metrization of various twistorial p –brane models were al-ready studied (see e. g. [22, 23]). It should be also men-tioned that our construction can be linked to the analysisbased on the use of Lorentz harmonics [19] as well as withthe formalism using d = 11 BPS preons [40, 41] describedby generalized OSp (1 |
64) supertwistor fields.
Acknowledgments
S.F. would like to thank Institute for TheoreticalPhysics, Wroc law University for kind hospitality and avery friendly creative atmosphere. He would like to thankBogoliubov–Infeld program for financial support. Thework of S.F. was partially supported also by the RFBRgrant 06-02-16684 and the grant INTAS-05-7928. [1] R. Penrose and M.A.H. MacCallum, Phys. Reports C6 ,241 (1972);R. Penrose, in “Quantum Gravity”, ed. C.J. Isham, R.Penrose and D.W. Sciama, p. 268, Oxford Univ. Press,1975.[2] A. Ferber, Nucl. Phys. B132 , 55 (1978).[3] T. Shirafuji, Prog. Theor. Phys. , 55 (1983).[4] N. Bengtsson and M. Cederwall, Nucl. Phys. B302 , 81(1988).[5] D. Sorokin, V. Tkach and D.V. Volkov, Mod. Phys. Lett. A4 , 901 (1989).[6] L.P. Hughston, Twistors and Particles , Lecture Notes inPhysics , Springer Verlag, Berlin (1979)[7] A. Bette, J. Math. Phys. , 2456 (1984).[8] S. Fedoruk and V.G. Zima, J. Kharkov Univ. , 39(2003) [hep-th/0308154] .[9] A. Bette, J.A. de Azc´arraga, J. Lukierski andC. Miquel-Espanya, Phys. Lett. B595 , 491-497 (2004) [hep-th/0405166] .[10] J.A. de Azc´arraga, S. Fedoruk, A. Frydryszak, J. Lukier-ski and C. Miquel-Espanya,
D73 , 105011 (2006) [hep-th/0510161] .[11] S. Fedoruk, A. Frydryszak, J. Lukierski and C. Miquel-Espanya, Int. J. Mod. Phys.
A21 , 4137 (2006) [hep-th/0510266] .[12] I. Bars and M. Picon, Phys. Rev.
D73 , 064002(2006) [hep-th/0512091] ; D73 , 064033 (2006) [hep-th/0512348] .[13] L.J. Mason, JHEP , 009 (2005) [hep-th/0507269] .[14] R. Boels, L. Mason and D. Skinner, JHEP , 014(2007) [hep-th/0604040] ; Phys. Lett.
B648 , 90 (2007) [hep-th/0702035] .[15] E. Witten, Commun. Math. Phys. , 189 (2004) [hep-th/0312171] .[16] N. Berkovits and E. Witten, JHEP , 009 (2004) [hep-th/0406051] .[17] S. Fedoruk and J. Lukierski, Phys. Rev.
D75 , 026004 (2007) [hep-th/0606245] .[18] T.R. Taylor,
Non-commutative Field Theory withTwistor Coordinates , arXiv: 0704.2071 [hep-th] .[19] I.A. Bandos and A.A. Zheltukhin, Class. Quant. Grav. , 609 (1995) [hep-th/9405113] .[20] I. Bandos, P. Pasti, D. Sorokin, M. Tonin and D. Volkov,Nucl. Phys. B446 , 79 (1995) [hep-th/9501113] .[21] I. Bandos, D. Sorokin and D. Volkov, Phys. Lett.
B352 ,269 (1995) [hep-th/9502141] .[22] I.A. Bandos, J.A. de Azc´arraga and C. Miquel-Espanya,JHEP , 005 (2006) [hep-th/0604037] .[23] D.V. Uvarov, Class. Quant. Grav. , 2711 (2006) [hep-th/0601149] ; Gauge symmetries of strings insupertwistor space , [hep-th/0606222] ; Supertwistorformulation for higher dimensional superstrings , [hep-th/0703051] .[24] W. Siegel, Nucl. Phys. B263 , 93 (1985).[25] M. Pavsic, Phys. Lett.
B197 , 327 (1987).[26] M. Gurses and F. Gursey, Phys. Rev.
D11 , 967 (1975).[27] A. Aurilia and E. Spallucci, Class. Quant. Grav. , 1217(1993) [hep-th/9305020] .[28] O.E. Gusev and A.A. Zheltukhin, JETP Lett. , 487(1996).[29] J.A. Nieto, Mod. Phys. Lett. A16 , 2567 (2001) [hep-th/0110227] .[30] P.A.M. Dirac, Proc. Roy. Soc. Lond.
A268 , 57 (1962).[31] Y. Nambu, Lectures at the Copenhagen Summer Sym-posium (1970);T. Goto, Prog. Theor. Phys. , 1560 (1971).[32] P.S. Howe and R.W. Tucker, J. Phys. A10 , L155 (1977).[33] A.M. Polyakov, Phys. Lett.
B103 , 207 (1981).[34] A. Sugamoto, Nucl. Phys.
B215 , 381 (1983).[35] L. Brink, P. Di Vecchia and P.S. Howe, Phys. Lett.
B65 ,471 (1976); Nucl. Phys.
B118 , 76 (1977).[36] V.A. Soroka, D.P. Sorokin, V.V. Tkach and D.V. Volkov,JETP Lett. , 526 (1990), Int. Journ. Mod. Phys. A7 ,5977 (1992). [37] A. Schild, Phys. Rev. D16 , 1722 (1977).[38] A. Karlhede and U. Lindstrom, Class. Quant. Grav. ,L73 (1986);J. Isberg, U. Lindstrom and B. Sundborg, Phys. Lett. B293 , 321 (1992) [hep-th/9207005] .[39] L. Brink and J.H. Schwarz, Phys. Lett.
B100 , 310 (1981). [40] I.A. Bandos, J.A. de Azc´arraga, J.M. Izquierdo andJ. Lukierski, Phys. Rev. Lett. , 4451 (2001) [hep-th/0101113] .[41] I.A. Bandos, J.A. de Azc´arraga, M. Picon and O. Varela,Phys. Rev. D69 , 085007 (2004) [hep-th/0307106][hep-th/0307106]