JHEP08(2008)002 is 2 T × R , rmation of July 11, 2008 June 10, 2008 August 1, 2008 Accepted: Received: s theory on Published: metric Type IIB plane wave. Type IIB plane wave in the ab o, [email protected] , and Filippo Passerini ab Published by Institute of Physics Publishing for SISSA Amir Jafari Salim a Gauge-gravity correspondence, M(atrix) Theories. We write down a maximally supersymmetric one parameter defo [email protected] Waterloo, Ontario N2L 3G1,E-mail: Canada Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5,Department Canada of Physics and Astronomy, University of Waterlo [email protected] b a Jaume Gomis, Matrix theory of type IIB plane wave from membranes Abstract: the field theory action of Bagger and Lambert. We show that thi invariant under the superalgebra ofIt the is maximally argued supersym thatdiscrete light-cone this quantization theory (DLCQ). holographically describes the Keywords: JHEP08(2008)002 9 1 3 (1.1) ) (1.2) ′ D , X ] ′ for the scalars C 2 , X ′ ravitons 8 B Bagger-Lambert the- , [4] for the Lagrangian , X
′ Ψ A grangian a mass term for in an arbitrary eleven di- , per [6] — incorporates the Chern-Simons like action. X go [1]. The realization that ations, and the observation 2-branes (see also the work of or the worldvolume theory of 3456 hould have generalized fuzzy- nes is described by non-abelian Tr([ ′ ring dualities to D-branes [2] and ¯ ΨΓ D ′ C ′ Tr B ′ µ A i 2 SO(4) invariant potential µε + 6 1
× I − ) – 1 – , X 2-branes was first considered by Bena [10]. I D M X , X ] C Tr 2 1 , X µ 1 2 B − , X = A X mass L Tr([ ABCD µε 6 1 − = and a Myers-like [11] flux-inducing SO(4) Recently, Bagger and Lambert have made an explicit proposal In this paper we construct a one parameter deformation of the 3.1 Deformed field theory vacua and type IIB plane wave giant g See also [12, 13]. The deformation of the theory on multiple flux 2 1 L A. Supersymmetry of deformed field theory B. Noether charges and supersymmetry algebra 11 mensional supergravity background wasbranes found in twenty eleven years dimensional supergravity a arethat related the by low st energy effective fieldsuper theory Yang-Mills of theory coincident D-bra [3],coincident naturally M2-branes. prompts the search f description of the low energyGustavsson limit [5]). of This a stack work ofrealization — coincident which in M builds [7] upon that theirfunnel the prior configurations theory pa [8] on coincident describedin M2-branes [9] by s that generalized the Nahm putative gauge equ field of theory theory [4] should which have is a all maximally the supersymmetric. eight We scalars add and to fermions their La 1. Introduction and discussion The supersymmetric worldvolume theory of a single M2-brane 2. Deformed supersymmetric field theory 3. Deformed theory as DLCQ of type IIB plane wave 5 Contents 1. Introduction and discussion JHEP08(2008)002 (1.4) provides the ’s [14], where 2 3 S T tes of the deformed × te down to capture has as its algebra of R . lgebra structure of [4] d theory are given by 2 n es in the Hilbert space T etry transformations of string field theory inter- n the potential (1.2) for = 0 correspond to configura- the bulk Type IIB string of adding the scalar mass field theory remains fully nsionally reduced to 1+1 ackground [16] with fixed × = 0 (1.3) agger-Lambert and in this A ered in [14]. Here we show ′ R d theory. ave, as expected from a holo- y supersymmetric, Poincare ic field theories, the precise ranes still remains to be un- A htcone quantization (DLCQ). , possibly an orientifold projection of 4 ven though several constructions A is proposed as the nonperturbative , X ′ 2 ra ane Lagrangian was proposed as the Matrix D , X M2-branes polarizing in the presence of flux T ve of eleven dimensional supergravity. D X as a collection of fuzzy × ′ X 5 R D ′ C ′ B ′ ABCD A – 2 – µǫ µǫ − − ]= ]= ′ C C , X of Type IIB string theory on the maximally supersymmetric , X B ′ 3 B , X A , X ′ X A [ topology. The supergravity description of these ground sta X [ 3 S [20]. We show that the deformed field theory on 4 We further argue that the deformed field theory compactified o and show that the theory is supersymmetric. The possibility We show that the supersymmetric ground states of the deforme or alternatively: We identify these states of the deformed theory with the stat It would be interesting to use the deformed field theory we wri and it would be interesting to extract the Type IIB lightcone The Lagrangian of the deformed theory is based on the same 3-a These states have yet another space-time interpretation as Or given the interpretation in [18, 19] for the known 3-algeb In [16] (see also [17]), an analogous deformation of the D0 br 5 4 3 into M5-branes with the maximally supersymmetric plane wave background. theory were found in [21] (see also [22]). symmetries the superisometry algebra of thegraphic Type dual theory. IIB plane The deformed w field theory on plane wave theory description of the maximally supersymmetry plane wa term and the potentialthat term if for we four give ofall a the the mass scalars to scalars was all that consid the the we Bagger-Lambert scalars theory can and [4] find fermions insupersymmetric. a such and deformation a turn This way of o that constructioninvariant the three the yields supersymm dimensional deformed a field theory. novel maximall Matrix theory [15] description formulation of the Type IIB string theory in the discrete lig a discrete set of states that have an interpretation of the Type IIBtions plane of wave D3-brane with giantlongitudinal gravitons zero momentum. in light-cone the energy, Type which IIB plane wave b stringy physics in the Typetheory IIB is plane weakly wave. coupled Indimensions the the deformed regime field when theory can be dime action vertices in the plane wave background from the(we reduce review it inpaper section 2). certainly provide Even thoughconnection new the with constructions construction the of of worldvolumederstood. B physics supersymmetr of Currently a coincident single M2-b 3-algebra structure is known e JHEP08(2008)002 2 T (2.1) (2.3) × R of a stack ]) (2.2) 6 K + , , X  J abcd . The Lagrangian uickly review the λef f µ , X and show that they a A I A 2 Ψ transverse to the M2- X J T ur deformed theory is d νcd I ][ A X × X K I c geometry in which the M2 f the Lagrangian and the imensions has been found. eformed theory on nds states of the deformed R mally supersymmetric La- n this paper as the Matrix µab X ht-cone energy, which corre- energy dynamics 6, 27] and the addition of a , X e there are subtleties identi- ersymmetric Type IIB plane uge field A elaxing the conditions on the J IJ ations while in appendix B we ndix A we present some details Γ b r fields , X efgb ¯ I Ψ , f b i g 4 X Φ + cda a µb f a ˜ Tr([ A 2 3 Ψ 3! µ − 1 · + a D 2 has precisely the same symmetry algebra as µ Φ λcd Γ µ 2 ≡ – 3 – a ∂ A T ν ¯ Ψ K g = ∂ i × 2 X a J R f µab Φ µ )+ A X I a I e D X X abcd µ f K c D  X )( J b . aI µνλ X ε cd µ I a X 2 1 A µ X d D + bcd ( a 2 1 efg f V f − − ≡ 0 limit, higher derivative corrections can be ignored. is the scalar potential = abcd a µb → ˜ f V A L p 1 l 12 = where and the covariant derivative of a field Φ is given by Establishing in more detail the M2 brane interpretation ofThe o plan of the rest of the paper is as follows. In section 2 we q where V In this 6 Bagger-Lambert theory [14]supersymmetry and transformations introduce that the givesgrangian deformation rise in o three to dimensions. a In new section maxi 3 we argue that the d have recently been considered [23, 24], and for the known cas fying the M2 content ofbranes the are theory embedded as [14, well as 23,3-algebra and 25, to the 18, construct spacetime 19]. newboundary The examples possibility to have of the been r theory considered has [2 been considered in [28]. important in understanding thetheory deformed description field of theory the Type found IIB i plane wave. provides the Matrix theorywave. description of We the show maximally that sup the theory on theory with the states inspond the to Type configurations IIB of plane D3-brane wave giantof with gravitons. zero the In lig calculation appe of thewrite deformed down supersymmetry the transform Noether charges of the deformed field theory on the Type IIB plane wave and identify the supersymmetric grou of M2-branes. It encodes the interactions of the eight scala is given by [4] branes, the worldvolume fermions Ψ and a non-propagating ga satisfy the Type IIB plane wave superalgebra. 2. Deformed supersymmetric field theory In [4], a newThe maximally authors supersymmetric have Lagrangian proposed in that this three theory d describes the low JHEP08(2008)002 , l al . mass ) (2.4) (2.6) ′ L 6 and ...C D , -branes 5 ⊕ , D , X 4 ] 1 ′ , C e constraint C ⊕ = 3 , X with generators ). Thus far the ′ 4 undeformed the- b n B A sformations of the n dimensions. This A , T = 0 (2.5) , X a ⊕ A ′ has the interpretation T A abcg . . . . The mass term X f µ ny 3-algebra with totally ras. The algebra indices flux lgebra ) where ⊕ etric. The new Lagrangian g ′ L = Tr( 4 Tr([ on of the Lagrangian of D0- e ′ 3]) in the context of def D ab ′ amental identity (2.5). akis [23] to obtain by compactification A, A f d to only four of the scalars has h C ⊕ A ′ − . , 4 B d ′ = ( A A T I flux d abdg L µε f 6 1 g abc + f − Ψ) (2.7) cef ) denote central elements in the algebra. The , mass f D i ]= c L – 4 – C + 3456 , X + , T ] are taken to be totally antisymmetric and to satisfy b ¯ C L ΨΓ acdg and , T f = a , X abcd g Tr( ˜ — and on a 3-product: B f T L µ [ abcd ǫ i n 2 bef , X does indeed reduce to the known Myers term. f = = A SO(4). The deformed theory is nevertheless invariant under )+ − flux X A I L × abcd generated by a background four-form flux of eleven dimension f , X bcdg Tr([ I dim f 7 g X ,... ABCD Tr( aef 2 f µε µ = 1 1 6 2 1 is the Bagger-Lambert theory in (2.1) and: a 10 and Ψ is an eleven dimension Majorana spinor satisfying th − − , L Ψ, where the Γ-matrices satisfy the Clifford algebra in eleve 9 is defined by , = = − 4 8 , A flux mass This theory is based on a novel algebraic structure [4], a 3-a The transverse index has been decomposed as The deformed theory (2.6) breaks the SO(8) R-symmetry of the In [4], the structure constants which generalizes the familiar Jacobi identity of Lie algeb The deformed Lagrangian now depends on the paramater We now find a deformation of the action and supersymmetry tran where Ψ= L We note that if we use the proposal made by Mukhi and Papageorg — where = 7 L 7 ′ a 012 the theory on D2 branes, that T A ory (2.1) down to SO(4) in the presence of background fluxes. the fundamental identity Γ are contracted with a prescribed non-degenerate metric supergravity, of the type found by Myers [11] (see also [12, 1 deformation of the Lagrangianbranes is considered analogous in to [16]. thebeen This deformati deformation considered when in restricte [14]. only examples of 3-algebras found are of the type of the scalar potential gives mass to all the scalars and fermions in the theory, whil where supersymmetric deformation we findantisymmetric structure in constants this which paper satisfies the applies fund to a action of Bagger and Lambertis [4] given that by is maximally supersymm JHEP08(2008)002 = 8 η (3.1) (2.9) (2.8) 012 . A = 0. Then the straint Γ ǫ 0 I g the constraint X abc I f Γ field theory (2.6) on to provide the Matrix ymmetry transforma- I 9 3456 n-linearly realized super- dx f that the action (2.6) is Γ I µ onstant of Type IIB string Poincare invariant three di- η dx , so that arized in appendix 0 − 0 e, extending the Matrix string lane wave see [33]. For the DLCQ + etry transformations of the de- ǫ T T persymmetry transformations  + was argued ident M2-brane theory. of the torus on which the deformed = µ sixteen nonlinearly realized supersymme- IJK 2 dx σ τ C µ ]Γ auge theory see [34]. See also [35 – 37]. + T Γ K × dx I R , X 3456 x I J Γ x 3 µ 2 , X µ − I – 5 – µ.  , − X [ a − 1 6 = 2 description of the maximally supersymmetric Type cdb SO(4) R-symmetry. dx − f 10 d + = 0 = 0 ǫ × I Ψ I a +5678 a dx I Γ Ψ = exp b c has a central element F µ n µ X X δ ˜ n Γ n I A = 2 = δ 0 we recover the supersymmetry transformations of the un- I n Γ A Ψ δ 2 I µ X → µ 0. Γ Γ ds ¯ ¯ ǫ ǫ i D i µ +1234 → F [38]. µ = = s I a Ψ = b ˜ δ is an eleven dimensional Majorana spinor satisfying the con µ i/g ˜ δX ˜ A η + ˜ δ 0 is a constant eleven dimensional Majorana spinor satisfyin are the three dimensional field theory coordinates. C ǫ . By setting µ = ǫ σ provides the Matrix theory τ 2 = T In this section we argue that the three dimensional deformed As in the case of flat space, the modular parameter The field theory with Lagrangian (2.6), (2.8) and with supers where now where The deformed action (2.6) is also invariant under sixteen no ǫ and At that time there was no Lagrangian description of the coinc For a different proposal for the Matrix theory of the Type IIB p The Bagger-Lambert theory (2.1) is also invariant under the 9 8 × 10 η 012 description of the plane wave in terms of a sector of a quiver g IIB plane wave background [20]: R theory [15] description of Typetheory IIB description string in theory [31, in 32] flat to spac Type IIB string theory. − Γ tries obtained by setting field theory is definedtheory determines the complexified coupling c tions (2.8), (2.9) definesmensional a field theory novel with maximally SO(4) supersymmetric formed theory are given by sixteen linearly realized supersymmetries. The supersymm 3. Deformed theory as DLCQ ofIn [29, type 30], IIB the theory plane of wave coincident M2-branes on symmetries if the 3-algebra deformed theory (2.1) foundinvariant under by the Bagger-Lambert supersymmetry [4]. transformations is The summ proo action (2.6) is invariant under the following non-linear su JHEP08(2008)002 AB . of the + tuition . As in H, J 8 P , 13 in a sec- 1 . We now 9) symme- R I , πR K raction SO(1 0 the plane wave + 2 miltonian and the I − → x ry. The non-manifest µ hirty-two supersymme- r the action of ≃ n of the Bagger-Lambert he lightlike identification and boosts 2.6) found in this paper is − o as ory background in the dis- ymmetries of the Type IIB . Therefore the non-central I ld theory (2.6) goes over to symptopia must realize the x lly extended Newton-Hooke , which generate time trans- , then the ′ + ory for multiple M2-branes is P tion we consider a string/M- 9 bra [20]. Unlike in flat space, , B of the contracted algebra down to ′ 1 P ies. . The Matrix theory descrip- A rs on the fields of the deformed R J ctively, and where the transverse he isometry algebra of elativistic symmetries of string theory in N/R and identification to the centrally extended = − AB x + , J P 8) symmetry algebra of Matrix string theory I , (1 – 6 – , K ). I ′ Gal A, A H, P = ( 11 I 8) [39], where the central extension corresponds to , (1 , just like the 9 SGal 8) are given by does not break any of these symmetries. It is useful to gain in AdS , (1 8). This algebra of symmetries is the non-relativistic cont πR , . (1 NH + 2 πR − x NH , which correspond in the deformed field theory (2.6) to the Ha + 2 ′ SO(4) R-symmetry charges of the three dimensional field theo B ≃ SO(4). 12 − ′ × x A × − J Matrix theory describes nonperturbatively a string/M-the In this paper we have constructed a one parameter deformatio It is therefore natural to propose that the deformed theory ( If we consider the DLCQ of Type IIB string theory in The Type IIB plane wave background (3.1) is invariant under t The deformed field theory (2.6) is manifestly invariant unde x The flux in (3.1) actually breaks the SO(8) rotation symmetry See [18, 19] forThis subtleties with algebra this has interpretation. appeared previously in the context of non-r ≃ 13 11 12 − SO(4) e.g. [40 – 43]. crete light cone quantizationtheory (DLCQ) background [39]. with In a this quantiza compactified lightlike coordinate background (3.1) reduces to flatthe space Bagger-Lambert just theory (2.1), as which thethe as deformed Matrix the fie theory candidate for the flat space. field theory that preserves all the thirty-two supersymmetr the Matrix theory description of the Type IIB plane wave. Als tor with quantized longitudinal momentum try algebra of Minkowski space is broken by the tion of a string/M-theorysame background symmetries with as some those prescribedx of a the asymptotic background with t in flat space arises in the non-relativistic contraction of t tries and under athe thirty-dimensional bosonic symmetry alge isometry algebra of Super-Galileo algebra on the action ofplane these wave symmetries background to (3.1) notice canalgebra that be the identified bosonic with the s centra the case of flat space, the central extension corresponds to lations, spatial translations, boostsindex has and been rotations decomposed respe as generators of symmetries that remain to be realized are the translations and SO(4) consider the following non-linear action of these generato JHEP08(2008)002 2 T ator (3.2) (3.3) (3.4) on which 2 T we write the Noether ) on the d theory is broken by B , Z generators along the ions (2.8) correspond to symmetries with the su- x 1) while the supercharges eformed Lagrangian (2.6) (2 ngredients to be the Matrix s in the way expected from ude that the deformed field 0 0 4, 30]. These central charges spond to the kinematical su- ping the longitudinal direction T T asi-invariant Lagrangians. The explanation for the existence of s and boosts: ) ) 0 0 µσ µσ µ . is a central element in the 3-algebra + 0 sin( cos( P T IJ IJ IJ and show that the commutation relations δ δ iδ J , 2 J v a T ]= = = 0 × – 7 – = 0 = J 0 we recover the usual Galilean transformations. I a I a R Ψ = 0 b Ψ = 0 b 8), or equivalenty under the superisometry algebra δ , K δ µ , → µ I δX ˜ δX ˜ A A (1 µ P δ δ [ 8), as central extensions of symmetry algebras are always , : (1 : SNH J J to just the translation algebra. The time translation gener P K NH 2 in T + × P R is the field theory time coordinate and 0 σ The original three dimensional Poincare symmetry of the fiel The supercharge generating the supersymmetry transformat Therefore the deformed field theory (2.6) has the necessary i is identified with the Type IIB Hamiltonian. The translation . Note that in the flat space limit and field theory (2.6) A where associated with symmetry transformations thatcentral result extension in appears qu in the commutator of translation compactification to Under the action ofchanges by the a total transformations derivative. This (3.2)the provides central and extension the field (3.3) theory the d can be identified withare central associated charges with of fundamental strings the andof superalgebra D1 the strings [4 Type wrap IIB plane wave (3.1). The geometrical action of SL H charges of the deformed field theory on of the Type IIB plane background (3.1) in the DLCQ. In appendi the theory is defined exchangesduality the [38]. fundamental and D1 string the dynamical supersymmetries ofgenerating the the Type IIB supersymmetry planepersymmetries transformations wave of (3. (2.9) the corre planepersymmetry wave. transformations found Thus in combiningtheory the the (2.6) section is bosonic 2 invariant under we concl theory description of the Type IIB plane wave (3.1). are those of the Type IIB plane wave (3.1). JHEP08(2008)002 10. , (3.5) (3.7) 9 , iton is 8 , 3-branes D = 7 ′ ravitons A ) is the sphere of l longitudinal mo- 3 Ψ = 0 in (2.8) and ˜ ˜ δ S r fruitful discussions. ( 6 and , , 14 giant graviton configu- 3 5 pace zero energy states , S = 0 tains in its Hilbert space 4 Research and Innovation. 3-brane [16]: s and their representation = 0 (3.6) , avitons. A giant graviton A y Grant. 3-brane configurations pre- IB plane wave (3.1). It is D ′ proposal of this paper. serve all the sixteen linearly A D t of Canada through Industry = 3 se ground states are described te for Theoretical Physics. Re- nearly supersymmetries of the itality during the final stages of ondition A hey break the non-linearly realized , X ′ = 0. D = 0, where , X . = 0 states of the DLCQ of the plane ′ D H . These X α − ′ ), with H ′ + x X D = 0 — that preserve half of the super- ′ carried by the C N/R at ′ µP H s + A, A B 3 ′ = ABCD P ˜ A S πg – 8 – = ( + µǫ µǫ or I P − = 2 − 3 ′ 2 S α L ]= ]= , and each state describes a configuration of ′ C C N . Therefore, the , X , X B ′ N/R B , X = A , X ′ + X A [ P in the plane wave geometry (3.1). The radius of the giant grav X [ 4 R 3 brane which wraps D When considering the DLCQ of the Type IIB plane wave, the tota The deformed field theory (2.6) also contains in its Hilbert s We identify these states of the deformed field theory with the The supersymmetry conditions of [33] were analyzed in [45]. 14 determined by the longitudinal momentum whose total longitudinal momentum is Type IIB string theorystates on with the zero plane wave light-cone background energy (3.1) — con where 3.1 Deformed field theory vacua and type IIB plane wave giant g mentum is quantized symmetry [16]. Theyin correspond (3.1) to is configurations a of giant gr wave are labeled by partitions of the first (second) serve half of therealized supersymmetries. supersymmetries while More theyplane precisely, break wave they all background. pre of the non-li that preserve half of theby supersymmetries constant of scalar the fields theory. satisfying The or alternatively: where we have split the transverse index rations of the Typetheory IIB is plane important wave. to Further further work understand on the 3-algebra Matrix theory Acknowledgments We thank Joaquim Gomis, JorgeJ.G. Russo would and like to Mark thank Van the Raamsdonkthis University fo of work. Barcelona This for hosp research wassearch supported at by Perimeter Perimeter Institute Institu isCanada supported and by by the the Governmen ProvinceJ.G. of also Ontario acknowledges through further the support Ministry by of an NSERC Discover preserve all the linearly realized supersymmetriessupersymmetries, while t just likestraightforward the to giant show that gravitons these in states the also have Type I These solutions automatically satisfy the supersymmetry c JHEP08(2008)002 (A.2) (A.3) (A.1) (A.7) (A.8) (A.4) . Since ǫ = ǫ , 012 ) ǫ . ) I ǫ 789(10) I Γ flux Γ µ Γ ′ ǫ traint Γ L ) µ ) (A.6) Γ C ǫ µ ǫ ′ ′ Γ δ B D IJK ′ s ) 3456 + ǫ A ) (A.5) Γ 3456 3456 ) (A.9) ǫ a Γ ¯ ǫ Γ ) ΨΓ ′ ¯ ǫ ΨΓ , − K ] ¯ 3456 bcd , C ΨΓ ′ D mass ′ I IJK Γ , f Γ C = B L I Γ ǫ ′ X K ′ d µ ¯ I ΨΓ A µ X δ D 3456 , , X X ] µ ′ ABC X ǫ, 3456 D Γ Γ J 1. We also have that c + I ¯ ′ C B ΨΓ D Γ , − D IJ X ¯ L ] ΨΓ ′ ¯ , ΨΓ Tr( I ′ b 3456 , µ , X , X C µ , . ] = Tr( ′ C ′ δ ] Γ ¯ δ X a iµ B ΨΓ 3456 C A = 0 up to total derivatives. It is trivial to B , K iµ ′ − 6 1 Γ + ] + , X cdb X A µ ′ , X L ǫ K , X − , = f − ǫ , X δ B A − d δ B flux – 9 – ǫ ǫ J Ψ) + I Tr([ I , X X Ψ Ψ) L = ′ , X ǫ = 0 = 0 I J Γ c Γ µ , X ′ ˜ , X δ δ D ¯ ǫ µ ′ , ε I a I A A a X , δ Ψ = b Γ Tr([ C a , i I , X + µ ′ 789(10) I µ X µ a X X X I 0123456789(10) I Γ Ψ δ B ˜ Γ µ 3456 ′ A µ I X δ X X ¯ ΨΓ Γ µ A µ Γ Γ mass δ Tr([ Tr([ Tr([ ¯ ¯ ǫ ǫ ABCD i i D L µ µε µ µ Tr( Tr( ǫ and Γ Tr([ 2 6 3 3 3 2 2 1 2 ABC δ µε µ µ i µ i i i = = = ǫ ∂ Γ 2 3 1 2 + i − − + + − i I a a 2 a i − = b Ψ L µ X − ǫ − = = = ǫ ǫ ǫ = ˜ δ δ A δ ǫ , we have that: = 012 D δ = flux Γ L mass flux ˜ L L µ ǫ L L ˜ δ δ ǫ δ = 0. The other terms are: δ + ABCD ε flux L mass µ are given in L δ is an eleven dimensional Majorana spinor subject to the cons ǫ + δ ǫ L In the last step of (A.6) we have used In [4] it has already been shown that = ˜ and where A. Supersymmetry of deformed field theory We first note that the susy variation (2.8) can be decomposed a where and which is implied by Γ see that L JHEP08(2008)002 η , − 3456 3456 = 3456 Γ (A.10) (A.14) (A.15) (A.11) (A.12) (A.13) ′ Γ η Γ 3456 ′ 6) gives ′ C . ′ C Γ ′ ′ B 012 ′ B ′ A a ABA A Γ Γ ABA Ψ Γ i ′ n i Γ i ′ ′ δ C i C A ′ traint Γ A , X 3456 ′ , X , X η ′ Γ B , X B , µ a B Lagrangian (2 ) σ B ǫ ¯ Ψ µ , X , X Γ ′ , X ) ′ ) iµ A ntities , X ǫ A ǫ ) rmations are given by 3456 A η A ǫ K 3456 X + X Γ 0 implies that the the deformed h X ′ Γ h Γ X h T µ 3456 C h + ′ 1 3 ) is also invariant under sixteen IJK . + Γ abcd  ) B persymmetries. − Γ + 3 − 3456 ′ µ I f e being central — and have ignored A Γ a 3456 σ 3456 0 µ ABC Γ Ψ , X 3456 IJ Γ 3456 3456 ¯ Γ T 3456 n ΨΓ A ′ Ψ A Γ Γ δ , ′ ¯ Γ ] ΨΓ I ¯ ¯ B ΨΓ ΨΓ J ′ 0 d ′ , B Γ , , ′ ] 3456 C ] ] ¯ A ¯ ǫ Ψ X A C Γ i ABC ABC Γ I c K K Γ µ iµ i Γ Γ , X i =0— X A ′ 3 1 , X   Tr( A + , X , X B b 2 B C C IJ − J J 0 – 10 – η µ , X Γ ′ ) , X  cd b , X ′ µ , X , X , X B ′ = f , X , X ¯ B σ Ψ I I A A , B B µ i 2 Γ , X X X X X ′ , X ) ′ , X , X mass + A a = 0 = 0 A A A 3456 L a X Ψ Tr([ Tr([ Tr([ Tr([ I a Γ a µ h X X X Ψ = exp b µ n δ h µ µ µ µ  3  Ψ) n X µ 3 1 δ 1 2 6 2 3 3 2 1 δ n − + ˜ n µ − A i i i i − − δ δ Γ e n µ − − + − ( a = = δ µ D ¯ Ψ K ( ∂ = ( Γ µ µ IJK µ ˜ Γ L Γ ∂ Γ ˜ 0 a δ i 2 3456 , ¯ ¯ Ψ Ψ Γ − i i 3456 Γ IJ = = 0 =  Γ  ˜ K L is an eleven dimensional Majorana spinor subject to the cons K n η δ , X J , X is a central generator of the 3-algebra. The variation of the J , X 0 I , X T Combining all the pieces together we get The proposed non-linearly realized supersymmetry transfo I X  X  where now and that where in the second step we used that where we have omitted the surface term in (A.9). Using the ide a total derivative. Therefore thenon-linearly deformed realized field supersymmetries. theory (2.6 and one can show that the right handfield side theory of (A.11) is vanishes. invariant under This sixteen linearly realized su JHEP08(2008)002 (B.2) (B.1) ) (A.16) ′ D  , X Ψ) ] 0 ′ . Ψ) Γ C Ψ)  ) 0 Γ ′ , X Ψ) λef IJK ′ 3456 0 B ′ Γ B A Γ Γ , A , ] Γ ¯ νcd , X Ψ AB K , ′ A Γ ¯ A Ψ , Tr( , X X ¯ µab Ψ J µ Tr( he deformed field theory on i 2 A e following transformations: i 4 η , X Tr( Tr([ 0 − ′ I i 4 efgb T ) D ′ X )+ I f
′ C g 0 ′ A )+ the three dimensional Poincare sym- σ B , X A Π 0 Tr([ ′
cda 2 I  , ) , A 1 6 Γ ) Π ′ f aI aI 0 a T 0 0 , X B 2 3  Ψ µε × − X X Ψ B i µσ i i 3456 X 6 1 µσ
+ R Ψ) Tr( X D 0 Γ D D , 2 i Ψ) I I a a I σ µ 0 µ a Tr( sin( Γ 0 λcd )+ cos( Tr( Γ a 2 1 − X X I X 0 Ψ Γ i i I A − I D i 0 ¯ – 11 – Ψ ν − Γ ) + D D i ′ D 2 X ∂ Tr( ) µ µX i . 2 2 1 1 , X 3456 0 B . The Hamiltonian of the theory is given by: ] V Γ B Γ − − 2 Γ Γ , µab Π a C + + Π I ) µ a , )+ T ¯ , = 0 = 0 A ′ 0 Ψ 0 )+ − Ψ X aI aI A A i 2 , X 3456 I a a J 0 µ Ψ = exp b Π Π µσ Γ B µ µσ X X abcd n µ X I a I a D I − D δ f ˜ 0 n , X b A Γ Π Π ] ( exp δ , X Γ sin( Tr( Tr( I n µ 2 2 1 1 cos( Ψ 0 a Tr( A δ I 0   I 0 + Γ ¯ µνλ cd Ψ + + Π X , X − 0 ε σ σ σ Π i 2 I I 2 2 2 b b  1 2 Ψ  A d d d , σ σ σ − X X σ Tr([ − 2 2 2 2 IJ a Z Z Z cd cd cdba d d d d 0 0 ˙ i i i f Ψ cdλ 0 0 ¯ A A Z ΨΓ − − − Z Z Z ˙ Γ Γ A ABCD a a ======cdba cdba ¯ ¯ Ψ Ψ µε Tr([ cdλ ′ f f I I q is the integral over the i i i + 1 6 2 2 4 Q I a I a B ′ P AB K P σ + +Λ + + + A J 2 J d are given by = Π = Π R 2 H H T When the deformed field theory is placed on Alternatively, one can write: × where metry is broken. In this case the theory is invariant under th B. Noether charges and supersymmetry algebra The charges that generateR the symmetry transformations of t JHEP08(2008)002 + P ′ (B.4) (B.3) (B.6) (B.5) IJ ′ ) B ′ B iδ D K . For the P ′ ′ = I a C ǫ C , X ′  ′ ] I X ′ A J A 0 δ C δ X Ψ) D I , K + + Γ I , X ′ = ′ ′ 3456 C P C te the commutator B I a Γ  3456 , K P d theory (2.6) satisfy ′ ′ Γ ¯ . , X Ψ B we need the canonical ′ B µ ′ I ′ A A A λef − Tr( δ iP δ X , A ǫ µ αβ s we get: − − − i 2 δ IJ νcd δ = on relation: = = Tr([ ab e Dirac brackets, which for the IJK ′ − A δ ab i  i Γ ) ) ′ D δ ′ and satisfies i ′ I )  ′ C C i µab ′ I C ′ ′ ,H σ ′ K B I A B σ B , X X ′ − I K A , J , X − i , J ′  ′ X efgb J i σ A A ( µε f σ ′ η. 2 ( g 6 1 P K 0 , X C Tr( δ 2 ′ h h I 2 T A − cda iδ µ
δ X f )+ ′ 2 1 0  AC = 0 0 D B δ σ – 12 – 6 1 } T T 0 + µνλ )] = B ) ) X i ) ǫ i Γ ′ 0 , X − ′ 0 ] ] V X 3 1 σ b + σ ǫ C ( ( ′ I µσ + µσ Ψ 3456 µ − J b β b B C Ψ Γ a )+ B I ′ [ ′ Γ ) Ψ Π , X µ Ψ J K B B µ , P BC , X ′ cos( Γ B sin( δ ) ) I δ λcd ′ I i − AB A i Ψ , X AC Γ A A AC σ IJ IJ A ] Γ Γ σ I i X δ , X i ( δ I ( 2 2 1 1 µ I X X iδ iδ Γ Γ a ∂ A α a ¯ ¯ ǫ ǫ + + D i i − − − − − − X X Ψ , X [ C µab C { Ψ ======A P K , I ] = Tr([      i i ′ IJ J J J K Ψ] = Ψ] = exp C AB AB Ψ abcd Ψ abi C 2 δ δ , , f ′ ¯ ( ΨΓ , X , X iµ − − , X ηq, 2 are the spatial coordinates on the membrane and Π B ǫQ, , X I I [¯ ABCD ′ AB [¯ ′ , = = = A µiλ ¯ ǫQ,X P J B AB K is the momentum density conjugate to ǫQ,A ′ ε Tr([ µε    [¯ J     J i A 2. I 1 1 2 4 6 h = 1  , are eleven dimensional spinor indices. J i + − + BC BC ,H h I = 1 , J , J P i α, β A  A P K where In order to calculate the algebra generated by these charges We now show that the Noether charges (B.1) of the deformed fiel To compute the action of the symmetries on the fields, we compu   canonical commutation relation for thecase spinors, of one must Majorana us spinors results in the following commutati momenta. Π where the Type IIB plane wave superalgebra. For the even generator where of the charges with the fields. We get JHEP08(2008)002 D , (B.9) (B.8) (B.7) , . ′ Phys. (1995) B , ′ I -branes A 2 75 K J . q Phys. Rev. M 0 αβ q. αβ , ′ ) ) q Γ q I B 0 0 ′ Γ Γ Γ . AB A lso [46] for a useful ′ 3456 I (2007) 045020 Γ Γ ]. C Γ Γ ′ (1996) 335 1 2 2 1 re: 3456 i i B − − − − (Γ J 789(10) ′ i (2004) 078 2 Γ D 75 = = = ′ D ′ µ ]= Phys. Rev. Lett.   i B B 460 A ′ 11 , − δ A ,q ,q arXiv:0709.1260 ,Q ′ BC I , H,q I − B (Γ J ′ AB P ′ K A J iµ C  JHEP ′ αβ  AD J ) , A δ hep-th/9910053 Phys. Rev. h + 0 , J -branes ]. ′ Γ − 2 ’s I Nucl. Phys. D AB 2 ]. ′ , M J (Γ AC B M Properties of the eleven-dimensional super i δ 2 J The noncommutative bion core αβ ) ]. − + 0 BD ′ (1988) 330. Γ (1999) 022 [ – 13 – δ = D brane system and a generalized Nahm’s equation ′ } + B 5 12 β 3456 q 185 J ′ ]. Γ M - dimensional theory with reduced supersymmetry BD ,Q C Q 2 ′ ′ arXiv:0711.0955 3456 α Q J 3 AB A B q hep-th/0412310 Γ ′ M δ JHEP { I (Γ AB A AC , Γ δ + Γ Γ Modeling multiple Gauge symmetry and supersymmetry of multiple iµ ′ hep-th/0004142 1 1 2 2 The iµ + D + ′ − − − + A ]. AD αβ P = = = J ′ J ]=0 [   i (2005) 136 [ C αβ ′ hep-th/9911136 Hδ Ann. Phys. (NY) (2008) 065008 [ ]. ]. ]. BC B iδ , ,Q ,Q ,Q δ δ ′ I Algebraic structures on parallel H,Q B [ = 2 = Superconformal Chern-Simons theories − − Dirichlet-Branes and Ramond-Ramond charges ′ P AB  Dielectric-branes A B 713 } } J = = (2000) 126006 [ D 77 Bound states of strings and p-branes  β β J  i h ′ The M-theory dual of a ,q ,Q D α ′ CD α q hep-th/9510017 C D 62 { Q , J (2000) 106009 [ { , J ′ AB hep-th/0611108 hep-th/9510135 hep-th/0411077 B Rev. [ [ Nucl. Phys. 61 Phys. Rev. [ 4724 [ membrane theory This is the superalgebra of the Type IIB plane wave [20] (see a The anticommutators of the supercharges are: The commutation relations between odd and even generators a ′ J A  J [9] J.H. Schwarz, [6] J. Bagger and N. Lambert, [7] A. Basu and J.A. Harvey, [5] A. Gustavsson, [8] N.R. Constable, R.C. Myers and O. Tafjord, [4] J. Bagger and N. Lambert, [3] E. Witten, [1] E. Bergshoeff, E. Sezgin and P.K. Townsend, [2] J. Polchinski, h [11] R.C. Myers, [10] I. Bena, summary of the superalgebra) References JHEP08(2008)002 ]. , , Nucl. , JHEP ]. IB Nucl. , , Nucl. , , B 500 . -branes 2 -branes M 2 arXiv:0804.0913 D (2008) 105 . 02 and (1997) 41 -branes on M-folds Nucl. Phys. BPS geometries 2 2 , arXiv:0803.3218 2 / A new maximally M M ]. (2002) 047 ]. 1 hep-th/9701025 JHEP (2008) 076 [ , , Matrix perturbation theory for B 497 01 onk, 05 ]. poulos, ]. superconformal Chern-Simons ]. (2008) 085 [ -branes M theory as a matrix model: a JHEP arXiv:0804.1114 2 , Strings in flat space and pp waves from , JHEP Aspects of multiple membranes 05 M , =8 ]. Nucl. Phys. -branes in weak background fields -branes in weakly curved backgrounds N , 0 hep-th/0202021 hep-th/0205185 Matrix string theory Dp D JHEP branes On relating multiple ]. 2 ]. 2, hep-th/9610043 ]. – 14 – M ]. ]. D Bubbling AdS space and hep-th/9702101 hep-th/0406145 to Multiple Multiple (2002) 013 [ 2 (2002) 056 [ M ]. 04 05 arXiv:0803.3242 (1997) 5112 [ Comments on multiple ]. (1997) 265 [ (2004) 021 [ Membranes on an orbifold A harmonic family of dielectric flow solutions with maximal Strings from matrices Rotational invariance in the M(atrix) formulation of type I arXiv:0803.3611 hep-th/9910052 JHEP JHEP hep-th/9904095 , , 12 D 55 arXiv:0803.3803 arXiv:0804.1256 Comments on the Bagger-Lambert theory and multiple (2008) 025 [ B 400 ]. . ]. ]. JHEP 05 hep-th/0409174 , (2008) 106 [ (2000) 703 [ (1999) 63 [ On the problem of multiple Phys. Rev. hep-th/9703030 (2008) 105 [ (2008) 038 [ , JHEP Proposals on nonperturbative superstring interactions , Phys. Lett. super Yang-Mills 05 05 B 802 B 573 B 558 , (2004) 025 [ =4 hep-th/9702187 arXiv:0712.3738 hep-th/0110242 theory [ (1997) 43 [ Phys. arXiv:0804.1784 Phys. theories JHEP conjecture supersymmetry [ 10 Phys. [ M-theory on a PP-wave N supersymmetric background of IIB superstring theory JHEP [30] T. Banks and N. Seiberg, [32] R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, [31] L. Motl, [29] S. Sethi and L. Susskind, [27] U. Gran, B.E.W. Nilsson and C. Petersson, [26] A. Morozov, [28] D.S. Berman, L.C. Tadrowski and D.C. Thompson, [14] J. Bagger and N. Lambert, [25] M. Van Raamsdonk, [15] T. Banks, W. Fischler, S.H. Shenker and L. Susskind, [18] N. Lambert and D. Tong, [24] M.A. Bandres, A.E. Lipstein and J.H. Schwarz, [22] I. Bena and N.P. Warner, [23] S. Mukhi and C. Papageorgakis, [13] W. Taylor and M. Van Raamsdonk, [21] H. Lin, O. Lunin and J.M. Maldacena, [17] K. Dasgupta, M.M. Sheikh-Jabbari and M. Van Raamsdonk, [19] J. Distler, S. Mukhi, C. Papageorgakis and M. Van Raamsd [16] D.E. Berenstein, J.M. Maldacena and H.S. Nastase, [20] M. Blau, J.M. Figueroa-O’Farrill, C. Hull and G. Papado [12] W. Taylor and M. Van Raamsdonk, JHEP08(2008)002 , , heory, a (1997) 91 ]. (1995) 13 ]. ]. B 490 ang-Mills theory B 360 . . (2002) 171601 89 BPS solutions of the tiny hep-th/0308200 hep-th/0202109 Nucl. Phys. ]. 2 hep-th/0603023 , Phys. Lett. / , 1 hep-th/0107067 hep-th/9704080 , , ]. (2003) 5225 [ ]. ]. ]. Phys. Rev. Lett. (2002) 126004 [ , 20 (2006) 085011 [ hep-th/0501001 Strings from quivers, membranes from moose D 65 Newton-Hooke algebras, non-relativistic branes and – 15 – D 73 Branes from matrices Classification of all Non-relativistic superstrings: a new soluble sector of ]. ]. Type II pp-wave matrix models from point-like hep-th/9508143 Newton-Hooke space-times, Hpp-waves and the hep-th/0507036 Exactly solvable model of superstring in plane wave (2005) 001 [ hep-th/0406214 hep-th/0606057 04 Phys. Rev. Phys. Rev. , , multiplet of type IIB superstrings ) JHEP ,Z Class. and Quant. Grav. Tiny graviton matrix theory: dLCQ of IIB plane-wave string t (1995) 252] [ (2005) 024 [ , hep-th/0205213 hep-th/0204147 (2004) 017 [ , (2006) 022 [ 12 09 SL(2 ]. ]. 08 String interactions in PP-waves B 364 An Another conjecture about M(atrix) theory JHEP Symmetries, matrices and de Sitter gravity Matrix strings in pp-wave backgrounds from deformed super Y JHEP , JHEP (2002) 022 [ (2002) 023 [ , 5 , S 08 05 × 5 hep-th/9612157 Erratum ibid. hep-th/0205174 graviton matrix theory Ramond-Ramond background [ AdS cosmological constant generalized pp-wave metrics [ gravitons [ JHEP JHEP conjecture [46] R.R. Metsaev and A.A. Tseytlin, [45] M.M. Sheikh-Jabbari and M. Torabian, [42] J. Gomis, J. Gomis and K. Kamimura, [41] G.W. Gibbons and C.E. Patricot, [44] T. Banks, N. Seiberg and S.H. Shenker, [43] J. Brugues, J. Gomis and K. Kamimura, [40] Y.-H. Gao, [39] L. Susskind, [38] J.H. Schwarz, [37] Y. Lozano and D. Rodriguez-Gomez, [36] G. Bonelli, [35] R. Gopakumar, [34] S. Mukhi, M. Rangamani and E.P. Verlinde, [33] M.M. Sheikh-Jabbari,