Supergravity, Supermembrane and M(atrix) model on PP-Waves

Transcription

1 CU-TP-074 hep-th/038 Supergravity, Supermembrane and Matrix) model on PP-Waves Norihiro Iizuka Department of Physics Columbia University, New York, NY 007 We study the back-reaction of large-n D0-brane charge on the maximally supersymmetric pp-wave background. This gives the type IIA geometry on which string theory is dual to the BMN Matrix model. We identify the energy scales where perturbation theory in this Matrix model is valid. We derive the supermembrane action on a general pp-wave background and through this, derive the Matrix model.

2 Introduction At the present time, two concrete examples of the holographic principle are well known. One is AdS/CFT and the other is the Matrix) model of M- theory []. The latter is conjectured to be the microscopic definition of dimensional M-theory in the DLCQ description and its action is described by the large-n limit of dimensional reduction of UN) gauge theory to 0 + dimensions: S 0 = dτt r Ẋi + [ X i,x j] + iψ T Ψ Ψ T γ [ i X i, Ψ ]) ) 4 i= i,j= i= Interest in this model was revived after the paper [] where the Matrix model on maximally supersymmetric pp-wave is derived containing extra mass terms: 3 3 S mass = dτt r i= µ ) X i 3 i=4 µ 6 ) X i + iµ 3 i,j,k= ɛ ijk X i X j X k iµ 4 ΨT γ 3 Ψ ) ) This corresponds to a background that is the maximally supersymmetric pp-wave metric [3, 4], ds =dx + dx 3 µ ) x i dx + 3 i= µ ) x i dx i=4 dx i. 3) i= F 4) = µdx + dx dx dx 3 Several papers about analysis and generalization of this action soon followed [6, 7, 8, 9, 0,,, 3, 4, 5, 6]. A big difference of this model from one on flat space is the non-existence of flat direction due to the mass terms which are induced by pp-wave geometry. This improves the situation, since we do not have to worry about the threshold bound state, which is a very difficult problem in the flat space case. On the other hand, it is well-known that the Matrix) theory action is the same action of type IIA large N D0-branes action in flat space background. This relation is clarified by the paper [7, 8] and its claim that by infinitely boosting a large-n IIA D0 system, the small spacelike compactified radius along x 0 is infinitely boosted to a large lightlike radius along x. The reason we need to take large-n limit is because we want to take lightlike circle to infinity while keeping p fixed. A similar story should hold in the

3 Matrix) model on the pp-wave background. In [6], IIA background metric on which D0-branes action is identical to BMN action is explicitly derived. They derived it by unboosting dimensional pp-wave metric and then compactifying this to 0 dimensions. The metric is written in string frame as, ds = g 00 dt + g AB dx A dx B. 4) g 00 = e 3 φ,g AB = δ AB e 3 φ e 4 3 φ = F A 0 = F F µ ) ) F x µ ) ) + x + x 3 + x x F 03 = µ F,H F 3 = µ The above metric makes sense only when curvature is small, that is, only when F. Actually by using this property [9], we can construct the D0-branes action in this weak background and it is the same action of BMN [6]. In [5] it was shown that the back-reaction of the large-n D0-brane charge on the geometry is important. In the flat space background case, we know large-n D0-branes action is dual to string theory on a warped AdS S 8 background, and depending on the energy scale we consider, we have different description of this 0+ dimensional theory. The phase diagram of this D0- branes system is shown in the beautiful paper [5]. Motivated by these developments, this paper establishes a similar correspondence in a non-trivial background 4), that is, we claim that string theory on the near-horizon geometry of N D0-branes in the background geometry 4) is dual to BMN action, and depending on the energy scale, we have different expressions of the theory as shown in [5]. The organization of this paper is composed of two parts. In section, we study the effect of the back-reaction of 0-brane charge on the maximally supersymmetric pp-wave and derive the IIA geometry on which string theory is dual to BMN Matrix model. This geometry is given by eq. 9) with flux ). Then we identify the energy scale where dual perturbation theory of the BMN Matrix model is valid [6]. In section 3, this is completely independent work to section, we try to derive the Matrix model on a general pp-wave background as opposed to the maximally supersymmetric choice in section.

4 We consider the supermembrane action on a general pp-wave background up to quadratic terms of fermion fields. Then we prove that there are no higher order corrections. Finally, we obtain the Matrix model on a general pp-wave background and we study its supersymmetry which has the expected form given the background. In the appendix, we summarize the relevant facts about general pp-waves. Relation to string theory. Type IIA background First we derive the background supergravity metric of N D0-branes in the metric 4). If we look at this system from an dimensional point of view, this is the system of gravitational waves having N units of momentum along x 0 in the pp-wave background 3). The gravitational wave with N units of momentum in flat dimensional space has the supergravity background ) C ds =dx + dx + dx + + d x. 5) x 7 F 4) =0 ) C 8 π 9 7 Γ α 5 gym N) F 4) is field strength of 3-form gauge potential and the constant C is checked from comparison of this metric when KK reduced to the 0 dimensional metric describing N D0-branes. Now because of enough supersymmetry, the superposition of pp-wave metric 3) and gravitational wave 5) is also a solution of Einstein equations. We combine them using the ansatz of plane wave, ds =dx + dx + H x) dx + + d x ) ) [ C C µ ) H F = x x 7 x x + ) µ ) x 3 + x ) ] x. 6) F +3 = µ where F is defined in eq. 4). The only nontrivial component of the Ricci tensor is R ++ i H µ except at x = 0). Because the curvature scalar vanishes, the superimposed solution satisfies the Einstein equation R ++ F +ijk F ijk + µ as expected. Knowing the dimensional metric, 3

5 it is straightforward to derive the 0 dimensional metric in string frame. The metric is, ds = g 00 dt + g AB dx A dx B. 7) g 00 = e 3 φ,g AB = δ AB e 3 φ e 4 3 φ =+ H A 0 = H + H F 03 = µ +H,H + H 3 = µ Note that this is the IIA supergravity solution describing HD0-branes, which are not supersymmetric [4].. The duality As usual in AdS/CFT duality, we must take the near-horizon geometry of the metric 7) so that the boundary theory decouples from the bulk theory. The decoupling limit we take is U α 3 i= x i =fixed, V α x i =fixed, 8) i=4 g YM = 4π g s α 3 =fixed, µα =fixed, α 0 The type IIA supergravity solution in this decoupling limit is ds = α [ g 0 dt + g 0 du + U dω ) + g 0 ) ] dv + V dω 5 9) where g 0 is well-defined: [ g 0 = 8 π 9 Γ 7 ) g ) YM N) α ) ] µ α U µ V U + V ) ) In this decoupling limit, the NSNS and RR field strengths are ) ) α H 3 = α µ α, F 03 = α µ ) 4

6 You can easily see that since both the metric and the NSNS field strength are O α ), their contributions are at the same order and give rise to the welldefined Nambu-Goto action, that is, α cancel out in S NG α d σ det G + B). And of course these fluxes are supported by a curvature term which is proportional to α µ in g 0. Now it is clear that this background has SO9) SO3) SO6) symmetry breaking because of the O α 3 µ) terms in the flux and the metric. Here we take parameter µ as 8). If we take different limit of µ, the results are very different. For example if we take µ =fixed,the effect of µ is negligible. The geometry is warped AdS S 8 space and the rotational symmetry is enhanced to SO9). If we take a limit where µ is bigger than O α ), the IIA metric doesn t make sense since the curvature becomes singular at g 0 = 0, i.e. from dimensional point of view, the sign of metric in dx 0 ) changes and becomes timelike, therefore compactification doesn t make sense at this point. The curvature of this metric is for simplicity taking U and V to be the same order) α R g 0 U ) So the curvature small condition is satisfied for U g YMN ) 3 if g YM N) 3 α µ) 3 3) U gym N) 9 α µ ) 9 if gym 3 N) α µ) 3 4) Also the dilaton small condition is satisfied for g YM N) 3 N 4 U g YM N) 3 α µ ) N 3 5) Note that the second inequality in eq. 5) is weaker than curvature small condition in eq. 3) and 4). We claim that IIA string theory on this background 9) with fluxes ) is dual to BMN action as long as we consider BMN action at energy scale region U: g YM N) 3 N 4 U g YMN ) 3 if g YM N) 3 α µ) 3 6) g YM N) 3 N 4 U g YM N) 9 α µ ) 9 if g YM N) 3 α µ) 3 7) You may notice that this background is very similar to the dilatonic version of the Polchinski and Strassler model [0] in the sense that mass deformation of the theory on the branes is induced by flux and also the flux makes The author thanks Koenraad Schalm for a useful discussion on this point. 5

7 the N 0-branes into a spherical membrane. But a few points are different. First in our case, mass deformation doesn t break any supersymmetry which corresponds to introducing both NSNS -form and RR 3-form in the background. Secondly, we have both pointlike 0-branes and spherical membrane at zero energy. In 0+ dimensional point of view, this simply corresponds to the fact that we have two classical vacua each of which corresponds to taking different vacuum expectation value. One of the big questions in Matrix) theory is how can we describe a transverse S 5 brane, and how many quantummechanical vacua exist for this BMN Matrix model. Classically we can see only two vacua, one being a pointlike graviton and the other a spherical membrane, represented by an N N irreducible representation of SU). But since the S 5 giant graviton exists in AdS 7 S 4 and AdS 4 S 7,andthis giant graviton survives after taking pp-wave limit, it should exist in BMN Matrix model. In [], two of these vacua are lifted by instanton effects and it is claimed that there is only one vacuum in this system. It would be nice to understand this problem from the BMN Matrix model point of view more precisely. An attempt to understand this problem using a toy model is found in []. In [6], it is discussed that in BMN Matrix) theory with µ,perturbation theory is a good description around each vacuum. So now we will R study at which range of U perturbation theory makes sense..3 Perturbative region of BMN action Following [6] we will study at what energy scale or parameter region the 0+ dimensional IIA BMN action is well described by perturbation theory. We may then compare this with the dual string theory description regime. The most interesting question is in the regime where supergravity make sense, can we describe BMN theory by perturbation theory in some parameter? In µ = 0 case, we know the answer is No [5]. But now we have an extra parameter µ, so let us re-analyze the question. In the IIA background metric 4), the DBI action is written as [ L = l3 s ) Tr [ U g s i + U i,u j] + ψ T ψ i ψ T γ [ i ψ,u i] 4 i= i,j= i= µ ) 3 ) U i µ ) ) ) + U i + µ i U i U j U k ɛ ijk µ ] 4 ψt γ 3 ψ 8) i= i=4 i,j,k= 6

8 Note that U i has dimensions of energy = length and ψ has dimensions of length 3. In this case the classical vacua are U i = 0 for all i =...9 and U i = µ 3 J i i =,, 3),U 4 = = U 9 = 0 9) where J i is representation of the SU) algebra satisfying, [ J i,j j] = iɛ ijk J k For convenience, we expand the field around the pointlike classical vacuum U i = 0 as follows [6]: U i gs l 3 s µy i i =...9),t µ τ 0) ψ gs Note that the rescaling factor gs has dimensions of energy so Y i is dimensionless, and so are the fermion fluctuations θ. Note that we re-scale every lsµ 3 field and t by a rescaling factor gs for bosons and gs for fermions. These ls 3µ ls 3 factors go to 0 in the decoupling limit. Note also we re-scale t to τ which is dimensionless, therefore all energy is measured in units of µ in this theory. After plugging this into the action 8), we get L 3 = [ L = Tr i= ) 3 3 i= l 3 s θ L = L + L 3 + L 4 Y i ) + θ T θ 4 θt γ 3 θ Y i ) + 6 ) ) ) ] Y i i=4 [ gs lsµ Tr i Y i Y j Y k ɛ ijk i θ T γ [ i θ, Y i]] i,j,k= i= ) ] gs [ L 4 = Tr[ Y i,y j] ls 3µ3 4 i,j= First we consider the expansion around the U i = 0 vacuum. We can identify the effective coupling constant in this 0+ dimensional theory as g s N g eff l = 4π gym N). ) s 3µ3 µ 3 7

9 Note that in the decoupling limit 8), this g eff 0 and also is independent of the scale of vacuum expectation value of the field. It is determined by the typical energy scale of this theory which is order of µt r < Y i > µ. N This is slightly different from the case where µ = 0 discussed in [5] where g g eff is given by YM N) and U is the vev of the theory. U 3 If we expand around the membrane vacuum 9), U i µ 3 J i gs + i ls 3µY i =,, 3) we get a similar result although in this case, L, L 3 and L 4 involve J i. Generally J i can take any representation. Since it is a little bit complicated to consider the general case, let us restrict the discussion in the case that J i is an irreducible N N representation. The correspondence between supergravity scale Usugra scale is given by, and Matrix model U sugra = N δu sugra = N 3 Tr i= 3 Tr [ <UN N i > ] N i= 3 i= [ J i Tr ) ] µn) [ ) ] ) gs l Y i gs s 3µ3 N N ls 3µ3 N g YM N µ 3 here the typical wave function YN N i of this theory is determined from L, andthisisbecausel 3 and L 4 are suppressed by g eff 0, see [6]. Therefore the parameter region of U where perturbative description holds are U sugra N g YM N µ 3. ) µn gym N U N µ 3 sugra µn + gym N. 3) N µ 3 In the region ), the ten dimensional description doesn t make sense because the dilaton is so large. In the region 3), we can t trust the metric 9) because this is the region where g 0 < 0. Therefore, as usual dualities, the two descriptions string theory and perturbation theory) doesn t hold at the same time, even though we have extra parameter µ. Itwouldbeniceto understand the correspondence for general representations of J i. 8

10 3 Matrix) model on general pp-wave Having understood the whole picture of supergravity, supermembrane, D0- branes and Matrix) model in maximally supersymmetric background, we study less supersymmetric pp-wave backgrounds. One of the surprising things in pp-wave is that there are plenty of pp-wave solutions which preserve supernumerary or fractional number of additional supersymmetries. So far as we know pp-wave solutions preserve N = 6 + N extra, where N extra =0,, 4, 6, 8, 0 [4, 5, 6, 7, 7, 8]. The goal of this section is to try to derive the DLCQ action of the Matrix model on these general ppwaves. We summarize the relevant known results about supersymmetry of fractional pp-waves in the appendix. The dimensional pp-wave we consider has metric [3], ds =dx + dx + H x i,x +) dx + + H x i,x +) = F 4) = The Einstein equations require, i,j,k= µ i = i= i= µ i xi 3! f ijkdx + dx ijk ) f ijk ) 3! i,j,k= i= dx i Here pp-waves always have at least 6 supercharges, plus k extra supersymmetries see eq. 3),33),34),35) in the appendix). 3. Supermembrane on general pp-wave The dimensional supermembrane action is [ S [Z ζ)] = d 3 ζ G Z) ] 6 ɛabc Π A a ΠB b ΠC c B CBA Z ζ)) where Z A = X M ζ),ξζ) ) are superspace embedding coordinates and B CBA the antisymmetric tensor gauge superfield, we also take light cone We use notation A = M,α) for curved space-time indices, here M = +,,i=,..., 9)) whereas tangent space vector indices r =+,,i=,..., 9). We are 9

11 gauge X + = τ, τ is the world-volume time coordinate 3,andG is given by G X, ξ) =det Π M a Π N b g MN ) = det Π r a Π s bη rs ). Here Π r a is the supervielbein pullback Π r a = a Z A E r A The supermembrane action on fully supersymmetric pp-wave background can be derived by using the coset space approach where we can express the supervielbein and tensor gauge superfield background in all orders of ξ [9]. In the fractional pp-wave case, we don t know the exact supervielbein and tensor field, although it is known to order ξ) [30]. In the case that the gravitino has 0 vev, supervielbein pullback Π r a is given by Π r a = a Z A EA r = a X e M r M ) 4 ξγ rst ξw Mst + ξγ r Ω M ξ + ξγ r a ξ + O ξ 4) Also the tensor fields pullback term is given by 6 ɛabc Π A a ΠB b ΠC c B CBA Z ζ)) = [ 6 ɛabc a X M b X N c X L C MNL ξγ rs Γ MN ξw rs ɛ abc ξγmn c ξ [ ax M b X N + ξγ N b ξ ) + 6 ξγ M a ξ ξγ N b ξ Here the nonzero components of wm rs, Ω M are given in eq. 30) in the appendix. In order to write down the explicit form of the fermion fields, we write the dimensional gamma matrices in terms of 9 dimensional gamma matrices as given in eq. 3) and remove the κ symmetry of this action by gauge fixing ξ = 4 Γ + ξ =0 0 Ψ T ), ξ 4 ) Ψ 0 Under this gauge condition, the supervielbein pullback and tensor gauge superfield pullback are written as Π + a = a X +, Π i a = a X i, H Π a = ax + a X + i ) 4 ΨT θψ + iψ T a Ψ, sloppy about transverse coordinates because that direction is flat. Also we use a =0,, for world-volume coordinate on membrane such that ɛ 0 =. 3 We choose world-volume coordinates τ,σ,σ ). ] L 3 ξγ MN Ω L ξ ] + O ξ 4) 0

12 6 ɛabc Π A a ΠB b ΠC c B CBA = 6 i,j,k= f ijk {X i,x j } PB X k i ) Ψ T γ i {X i, Ψ} PB Here {X, Ψ} PB = ɛ 0ab a X b Ψ, and θ 3! f ijkγ ijk is the 6 6 SO9) gamma matrix. Therefore the action is given by S = d 3 σ i= again up to O Ψ) 4. Ẋi i= µ i Xi 4 i= {X i,x j } PB 6 i,j= i,j,k= +iψ T Ψ i 4 ΨT θψ i f ijk {X i,x j } PB X k Ψ T γ i {X i, Ψ} PB )4) So far we have derived this action by truncating the full action, and in principle we should include terms of higher order in Ψ. But even though we derived this action by the truncation of O Ψ) 4 terms, we can argue that this is the exact action on general pp-wave background. The proof is as follows. 4 The truncated terms have two parts, one being higher order terms in the supervielbein pullback, the other higher order terms in the tensor gauge superfield pullback. Let s consider the supervielbein first. The supervielbein is defined as Π r i = iz A EA r,and iz A is proportional to i X M or i ξ, while EA r is superfield of elfbein which is made by some complicated function of the quantities ξ, Γ r, C ijk, F +ijk, M, w MNL e rn e sl wm rs,γm NL and all other geometrical functions with indices of curved space M, N,..., i.e. some derivative of e r M ) e rn. But note that C ijk, F +ijk, M, w MNL e rn e sl wm rs,γm NL and whatever function we obtain from the derivative of elfbein, they cannot have curved space-time lower index M =. This is because the pp-wave has null Killing vector k M, which has only one nonzero component along upper index M = and lower index M = +, and the field strength is proportional to k M = k +. In the same way all upper curved space indices can t have M =+. On the other hand, because of the gauge condition Γ + ξ =0, ξγ r...γ s ξ can be nonzero if and only if we have one tangent space upper index r = and no upper index r = + for the Gamma matrices between ξ and ξ. Since e r=,m 0ifandonlyifM = +, in order to contract the curved space-time index M, we need upper M = + for each ξγ r...γ s ξ term, and the only component which can have upper M = + to contract is a X M=+) = δa 0. Since the supervielbein pullback is at most linear in a X M, we conclude that ξ is at most bilinear in ξ, therefore that there are no higher order corrections in ξ. 4 Similar argument about Green-Schwarz action on general pp-wave background which doesn t have supernumerary supersymmetry is found in [3]. i=

13 In the same way we can show that there are no higher order corrections to the ɛ abc Π A a Π B b ΠC c B CBA Z ξ)) term in the action. Since each bilinear term ξγ r...γ s ξ needs one upper index r = and e r=,m 0iffM = +, we need a X M=+) = δ 0 a from a Z A. But because ɛ abc a X M=+ b X N=+ =0,wecan have at most linear terms in i X M=+. Therefore these terms are at most bilinear in ξ so the action 4) which we obtained is exact. 3. Matrix model and its supersymmetry Once we obtain the supermembrane action exactly, it is straightforward to derive the Matrix model action by regularizing according to the usual prescription [3]. X M ζ) X M N N Ψζ) Ψ N N d σ Tr {, } PB i [, ] S = dτt r i= Ẋi i= µ i Xi + 4 i,j= [ X i,x j] + i 3 +iψ T Ψ i 4 ΨT θψ i,j,k= f ijk X i X j X k Ψ T γ [ i X i, Ψ ]) 5) i= Note that on general pp-wave, the mass term is time dependent. Remember that in BFSS Matrix model description of M-theory, we have 6 dynamical supersymmetries of the action, and that the supersymmetry transformation spinor is the one for the background which survives after we introduce momentum. That is, the 6 components of the background Killing spinor which satisfy Γ 0 ɛ = ɛ Γ ɛ = 0) are realized as the dynamical supersymmetries of the flat space Matrix theory. The same story should hold for a Matrix model on a pp-wave background. Actually one can see that the dynamical supersymmetry transformation spinor of BMN action has exactly this form as given in eq. 3) χ in the appendix. Note that because Γ + ɛ =Γ + χ =0 χ =0,theχ + components are not realized as dynamical supersymmetry, but rather realized as kinematical supersymmetry of the Matrix model on pp-wave.) In the same way we expect the above action has k supersymmetries as expected from the background pp-wave geometry. Now it is straightforward to check the condition of supersymmetry of this action.

14 In order to analyze the supersymmetry of this action, it is enough to analyze just the Abelian case. We therefore consider the Abelian action ) S = dτ µ i xi +Ψ T τ Ψ a 4 ΨT θψ. 6) i= ẋi i= Even though we expect a = from an derivation, here we take a to be an unknown constant. Taking the ansatz of supersymmetry transformation as, δx i =Ψ T γ i N i ε no sum over i) δψ = Ẋi γ i N i ε + µx i γ i M iε) i= ε = ε τ) =e µmτ ε 0 We wish to determine the matrices N i, M i, M as well as a. Terms of O µ 0 ) automatically cancel. Terms of O µ )give and µm i = µn i M + a 4 θi N i 7) µ =0. Here θ i γ i θγ i. Finally the O µ )termsgive µ N i M aµ a ) ) ) θi N i M + θ i + µ i N i 4 ε = 0 8) for all i =...9. Let s consider some solutions of this equation. Setting the matrices N i = for all i =...9, equation 8) becomes µ M aµ a ) ) θi M + θ i + µ i ε =0. 4 By setting M b θ,withb some unknown constant, this equation reduces to µ b θ ab a ) ) θi θ + θ i + µ i ε =0. 4 You can see that the solution a = ±, b = exactly coincides with eq. 33). Also the form of the spinor is given by ε = e µ θ and again this is exactly the expected form of the k Killing spinor in 3). So far we have 3

15 found that some supersymmetry transformations of the Matrix action are inheritated from the background action. Note also that this action has kinematical N = 6 supersymmetry under which fields transform as: δx i =0 δψ =exp + x + dx + θ x +)) η 4 for some constant spinor η. This is again expected from the supersymmetry of the background. 4 Conclusions We have studied the back-reaction of the large-n momentum on the maximally supersymmetric pp-wave geometry, and derived the type IIA geometry on which string theory is dual to the BMN action. We also identified the SUGRA energy scales where a perturbative description of the BMN Matrix model is valid. We succeed in deriving the supermembrane action and the Matrix model on a general pp-wave background and checked that the supersymmetry of this action is as expected from the background. There are several points which are not fully understood. It would be nice to understand the amount of supersymmetry in the geometry 9), and also the SUGRA/Matrix model energy scale correspondence for the general representation of J i as we discussed in section.3. How many quantum vacua does the BMN theory have is very interesting question. Finally it would be interesting to analyze the light cone time-dependent Matrix theory which is dual to M-theory on a time dependent pp-wave. Acknowledgments The author is very grateful to Mark Jackson, Koenraad Schalm and especially Dan Kabat for helpful discussion and encouragement. This research is supported by DOE under contract DE-FG0-9ER A Fractionally supersymmetric pp-waves Here we summarize the known results of dimensional general pp-waves which posses 6 N 3 SUSY. 4

16 The dimensional pp-wave has metric ds =dx + dx + H x i,x +) dx + + i= dx i H x i,x +) = F 4) = i= i,j,k= µ i = 3! Generally µ i and f ijk are functions of x +. The Killing spinor ɛ should satisfy i= µ i xi 3! f ijkdx + dx ijk i,j,k= f ijk ) D M ɛ = M ɛ +Ω M ɛ =0. 9) Ω M = ) Γ PQRS M 8δM P 88 ΓQRS F PQRS for M =0,,...,. In a pp wave background, the non-zero components of the elfbein e rm, spin connection wm rs and Ω M are e r=+,m=+ = H,e r=+,m= =,e r=,m=+ =,e r=i,m=j = δ ij 30) w i + = ih = µ i xi. Ω + = ΘΓ Γ + +) Ω i = 4 3ΘΓ i +Γ i Θ) Γ where Θ is defined as Θ ijk= 3! f ijkγ ijk. In order to discuss the supersymmetry of these pp-wave backgrounds, it is convenient to decompose the 3-component Killing spinor ɛ in terms of two 6-component SO9) spinors as ɛ = i= ) ) x i χ+ Ω i χ, χ χ 5

17 Here we extract x i dependence from ɛ explicitly, therefore χ χ+ depends only on x +. One can check that this x i dependence satisfies 9). χ ) Decomposing the dimensional gamma matrices in terms of SO9) gamma matrices, Γ i = γ i σ 3 i =...9), Γ 0 = iσ, Γ = σ 3) Γ =Γ + = Γ 0 +Γ ) = ) 0 i 0 0 Γ + =Γ = Γ 0 +Γ ) = ) 0 0 i 0 and define the 9 dimensional gamma matrix θ Θ i,j,k= 3! f ijkγ ijk = i,j,k= 3! f ijkγ ijk σ 3 θ σ 3. Therefore Γ + ɛ =0meansχ =0sinceΓ + ɛ =Γ + χ. Then the Killing spinor which satisfies 9) is expressed as χ + = exp + ) x + dx + θ ψ +, χ = exp ) x + dx + θ ψ. 3) 4 Here ψ ± is a constant spinor. Equation 9) gives no constraint on ψ + but gives a constraint on ψ,that [ 44µ i + 3γ i θγ i + θ ) 3γ i + θγ i + + θ )] ψ = 0 33) for all i =...9. Here + =. The condition that there exist supernumerary supersymmetry is that there exists non-trivial ψ which satisfy this x + equation, that is, the determinant must be zero, or Π 8 I={ µ i ) ρi,i) + + ρ i,i ) } = 0 34) for all i =...9. Here we choose the basis so that the 6 6 matrix 3γ i θγ i + θ) is skew diagonal and its skew eigenvalues are ρ i,i where I =,..., 8. This shows that f ijk should be x + -independent in order to have supernumerary supersymmetry. Then 34) can be zero if 44µ i = ρ I i = = ρ I k,i 35) is satisfied for all i =...9. Therefore pp-wave backgrounds always possess 6 supersymmetries from ψ +,andksupersymmetries from ψ when the background has no x + -dependence. ψ has 6-k) zero components because of the 6 k nontrivial constraints 33). 6

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