Superembedding approach to M0-brane and multiple M0-brane system
aa r X i v : . [ h e p - t h ] M a r January, 2010
Superembedding approach to M -brane andmultiple M –brane system Igor A. Bandos † IKERBASQUE, the Basque Foundation for Science, andDepartment of Theoretical Physics, University of the Basque Country,P.O. Box 644, 48080 Bilbao, Spain
We study the possibility to describe multiple M0–brane system in the frame ofsuperembedding approach. The simplest framework is provided by the maxi-mally supersymmetric non-Abelian SU ( N ) Yang–Mills supermultiplet on the d = 1 N = 16 superspace the embedding of which to the target D = 11 super-gravity superspace is determined by the so-called superembedding equation,characteristic of the worldline superspace of a single M0-brane. We use it toobtain a covariant generalization of the Matrix model equations describing themultiple M0–system in flat target superspace. Also at Institute for Theoretical Physcs, NSC Kharkov Institute of Physics & Technology, UA 61108,Kharkov, Ukraine. E-mail: igor [email protected], bandos@ific.uv.es
Introduction
The action for M0-brane, the Brink-Schwarz action for massless eleven dimensional su-perparticle, was discussed for the first time by Bergshoeff and Townsend in [1], were itwas used as a starting point to obtain the 10D type IIA D0-brane action, the p = 0representative of the D p -branes (Dirichlet p –branes) [2, 3, 4].Although the rˆole of D p -branes and multiple D p -brane system in String/M-theory [5]was appreciated in middle 90th [6], the problem of Lorentz covariant and supersymmetricaction which governs dynamics of the system of N D p -branes still remains unsolved. Evenbefore the nonlinear action for single D p –brane were found in [7, 8, 9, 1], it was understoodthat at the very low energy approximation this should reduce to the action of maximallysupersymmetric Yang–Mills theory (SYM) [10, 11], which can be obtained by dimensionalreduction of D = 10 SYM down to d = p + 1. The purely bosonic action for multipleD p -brane system was proposed by Myers [12]; he derived it by chain of dualities startingfrom the 10D non-Abelian DBI–action with symmetric trace prescription proposed byTseytlin [13]. However, more than decade of attempts to make supersymmetric andLorentz covariant generalization of the action [12] and/or [13] was not successful (althougha particular progress was achieved in the cases of low dimensions D , low dimensionalbranes and low co-dimensional branes [14, 15]).The only exception is the boundary fermion approach by Howe, Lindstr¨om and Wulff[16, 17] which does provide supersymmetric and covariant description of Dirichlet branes,but on the ’pure classical’ (or ’menus one quantization’) level in the sense that to arrive atthe description of multiple D-brane system in terms of the variables corresponding to thestandard single D p –brane action [8, 9, 1] (usually considered as a classical or quasi-classicalaction) one has to perform a quantization of the boundary fermion sector. The completequantization of the model [16, 17] should produce not only worldvolume fields of multi-ple D p -brane system but also bulk supergravity and higher stringy modes. The partialquantization of only the boundary fermion sector allowed Howe, Lindstr¨om and Wulff toreproduce the purely bosonic Myers action [12], but the Lorentz invariance was lost on thisway. The supersymmetric action [17], involving quite interesting and novel prescription ofintegration, possesses the κ –symmetry with parameters dependent of boundary fermions.This implies that, upon the quantization, the action with non-Abelian κ –symmetry shouldappear. The previous attempts to incorporate non-Abelian κ –symmetry gave negative re-sults [18]. Probably, the further development of the boundary fermion approach will help,in particular, to resolve this problem as well as other problems which hamper the wayto constructing the supersymmetric and Lorentz covariant (diffeomorphism invariant) ac-tion for multiple D p –brane system similar to the DBI plus WZ action for single D p –brane[8, 9, 1] . It also might happen that the boundary fermion action of [17] is the best whatone can write to describe multiple D p –brane system without involving additional fieldscarrying the bulk supergravity and/or other stringy degrees of freedom. In both cases inthe middle time it looks reasonable, not diminishing the importance of the development ofboundary fermion approach, to search for a more straightforward, probably approximatebut covariant and supersymmetric description, going beyond the U ( N ) SYM approxima-tion, and formulated in variables similar to the ones used to write single D p –brane action DBI is for Dirac–Born–Infeld and WZ is for Wess–Zumino. The action of [8, 9, 1] is given by the sumof Dirac–Born–Infeld, kinetic term and of the Wess–Zumino term describing coupling to the RR gaugefields, the antisymmetric tensor (differential form) potentials C p +1 , C p − , . . . .
28, 9, 1].Recently [19] a way to search for multiple D p -brane equations in the frame of su-perembedding approach [20, 21, 22, 23, 24, 25, 26] was proposed and elaborated for thecase of D0-branes. Roughly speaking the proposition of [19] is to describe the N D p –brane system by imposing a suitable set of constraints on field strengths of the maximal d = p +1 dimensional SU ( N ) SYM super-1-form potential which lives on the worldvolumesuperspace W ( p +1 | with d = p + 1 bosonic and 16 fermionic directions the embedding ofwhich into the type II supergravity superspace is determined by the so–called superem-bedding equation. The latter universal equation describes completely the dynamics ofsingle D p -brane with p < C q with order q >p + 1) which do not interact with a single D p -brane. In the case of flat target superspaceour multiple D0–brane equations coincide with the result of dimensional reduction of 10DSYM equation to d = 1 [19]. This is important because it shows the relation with Matrixmodel [29] the description of which uses the d = 1 reduction of maximal SYM action asa starting point.However, one of the most important property of the Matrix model is the 11D invari-ance. The possibility of the restoration of the 11D Lorentz invariance in the system of10D D0-branes as described by 1d reduction of the 10D SYM action, was one of the keypoints in [29] which allowed the M(atrix) model conjecture on that the Matrix modelconsidered as multiple D0-brane system provides a nonperturbative description of theunderlying String/M-theory. In this respect the natural question is whether one can showthe restoration of 11D Lorentz invariance in the framework of superembedding descriptionof multiple D0-brane proposed in [19].It seems that the simplest way to check this is to try to develop the similar descrip-tion of 11D multiple M0–brane system. Indeed, as far as D0–brane can be obtained by‘dimensional reduction’/duality of M0-brane [1], it is natural to expect that multiple D0-brane system can be obtained by ‘dimensional reduction’/dualization of the system ofmultiple M0-branes, if such exists. Then one can also expect that, as in the case of singlebranes, this ‘dimensional reduction’/dualization is invertible, and the inverse transfor-mation would give the desired restoration of the 11D Lorentz invariance in the multipleD0-brane system described by superembedding approach of [19].Furthermore, the possible existence of multiple M0-brane system may be interestingon its own. The fact that, in distinction to D-particles (D0-branes), M-particles (M0-s)are massless, suggests a possibility of appearance of novel features and/or novel problemswhen studying multiple M0-system. The aim of the presence letter is to check whetherone can develop a description of such a system in the frame of superembedding approachon the line of [19].We begin in the next Sec. 2 by a brief review of superembedding approach on theexample of single M0-brane. This allows us to fix the notation and to provide the basisfor the superembedding description of multiple M0 system which we develop in Sec. 3.The concluding Sec. 4 contains discussion and outlook.3 Superembedding approach to a single M -brane. D = 11 superspace Σ (11 | We denote the supervielbein of target D=11 superspace Σ (11 | by E A := dZ M E M A ( Z ) = ( E a , E α ) , (cid:26) α = 1 , . . . , ,a = 0 , , . . . , , . (2.1)and its local coordinates by Z M = ( x m , θ ˇ α ) , ˇ α = 1 , . . . , , m = 0 , , . . . , , . (2.2)The supervielbein (2.1) describes 11D supergravity when it obeys the set of superspaceconstraints [31] the most essential of which are collected in the following expression forthe bosonic torsion two form: T a := DE a = − i E ∧ Γ a E . (2.3)In (2.3) E ∧ Γ a E := E α ∧ Γ aαβ E β , the symmetric matrices Γ aαβ = (Γ a C ) αβ = Γ aβα are real,while the 11D Dirac matrices (Γ a ) αβ and charge conjugation matrix C are imaginary inour mostly minus notation η ab = diag (+ , − , . . . , − ). Below we will mostly use Γ aαβ andtheir counterparts with upper case indices, the real matrices ˜Γ aαβ = (Γ a C ) αβ = ˜Γ aβα .Studying Bianchi identities with (2.3) one finds the structure of fermionic torsion and SO (1 ,
10) curvature two form [31] T α = − i E a ∧ E β (cid:18) F ac c c Γ c c c αβ + 18 F c c c c Γ ac c c c αβ (cid:19) + 12 E a ∧ E b T baα ( Z ) , (2.4) R ab = E α ∧ E β (cid:18) − F abc c Γ c c + i .
5! ( ∗ F ) abc ...c Γ c ...c (cid:19) αβ ++ E c ∧ E α (cid:0) − iT abβ Γ cβα + 2 iT c [ a β Γ b ] βα (cid:1) + 12 E d ∧ E c R cdab ( Z ) . (2.5)Here F abcd = F [ abcd ] ( Z ) is the bosonic superfield strength of the 3-form gauge field of the11D supergravity. It obeys D [ a F bcde ] = 0 and D α F abcd = − T [ abβ Γ cd ] βα . –brane W ( p +1 | of M-branes The standard formulation of M–branes (M p -branes with p = 0 , ,
5) deals with embeddingof a purely bosonic worldvolume W p +1 (worldline for M0-case of [1]) into the targetsuperspace Σ (11 | . The superembedding approach to M-branes [20, 22], following theso–called STV (Sorokin–Tkach–Volkov) approach to superparticles and superstrings [32](see [25] for review and further references) describes their dynamics in terms of embedding Here and below we write explicitly the exterior product symbol ∧ . The exterior product of a q -formΩ q and a p -form Ω p has the property Ω q ∧ Ω p = ( − pq Ω p ∧ Ω q if at least one of two differential formsis bosonic; when both are fermionic, an additional ( −
1) multiplier appears in the r.h.s. . The exteriorderivative acts on the products of the forms ‘from the right’: d (Ω q ∧ Ω p ) = Ω q ∧ d Ω p + ( − p d Ω q ∧ Ω p . worldvolume superspace W ( p +1 | with d = p + 1 bosonic and 16 fermionic directionsinto the target superspace Σ (11 | .The embedding can be described in terms of coordinate functions ˆ Z M ( ζ ) = (ˆ x m ( ζ ) , ˆ θ ˇ α ( ζ )),which are superfields depending on the local coordinates of W ( p +1 | ( ζ M = ( τ, η ˇ q ), withˇ q = 1 , . . . ,
16 in the case of M0-brane) W ( p +1 | ∈ Σ ( D | : Z M = ˆ Z M ( ζ ) = (ˆ x m ( ζ ) , ˆ θ ˇ α ( ζ )) , (2.6) m = 0 , , .., ,
10, ˇ α = 1 , ..., superembedding equa-tion which completely determines the M p -brane dynamics. –brane To write superembedding equation for the case of M0–brane, let us denote the superviel-bein and local coordinates of the corresponding worldvolume superspace W | by e A = dζ M e M A ( ζ ) = ( e , e + q ) , ζ M = ( τ, η ˇ q ) , (cid:26) q = 1 , . . . , , ˇ q = 1 , . . . , , (2.7)where q is a spinor index of SO (9), + denotes the weight (‘charge’) with respect to thelocal SO (1 ,
1) group and the bosonic vielbein has the weight two, e := e ++ . Notice thatin our notation the upper plus sign is equivalent to the lower minus sign and vice-versa,so that one can equivalently write, for instance, e + q = e q − and e = e = := e −− .Now, let us denote the pull–back of supervielbein E A ( Z ) of target superspace, Eq.(2.1), to W | by ˆ E A := E A ( ˆ Z ). The general form of its decomposition of the basis ofworldvolume supervielbein readsˆ E A := E A ( ˆ Z ) = d ˆ Z M E M A ( ˆ Z ) = e ˆ E A + e + q ˆ E + qA . (2.8)The superembedding equation states that the pull–back of bosonic supervielbein to theworldvolume superspace has no projection on the fermionic supervielbein, i.e. it readsˆ E + qa = 0 . (2.9)After some algebra, one can show that the equivalent form of the superembedding equation(2.9) is given by [20, 30] (cid:26) ˆ E a u ai = 0 , ˆ E a u = a = 0 , (2.10)and also by ˆ E a = 12 e u a = , (2.11)where u a = u a ( ζ ), u a = = u a −− ( ζ ) and u ai = u ai ( ζ ) ( i = 1 , . . . ,
9) are moving framevector superfields. These are elements of pseudo–orthogonal moving frame matrix U ( b ) a = (cid:18)
12 ( u a + u a = ) , u ai ,
12 ( u a − u a = ) (cid:19) ∈ SO (1 , , (2.12)5hich is to say u a = ( ζ ) = u a −− ( ζ ) and u a ( ζ ) = u a ++ ( ζ ) are light–like 11-vectors withzero weight contraction u a = u a normalized to be equal to 2, and u ai ( ζ ) are 9 orthogonaland normalized 11–vectors superfields which are orthogonal also to u a ±± ( ζ ), u a u a = 0 , u = a u a = = 0 , (2.13) u a u a = = 2 , (2.14) u a u a i = 0 , u = a u a i = 0 , u ia u a j = − δ ij . (2.15) Using (2.14) one finds that Eqs. (2.11) contains, in addition to (2.10), also the conventionalconstraint which defines the bosonic supervielbein of the worldline superspace W (1 | tobe induced by (super)embedding, e = ˆ E := ˆ E b u b . (2.16)To specify further the geometry of W (1 | one has to define as well the SO (1 , ⊗ SO (9)connection and the fermionic supervielbein e + q induced by superembedding . It is con-venient to define this connection by the equations specifying the action of the (exterior) SO (1 , ⊗ SO (1 , ⊗ SO (9) covariant derivative on the moving frame vectors Du b = = u bi Ω = i , Du b = u bi Ω i , Du bi = 12 u b Ω = i + 12 u b = Ω i . (2.17)These equations involve the generalization of the SO (1 , SO (1 , ⊗ SO (9) covariant Cartan forms(where generalization means the use of SO (1 ,
10) Lorentz covariant derivative instead ofthe usual derivatives in the definition of proper Cartan forms). In the case of M0-brane,which is massless superparticle, the form Ω i is actually not covariant under the whole setof bosonic gauge symmetries (SO(1,10) ⊗ { [SO(1,1) ⊗ SO(9)] ⊂× K } in the case of curvedtarget superspace) but transforms as connections under K transformations (see footnote4). In contrast, Ω = i is covariant and generalizes the SO (1 , SO (1 , ⊗ SO (9)] ⊂× K Cartan form.The forms Ω i , Ω = i obey the generalized Peterson-Codazzi equations (see [20, 30]) D Ω = i = ˆ R = i , D Ω i = ˆ R i ( ˆ R i := ˆ R ab u a u bi , ˆ R = i := ˆ R ab u = a u bi ) . (2.18)The induced Riemann curvature 2-form ( d Ω (0) from DDu a = +( − ) d Ω (0) u a + ˆ R ab u b )and the curvature of normal bundle of the worldline superspace ( G ij from DDu ia = u ja G ji + ˆ R ab u bi ) are defined by the Gauss and Ricci equations d Ω (0) = 14 ˆ R = + 14 Ω = i ∧ Ω i , ˆ R = := ˆ R ab u = a u b , (2.19) G ij = ˆ R ij − Ω = [ i ∧ Ω j ] , ˆ R ij := ˆ R ab u ia u bj . (2.20) The true group of gauge invariance is the small group of massless superparticle, which is the so-calledBorel subgroup [ SO (1 , ⊗ SO (9)] ⊂× K of the Lorentz group SO (1 , K transformationsare defined by δu = 2 k i u i , δu i = k i u = (see [28] and refs. therein for details). However, whendeveloping superembedding approach and twistor-like formulations for massless superparticles [30, 28], itis convenient to use the formalism where only the gauge symmetry under [ SO (1 , ⊗ SO (9)] subgroupof [ SO (1 , ⊗ SO (9)] ⊂× K is manifest. T a := − i ˆ E ∧ Γ a ˆ E = 12 De u a = + 12 e ∧ Ω = i . (2.21) To move further, we need to define the fermionic supervielbein of the worldline superspace W (1 | . It is convenient to do this by specifying the pull–back of the target superspacefermionic supervielbein form to W (1 | ˆ E α = e + q v − αq + e χ − q v + αq . (2.22)This expression involves so–called spinor moving frame variables or spinorial harmonics(see [20, 30, 28] and refs. therein). These can be considered as 32 ×
16 blocks of the
Spin (1 ,
10) valued moving frame matrix V ( β ) α = (cid:0) v − αq , v + αq (cid:1) ∈ Spin (1 , , (2.23)and are related to the moving frame vectors by the equations v + q Γ a v + p = u a δ qp , v + αq v + q β = ˜Γ aαβ u a , (2.24) v − q Γ a v − p = u a = δ qp , v − αq v − q β = ˜Γ aαβ u a = , (2.25) v − q Γ a v + p = − u ai γ iqp , v − ( αq γ iqp v + p β ) = − ˜Γ aαβ u ai , (2.26)where γ iqp = γ ipq are the SO (9) Dirac matrices. Notice that d=9 charge conjugation matrixis symmetric which allowed to choose it equal to δ qp and to do not distinguish the upperand lower case indices of Spin (9).Eqs. (2.24)–(2.26) can be obtained from the condition of the Dirac matrix conservationunder Lorenz rotations described by moving frame superfields, V Γ a V T = u ( b ) a Γ ( b ) and V T ˜Γ ( a ) V = ˜Γ b u ( a ) b , by using a suitable SO (1 , ⊗ SO (9) invariant representation for theDirac and charge conjugation matrices. These equations, together with the conditionof the charge conjugation matrix preservation under Lorentz rotation described by themoving frame variables, V CV T = C , are the constraints which define the spinor movingframe variables. The latter implies that the elements of the inverse moving frame matrix, V ( β ) α = (cid:0) v αq + , v αq − (cid:1) ∈ Spin (1 , , (cid:26) v − αq v αp + = δ qp = v + αq v αp − ,v − αq v αp − = 0 = v + αq v αp + , (2.27)are related with v + αq by v ± αq = ± iC αβ v βq ± . (2.28)The SO (1 , ⊗ SO (1 , ⊗ SO (9) covariant derivatives of the moving frame variables areexpressed through the generalized Cartan forms of Eqs. (2.17) by the following equations(see [28] and refs. therein for the details of derivation) Dv − αq = − Ω = i v + αp γ ipq , (2.29) Dv + αq = − Ω i v − αp γ ipq . (2.30)7sing (2.27) we can split Eq. (2.22) on two equations e + q = ˆ E + q := ˆ E α v αq + , (2.31)ˆ E − q := ˆ E α v αq − = e χ − q , (2.32)the first of which manifests the conventional constraint defining fermionic supervielbeinof the worldline superspace W (1 | to be induced by the superembedding.The second equation, (2.32), states that v αq − projection of pull–back to W (1 | ofthe target superspace fermionic supervielbein, ˆ E − q , has no projection on the fermionicsupervielbein. One can start from the most general decomposition ˆ E − q := ˆ E α v αq − = e + p h = pq + e χ − q . However then, checking the selfconsistency condition (2.21) for thesuperembedding equation, one arrives (in its dim 1 , ∝ e + q ∧ e + p u a component) at h = ( h = ) T = 0. In the case of real number valued matrix elements the only solution ofthis equation is h = qp = 0. In the supersymmetric case one can consider a nonzero solutionbut with all the matrix elements being nilpotent. The rˆole of such a solution and ofits nilpotent Grassmann even parameters is presently unclear so that below we restrictourselves by considering the trivial solution h = qp = 0 only. Moreover, we have allowedourselves a shortcut and have used this solution in Eqs. (2.22) and (2.32), specifying thepull–back of the target space fermionic supervielbein form, from the very beginning. –brane equa-tions of motion from superembedding Substituting Eq. (2.22) into the integrability conditions for superembedding equation,Eq. (2.21), one finds, besides the expression for the worldvolume bosonic torsion 2-form, De = − ie + q ∧ e + q , (2.33)also the expression for dim 1/2 ( ∝ e + q ) component of the generalized Cartan form Ω = i entering Eqs. (2.17) and (2.29), Ω = i + q = − iγ iqp χ p − . This implies thatΩ = i + q = − ie + q γ iqp χ p − + e Ω = i , (2.34)where χ p − := ˆ E α v αp − , Ω = i := D u = a u ai = D ˆ E a u ai . (2.35)Eqs. (2.35), which resume some consequences of Eqs. (2.22), (2.17) and (2.11), show thatthe bosonic and fermionic equations of motion for M0–brane can be formulated in termsof the differential form Ω = i (see [30] for similar statements in D = 10 superparticle case).At this stage it is useful to calculate the pull–back of the fermionic torsion (2.4),ˆ T α = − e ∧ e + q ˆ F = ijk γ ijkqp v − αp , ˆ F = ijk := F abcd ( ˆ Z ) u a = u bi u cj u dk , (2.36)and substitute it in the integrability conditions for the fermionic differential form equation(2.22), D ( ˆ E α − e + q v − αq − e χ − q v + αq ) = 0 . (2.37)8he lowest dimensional (dim 1) component of the v αq − projection of this equation resultsin requirement that ( χ − γ i ) ( r γ ip ) q = δ pq χ q − . This equation has only trivial solution,which implies that the superembedding equation results in the fermionic equations ofmotion for the M0–brane, χ − q = 0. The next, dim 3/2 component of the v αq − projectionof Eq. (2.37) results in D + p χ q − = − γ ipq Ω −− i , so that the above fermionic equationimplies the bosonic equation of motion Ω −− i = 0. To resume, we have shown howthe superembedding equation leads to the dynamical bosonic and fermionic equations ofmotion of the M0–brane, χ p − := ˆ E α v αp − = 0 , (2.38)Ω = i := D u = a u ai = D ˆ E a u ai = 0 . (2.39)Let us stress that the above relations are equations of motion for M0-brane in an arbitrary D = 11 supergravity background. When passing from flat to curved target superspace,no explicit fluxes appear in the right–hand side of these equations ( cf. D = 10 D0-branecase [19]), but the bosonic and fermionic supervielbein, the pull–back of which to W (1 | enter (2.38), acquire the contribution form the supergravity fields.Notice that, besides the equations of motion, the investigation of Eq. (2.37) (of its v αq + projection) allows to find the fermionic torsion of the worldvolume geometry inducedby superembedding, De + q = − e ∧ e + q ˆ F = ijk γ ijkqp , ˆ F = ijk := F abcd ( ˆ Z ) u a = u bi u cj u dk . (2.40)The influence of the 4-form flux on the induced geometry of W (1 | is seen in this equationand also in the expression for the curvature of normal bundle (see (2.20), G ij = ˆ R ij = e + q ∧ e + p (cid:18) i F = ijk γ kqp + i
18 ˆ F = klm γ ijklmqp (cid:19) + ie ∧ e + q ˆ T = [ i − p γ j ] pq (2.41)( γ i ...i qp = ǫ i ...i j ...j qp γ j ...j qp , ˆ T = i − q := ˆ T abβ u a = u bi v βq − ). The Riemann curvature of W (1 | (see (2.19)) vanishes on the mass shell, d Ω (0) = 0. -equations in the frameof superembedding approach In this section we will report first results of the search for multiple M0-brane equationsin the frame of superembedding approach, which can be considered as the search forrestoration of D = 11 Lorentz invariance in the multiple D0-brane equations of [19]. Asfar as the latter are obtained by considering the constraints of maximally supersymmetric SU ( N ) YM gauge theory on the worldline superspace of D0-brane, it is natural to searchfor multiple M0–brane equations by studying the possible constraints imposed on thesuperfield strength G = dA − A ∧ A = 12 e + q ∧ e + p G −− q p + e ∧ e + q G + q (3.42)9f the su ( N )-valued 1-form gauge potential A = e A + e + q A + q defined on d = 1 N = 16superspace W (1 | . It is also natural to assume that the embedding of W (1 | into the11D target superspace is determined by superembedding equation (2.11),ˆ E a = 12 e u a = (3.43)(see [19] for more discussion). Then this superspace describing the ‘center of mass’ motionof the multiple M0–system is described by the superfield equations of Sec. 2.We have put quotation mark on ‘center of mass’ as M0-branes are massless superparti-cles, so that one should rather say ’center of energy’. As the superembedding equation for W (1 | ⊂ Σ (11 | puts the theory on mass shell, one finds that this center of energy motionis characterized by light-like geodesic in the bosonic body of the target 11D superspace.As an analogy, one can imagine the system of multiple M0-s as the beam of light, thetrajectory of which is light-like, but with an interaction between the (originally massless)constituents which is assumed to be described by the above d = 1 N = 16 SU ( N ) gaugesupermultiplet. d = 1 N = 16 SU ( N ) SYM on W (1 | The natural candidate for the constraints, which is also suggested by the D0-brane de-scription of [19], reads G −− q p = iγ iqp X i . (3.44)The nanoplet of su ( N )-valued superfields X i (= X i ) provides a natural candidate for thedescription of the relative motion of the M0–brane constituents. The lowest dimensional(dim 1, ∝ e + q ∧ e + p ) component of the Bianchi identities DG = 0 requires that X i obeysthe following superembedding–like superfield equation D + q X i = 4 iγ iqp Ψ q , (3.45)and identify the fermionic su ( N ) valued (Hermitian traceless) matrix superfield Ψ q withthe dim 3/2 field strength in (3.42), G + q = − i Ψ q . (3.46)The highest dimensional (dim 3/2, ∝ e ∧ e + q ) component of this Bianchi identity is satis-fied identically when the consequences of the integrability conditions for the superembedding–like equation (3.45) are taken into account. These include 8 iγ ir ( p D + q ) Ψ r = 4 iδ qp D X i + iγ jqp [ X j , X i ] + X j G ji + q + p which, after some algebra, results in D + q Ψ p = 12 γ iqp D X i + 116 γ ijqp (cid:18) [ X i , X j ] −
43 ˆ F = ijk X k (cid:19) − γ ijklqp X i ˆ F = jkl . (3.47) Although below we do not specify the SO (1 ,
1) weights of the matrix superfields, it is convenient toremember that, e.g. X i := X i ≡ X ++ i ≡ X −− i =: X = i . To arrive at this conclusion, one uses the d = 9 γ –matrix identity γ i ( pq γ ir ) s = δ ( pq δ r ) s . Among theirother useful properties are δ p ( r δ s ) q = δ pq δ rs + γ ipq γ irs + · γ ijklpq γ ijklrs and γ ijkl ( pq γ ijklr ) s = 14 · δ ( pq δ r ) s . .2 Equations of motion for multiple M -brane system in flattarget D = 11 superspace For simplicity, we restrict our discussion below by the case of multiple M0-brane systemin flat target superspace, where, in particular, ˆ F = jkl = 0, De + q = 0, G ij = 0 and Eq.(3.47) simplifies to D + q Ψ p = 12 γ iqp D X i + 116 γ ijqp [ X i , X j ] . (3.48)The investigation of the more general case of curved supergravity superspace, includingthe study of the dielectric brane effect for the multiple M0–system, will be the subject ofthe separate study [33].Now, the integrability condition for Eq. (3.48) ( D +( p D + q ) Ψ r = ... ) reads( δ qp δ rs − γ ir ( p γ iq ) s ) D Ψ q = (cid:16) γ ir ( p δ q ) s − γ ipq γ irs − γ ijr ( p γ jq ) s (cid:17) (cid:2) X i , Ψ s (cid:3) . (3.49)The δ qp trace part of (3.49) gives the fermionic equations of motion D Ψ q + γ iqp (cid:2) X i , Ψ p (cid:3) = 0 , (3.50)while the other irreducible parts are satisfied identically after Eq. (3.50) is taken intoaccount. Acting on Eq. (3.50) by Grassmann covariant derivative D + p and decomposingthe result on irreducible representations of SO (9) one finds the bosonic equation of motion D D X i = (cid:2) X j , (cid:2) X j , X i (cid:3)(cid:3) + iγ iqp { Ψ q , Ψ p } , (3.51)as well as the constraint (cid:2) X i , D X i (cid:3) = − i { Ψ q , Ψ q } , (3.52)This latter is the counterpart of the Gauss constraint of the multiple D0-brane systemobtained from the superembedding description in [19].Thus our superembedding description is consistent and produces the dynamical equa-tions for the multiple M0-brane system in flat target superspace given in Eqs. (2.38),(2.39), (3.50) (3.51) and (3.52). Our approach can be considered as a covariant general-ization of the Matrix model. The study of the system of equations for multiple M0-systemin the nontrivial D = 11 supergravity background, including the description of dielectricbrane effect in this case, will be discussed elsewhere [33]. In this letter we have studied a possibility to describe multiple M0-brane system in theframe of superembedding approach. One of the motivation for such a study is thatexistence of such a system can be considered as an indication of restoration of the 11 D Lorentz symmetry in the superfield description of multiple D0–brane system of [19]. It isalso of interest in the context of searching for a complete, covariant and supersymmetricgeneralization of the Matrix model equations.11e have studied the simplest version of the superembedding approach based on themost natural 1d N = 16 SYM constraints imposed on the su ( N ) valued 2-superformfield strength defined on superspace W (1 | with one bosonic and 16 fermionic directions.The embedding of W (1 | into the target 11 D superspace Σ (11 | obeys superembeddingequation characteristic for a single M0-brane. We have shown that such a descriptionis consistent and produces the interacting bosonic and fermionic equations of motion forthe (super)field describing relative motion of the constituents of multiple M0–system. Al-though the basic equations of our formalism and some of their consequences have beenwritten in general D = 11 supergravity background, the equations of motion have beenpresented for the case of multiple M0-brane system in flat target superspace only. Thedetailed study of the multiple M0 equations in the general curved 11D supergravity su-perspace, which shall provide a covariant generalization of Matrix model equation to thecase of nontrivial supergravity background, will be the subject of future paper [33]. Acknowledgments
The author thanks Djordje Minic and Martin Kruczenski for useful conversations as wellas Thomas Curtright, Jo Ann Curtright and Luca Mezincescu for the hospitality at theMiami 09 Conference in Fort Lauderdale, where this work was essentially completed.The partial support by the research grants FIS2008-1980 from the Spanish MICINN and38/50–2008 from Ukrainian National Academy of Sciences and Russian Federation RFFIare greatly acknowledged.
Notice added
After the first version of this letter had appeared on the net the author became awareabout the paper [34] where a counterpart of the Myers action for the case of multipleM0-brane (‘multiple M-theory gravitational waves/gravitons’) was proposed and studied.This (purely bosonic) action was obtained starting from the counterpart of the Myersaction for multiple type IIA gravitons [35].
References [1] E. Bergshoeff, P.K. Townsend,
Super- D -branes , Nucl.Phys. B490 (1997) 145–162,[hep-th/9611173].[2] A. Sagnotti,
Open strings and their symmetry groups , in:
NATO Advanced Summer In-stitute on Nonperturbative Quantum Field Theory (Cargese Summer Institute) Cargese,France, Jul 16-30, 1987 , G. ’t Hooft, A. Jaffe, G. Mack, P.K. Mitter, R. Stora Eds., PlenumPress, 1988, pp. 521-528 [Preprint ROM2F-87-25 available as arXiv:hep-th/0208020].[3] P. Horava,
Strings on world sheet orbifolds , Nucl. Phys.
B327 , 461 (1989);
Backgroundduality of open string models , Phys. Lett.
B231 , 251 (1989).[4] J. Dai, R. G. Leigh and J. Polchinski,
New Connections Between String Theories , Mod.Phys. Lett. A4 , 2073 (1989); R. G. Leigh, Dirac-Born-Infeld Action from Dirichlet SigmaModel , Mod. Phys. Lett. A , 2767 (1989).
5] C.M. Hull and P. K. Townsend,
Unity of superstring dualities , Nucl. Phys.
B438 , 109-137(1995) [hep-th/9410167];E. Witten,
String theory dynamics in various dimensions , Nucl. Phys.
B443 , 85 -126 (1995)[hep-th/9503124];J.H. Schwarz,
Second Superstring Revolution , Nucl.Phys.Proc.Suppl.
B55 , 1-32 (1997)(hep-th/9607067);P.K. Townsend,
Four Lectures on M–theory , In :
Summer School High energy physics andcosmology, Trieste 1996 (Eds. E. Gava, A. Masiero, K.S. Narain, S. Randjbar-Daemi, Q.Shafi), The ICTP Series in Theoretical Physics, Vol. 13, World Scientific, 1997, Singapore,pp. 385-438 [hep-th/9612121].[6] J. Polchinski,
Dirichlet-Branes and Ramond-Ramond Charges , Phys. Rev. Lett. , 4724(1995) [hep-th/9510017].[7] P. K. Townsend, D-branes from M-branes , Phys. Lett.
B373 , 68-75 (1996) [hep-th/9512062].[8] M. Cederwall, A. von Gussich, B.E.W. Nilsson, A. Westerberg,
The Dirichlet super-three-branes in ten-dimensional type IIB supergravity , Nucl.Phys.
B490 (1997) 163–178[hep-th/9610148]; M. Aganagic, C. Popescu, J.H. Schwarz,
D-brane actions with local kappasymmetry , Phys.Lett.
B393 (1997) 311–315, [hep-th/9610249].[9] M. Cederwall, A. von Gussich, B.E.W. Nilsson, P. Sundell and A. Westerberg,
The Dirichletsuper-p-branes in ten-dimensional type IIA and IIB supergravity , Nucl.Phys.
B490 (1997)179–201 [hep-th/9611159]; M. Aganagic, C. Popescu, J.H. Schwarz,
Gauge–invariant andgauge–fixed D-brane actions , Nucl.Phys.
B490 (1997) 202, [hep-th/9612080].[10] E. Witten,
Bound states of strings and p-branes , Nucl. Phys.
B460 , 335 (1996)[hep-th/9510135].[11] U. H. Danielsson, G. Ferretti and B. Sundborg,
D-particle Dynamics and Bound States ,Int. J. Mod. Phys.
A11 , 5463 (1996) [hep-th/9603081]; D. N. Kabat and P. Pouliot,
A Comment on Zero-brane Quantum Mechanics , Phys. Rev. Lett. , 1004-1007 (1996)[hep-th/9603127]; M. R. Douglas, D. N. Kabat, P. Pouliot and S. H. Shenker, D-branes andshort distances in string theory , Nucl. Phys.
B485 , 85-127 (1997) [hep-th/9608024].[12] R. C. Myers,
Dielectric-branes , JHEP , 022 (1999) [hep-th/9910053].[13] A. A. Tseytlin,
On non-abelian generalisation of the Born-Infeld action in string theory ,Nucl. Phys.
B501 , 41-52 (1997) [hep-th/9701125].[14] D. P. Sorokin,
Coincident (super)-Dp-branes of codimension one , JHEP , 022 (2001)[hep-th/0106212]; Space-time symmetries and supersymmetry of coincident D-branes ,Fortsch. Phys. , 973 (2002); S. Panda and D. Sorokin, Supersymmetric and kappa-invariant coincident D0-branes , JHEP (2003) 055 [hep-th/0301065].[15] J. M. Drummond, P. S. Howe and U. Lindstrom,
Kappa-symmetric non-Abelian Born-Infeldactions in three dimensions , Class. Quant. Grav. , 6477 (2002) [hep-th/0206148].[16] P. S. Howe, U. Lindstrom and L. Wulff, Superstrings with boundary fermions , JHEP ,041 (2005) [hep-th/0505067];
On the covariance of the Dirac-Born-Infeld-Myers action ,JHEP , 070 (2007) [hep-th/0607156];[17] P. S. Howe, U. Lindstrom and L. Wulff,
Kappa-symmetry for coincident D-branes , JHEP , 010 (2007) [arXiv:0706.2494 [hep-th]].
18] E. A. Bergshoeff, A. Bilal, M. de Roo and A. Sevrin, “Supersymmetric non-abelian Born-Infeld revisited,” JHEP , 029 (2001) [arXiv:hep-th/0105274].[19] I. A. Bandos, “On superembedding approach to multiple D-brane system. D0 story,” Phys.Lett. B , 267 (2009) [arXiv:0907.4681 [hep-th]].[20] I. A. Bandos, D. P. Sorokin, M. Tonin, P. Pasti and D. V. Volkov,
Superstrings and super-membranes in the doubly supersymmetric geometrical approach , Nucl. Phys.
B446 , 79-118(1995) [hep-th/9501113].[21] P. S. Howe and E. Sezgin,
Superbranes , Phys. Lett. B , 133 (1997) [hep-th/9607227].[22] P. S. Howe and E. Sezgin,
D = 11, p = 5 , Phys. Lett.
B394 , 62 (1997) [hep-th/9611008].[23] I. A. Bandos, D. P. Sorokin and M. Tonin,
Generalized action principle and superfield equa-tions of motion for D = 10 D p -branes , Nucl. Phys. B497 , 275-296 (1997) [hep-th/9701127].[24] C. S. Chu, P. S. Howe and E. Sezgin,
Strings and D-branes with boundaries , Phys. Lett.
B428 , 59-67 (1998) [hep-th/9801202]; C. S. Chu, P. S. Howe, E. Sezgin and P. C. West,
Open superbranes , Phys. Lett.
B429 , 273–280 (1998) [hep-th/9803041].[25] D. P. Sorokin,
Superbranes and superembeddings , Phys. Rept. , 1 (2000)[hep-th/9906142].[26] I. A. Bandos, “Superembedding approach to Dp-branes, M-branes and multiple D(0)-branesystems,” arXiv:0912.2530 [hep-th].[27] R. Emparan,
Born-Infeld strings tunneling to D-branes , Phys. Lett. B , 71 (1998)[arXiv:hep-th/9711106].[28] I. A. Bandos, “Spinor moving frame, M0-brane covariant BRST quantization and intrinsiccomplexity of the pure spinor approach,” Phys. Lett. B , 388 (2008) [arXiv:0707.2336[hep-th]]; “D=11 massless superparticle covariant quantization, pure spinor BRST chargeand hidden symmetries,” Nucl. Phys. B , 360 (2008) [arXiv:0710.4342 [hep-th]].[29] T. Banks, W. Fischler, S. H. Shenker and L. Susskind,
M theory as a matrix model: Aconjecture , Phys. Rev.
D55 , 5112-5128 (1997) [hep-th/9610043].[30] I. A. Bandos and A. Y. Nurmagambetov, “Generalized action principle and ex-trinsic geometry for N=1 superparticle,” Class. Quant. Grav. , 1597-1621 (1997)[arXiv:hep-th/9610098].[31] E. Cremmer and S. Ferrara, Formulation of eleven-dimensional supergravity in superspace ,Phys. Lett.
B91 , 61 (1980);L. Brink and P.S. Howe,
Eleven-dimensional supergravity on the mass-shell in superspace ,Phys. Lett.
B91 , 384 (1980).[32] D. P. Sorokin, V. I. Tkach and D. V. Volkov,
Superparticles, twistors and Siegel symmetry ,Mod. Phys. Lett. A4 (1989) 901-908.[33] I.A. Bandos, “Dielectric coupling of multiple M0 to supergrvaity fluxes from superembed-ding approach”, paper in preparation.[34] B. Janssen and Y. Lozano, “A microscopical description of giant gravitons,” Nucl. Phys. B658 , 281–299 (2003) [arXiv:hep-th/0207199];[35] B. Janssen and Y. Lozano, “On the dielectric effect for gravitational waves,” Nucl. Phys.
B643 , 399–430 (2002) [arXiv:hep-th/0205254]., 399–430 (2002) [arXiv:hep-th/0205254].