Lorentz Symmetry of Supermembrane in Light Cone Gauge Formulation

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AXa. E)XD · XF (cHC. F ϕH + cλC. F ϕλ). −fDAE(Xa. AXb. E − Xb. AXa. E)XD · XF ϕF . (31). 3In this calculation we have used the relation: ((M−a)(0), (M−b)(0))DB.
YITP-97-19 May 1997

arXiv:hep-th/9705005v2 22 May 1997

Lorentz Symmetry of Supermembrane in Light Cone Gauge Formulation

Kiyoshi EZAWA∗1 , Yutaka MATSUO2 , Koichi MURAKAMI3 Yukawa Institute for Theoretical Physics Kyoto University, Sakyo-ku, Kyoto 606-01, Japan

Abstract We prove the Lorentz symmetry of supermembrane theory in the light cone gauge to complete the program initiated by de Wit, Marquard and Nicolai. We give some comments on extending the formulation to the M(atrix) theory.

hep-th/9705005 JSPS fellow e-mail address : [email protected] 2 e-mail address : [email protected] 3 e-mail address : [email protected]



1

1

1

Introduction

After the discovery of the string duality, our perception of the string theory was drastically changed. What used to be the obscure inhabitants of the string theory, the p-branes, turned out to be the key ingredients of the non-perturbative physics. M-theory is supposed to be one of the most symmetric form of the “string” theory. However, because of our ignorance of the quantization of the p-branes, the very definition of the theory has been largely unknown. By critical use of the simplification due to the infinite momentum frame, BFSS [1] proposed a constructive definition of the M-theory. The momentum along the eleventh dimension is identified with the zero-brane charge. The infinite boost kills the degree of freedom which has zero (fundamental string) and negative (anti-zero brane) charges. The resulting Lagrangian is made up only with the zero-branes described by the large N limit of the SU(N) Yang-Mills theory. BFSS have indicated two major evidences which support their idea. 1. The matrix theory Lagrangian coincides with that of supermembrane proposed by de Wit, Hoppe and Nicolai (dWHN) [2] if one replaces the gauge group from SU(N) to the area preserving diffeomorphism (APD) in two dimensions. 2. The scattering of the zero-branes coincides with the prediction of the eleven dimensional supergravity. As usual, the subtlety in the infinite momentum frame is the Lorentz symmetry. This problem is very difficult to analyze in the matrix theory since the momentum exchange in the eleventh dimension means the exchange of zero-brane charge. We need to treat the quantum process which changes the size of matrices1 . On the other hand, the analysis of the similar problem in the dWHN model is accessible since we know the covariant Lagrangian in eleven dimensions. Indeed this program was nearly accomplished by de Wit, Marquard and Nicolai (dWMN)[4]. They defined the Lorentz generators and have shown that they commute with the Hamiltonian of the system. In their proof, they essentially used various identities of the APD tensors. The purpose of this technical note is to complete this program, namely to give the direct computation of the algebra of the Lorentz generators. Our result supports the Lorentz symmetry after the cancellation among numerous non-trivial factors. We need to prove some additional identities of the APD tensors to finalize our result. In section two, we briefly review the result of dWMN to make this note self-contained. In section three, we summarize our proof of the Lorentz invariance. In section four, we give a discussion on the possible extension of our result to the M(atrix) theory. One of the generators of Lorentz algebra depends essentially on the metric of the membrane world volume. Therefore, it is not invariant under the APD and it causes some nontriviality. We argue that this fact might give a hint to eleven dimensional definition of M-theory. Explicit computations and technical 1

Beautiful treatment of this issue is recently proposed by Polchinski and Pouliot [3] by considering scattering of two membranes where zero-brane charge can be treated as the monopole charge on the world brane. In this setting, the zero-brane exchange can be calculated by the instanton calculus on the world brane.

2

comments are provided in the appendix. In appendix A, we describe the identities between the APD tensors. In appendix B, we summarize properties of the Clifford algebra of SO(9). The identities in these sections are used in appendices C and D to prove the Lorentz algebra.

2

Summary of dWHN model

DWHN model [2] is defined as a 0 + 1 dimensional supersymmetric Yang-Mills system whose gauge group is the APD of a fixed two dimensional manifold. The Lagrangian (slightly modified from the original definition) is √

o n 1 −1 ~ 2 + i θD0 θ − 1 X a , X b 2 + i θγa {X a , θ} , w L = (D0 X) 2 2 4 2

(1)

where the definitions of the notation are following. X a (t, σ r ), θα (t, σ r ) (a = 1, . . . , 9, α = 1, . . . , 16, r = 1, 2) are the quantum mechanical variables whose internal degree of freedom is described by two parameters σ. The indices a and α are respectively the vector and the spinor degrees of freedom of SO(9). wij is the 2×2 metric tensor for the parameter space and rs w is its determinant. The curly bracket, {A, B} ≡ √ǫ ∂r A(σ)∂s B(σ), and the covariant w(σ)

derivative, D0 X a = ∂0 X a − {ω, X a}, D0 θ = ∂0 θ − {ω, θ} , define the gauge transformation based on the APD, δX a = {ξ, X a } ,

δθ = {ξ, θ} ,

δω = ∂0 ξ + {ξ, ω} .

The canonical Hamiltonian is H = −

Z

1 = P0+

d2 σP − (σ) Z

q





i 1 n a b o2 1 d σ w(σ) w −1 P~ 2 + X ,X − θγa {X a , θ} . 2 4 2 2

(2)

~ The non-vanishing Dirac brackets are P~ denotes the canonical momentum conjugate to X. 

X a (σ), P b(ρ)



DB

= δ ab δ (2) (σ, ρ),

(θα (σ), θβ (ρ))DB = − q

i w(σ)

δαβ δ (2) (σ, ρ).

(3)

The Gauss law constraints associated with the APD can be written as ϕ(σ) ≈ 0 and ϕλ ≈ 0 where ϕ(σ) ≡ ϕλ ≡

(

P a (σ) a − , X (σ) w(σ)

)

i {θ(σ), θ(σ)} , 2   Z iq ~ w(σ)θ(σ)∂r θ(σ) . − d2 σΦ(λ)r (σ) P~ (σ) · ∂r X(σ) + 2 −

Φ(λ)r is the basis of the harmonic vectors in the parameter space[4]. 3

(4)

The light cone directions are expressed through X ± = √12 (X 10 ± X 0 ) where one of them is identified with the world volume time variable X + (τ ) = X + (0) + τ . The other one is defined by   i 1 1 ~ P~ (σ) · ∂r X(σ) + θ(σ)∂r θ(σ) . (5) ∂r X − (σ) = − +  q P0 2 w(σ)

The integrability conditions of this differential equation coincide with the Gauss law constraints. When integrated, it gives 1 X (σ) = q − + P0 −



Z





iq ~ d ρG (σ, ρ) P~ (ρ) · ∂r X(ρ) + w(ρ)θ(ρ)∂r θ(ρ) , 2 2

r

(6)

where the integration constant satisfies (q − , P0+ )DB = 1 and Gr (σ, ρ) is the Green function defined by Drρ Gr (σ, ρ) = −(w(σ))−1/2 δ (2) (σ, ρ) + 1. This system has supersymmetry generated by ! √ n Z o 1 w a b 2 a + X , X γab θ, d σ P γa + Q = q 2 P0+ Q− =

q

P0+

Z

q

d2 σ w(σ)θ.

(7)

The Lorentz generators are defined by,

M +− M

+a

M −a

Z





i d2 σ −P a X b + P bX a − θγ ab θ , 4 Z   2 + − − + = d σ −P X +P X ,

M ab =

=

Z

d2 σ(−P + X a + P a X + ),

(8)

! √ Z i w i ab abc 2 a − − a θγ θPb − {Xb , Xc } θγ θ . = d σ P X −P X − 4P0+ 8P0+

DWMN [4] proved that these generators satisfy, d ∂ M= M + (M, H)DB = 0, dτ ∂τ

(9)

namely the conservation of these charges. Although this is a nontrivial consistency check, it is obviously important to prove that the Dirac brackets between these charges indeed satisfy the Lorentz algebra. As in [4], we carried out our computation by using mode expansion in two dimensional parameter space. To define the basis, we pick the covariant Laplacian in the parameter space and define the basis as its eigenfunctions, ∆Y0 = 0,

∆YA = −ωA YA

4

(10)

where ωA > 02 . The index A will take value in positive integers. When we need to treat both zero- and non-zero modes we will use indices I,J. We may require them to satisfy the orthonormal condition, Z

q

d2 σ w(σ)Y I (σ)YJ (σ) = δ I J

Y I ≡ YI∗ = η IJ YJ .

(11)

The completeness condition is X A

1 δ (2) (σ, ρ) − 1. Y A (σ)YA (ρ) = q w(σ)

(12)

P

All fields will be expanded in terms of YI such as, X a (σ) = I X aI YI (σ). The Green function which appeared in the definition of X − is then expanded as, Gr (σ, ρ) =

X A

1 A Y (σ)∂ r YA (ρ). ωA

(13)

The structure constant of the APD is given by fABC ≡ We also define,

Z

q

d2 σ w(σ)YA (σ) {YB (σ), YA (σ)} ,

dABC = cABC =

Z

2

q

{YA , YB } = fABC Y C .

(14)

d σ w(σ)YA (σ)YB (σ)YC (σ)

−2

Z

q

w rs d σ w(σ) ∂r YA YB ∂s YC . ωA 2

(15)

The tensor cABC is motivated to express the mode expansion of the Green’s function and is indispensable to express X − . Whereas the tensors f and d are invariant under APD, c is not invariant because it depends explicitly on the metric. DWMN have argued that there is no modification which makes it invariant. In this sense, it is a challenge to find the analogue of this constant when we treat the M(atrix) theory. We will come back to this issue later. In terms of the coefficients of mode expansion, the Dirac brackets are rewritten as, (XAa , PBb )DB (θαA , θβB )DB (q − , P0+ )DB (X0a , P0b)DB (θα0 , θβ0 )DB

= = = = =

δ ab ηAB , −iδα,β ηAB , 1, δ ab , −iδα,β .

(16)

Let us write down the mode expansion of various conserved charges. The elements of the APD are given by, ϕA ϕλ 2





~ B · P~ C − i θB θC = fABC X 2   i B C B ~C ~ = fλBC X · P − θ θ 2

We consider the case when the parameter space is compact and the spectrum is discrete.

5

(17)

As dWMN indicated, to describe Lorentz generators, it is convenient to separate the zeromode and others. H = M2 = M ab = M +− = M +a =

M2 P~02 + 2P0+ 2P0+ 1 P~A2 + (fABC XaB XbC )2 − ifABC θA γ a XaB θC 2 i i a b −P0 X0 + P0bX0a − θ0 γ ab θ0 − PAa X bA + PAb X aA − θA γ ab θA 4 4 −P0+ q − − Hτ, −P0+ X0a + τ P0a .

(18)

(19)

We write M −a in the following form, M

−a

= (M

−a (0)

)

where





1 f−a , e+ + 1 M fab − i θ γ a Q + + P0b M 0 P0 2 P0+

(M −a )(0) = q − P0a + X0a H −

(20)

i ab b + θ0 γ θ0 P0 , 4P0

B C ab A e + = (P a + 1 f Q ABC Xa Xb γ )θ , A 2 fab = −P a X bA + P b X aA − i θ γ ab θ A M A A A 4 b b f−a = 1 dABC X a (P ~B · P~C + 1 (fBDE XD M XEc )(fCF G XFb XGc ) − ifCDE XD θB γb θE ) A 2 2 1 i ~ C + i θB θC ) − dABC PAb θB γ ab θC + cABC PAa (P~B · X 4 2 2 i ABC DE dA XBb XCc θD γ abc θE . (21) − f 8

3

Lorentz symmetry in supermembrane

In this section we summarize our proof of the Lorentz symmetry. The detail is explained in appendices C and D. What we want to do is to show that we have the eleven dimensional Lorentz algebra: (M µν , M ρσ )DB = η µρ M νσ + η νσ M µρ − η µσ M νρ − η νρ M µσ ,

(22)

where the indices µ, ν, ρ and σ run the eleven dimensional space-time indices +, −, 1, · · · , 9. DWMN have shown ˜ ab , M2 )DB = 0, (M ˜ + , M2 )DB = 2θA ϕA , (Q





i a ABC ˜ −a , M2 )DB = − fB DE X a X ~ ~ ϕA + cλBC ϕλ ). (M D C · XE − θB γ θC (c 2 6

(23) (24) (25)

The RHS vanishes in the physical subspace. By using these relations and the Dirac bracket (q − , P+0 )DB = 1, we can easily prove the Lorentz algebra(22) except (M −a , M −b )DB = 0, modulo the first class constraints. Proof of the only nontrivial part (M −a , M −b )DB = 0 goes as follows. By separating zero and non-zero modes we find 3 (M −a , M −b )DB =

1 (P0+ )2

(C + D),

(26)

where ˜ ab − i Q ˜ + γ ab Q ˜ + ) + (M ˜ −a , M ˜ −b )DB , C = (−M2 M 4 i i a ˜ + ˜ −b ˜ +, M ˜ −a )DB . D = − θ0 γ (Q , M )DB + θ0 γ b (Q 2 2

(27) (28)

In the appendices C and D we will show that the following relations hold modulo the first class constraints, ˜+

˜ −a

(Q , M

C = 0, )DB = 0.

(29) (30)

Here we only quote the final result. 1. The first equation (29):  i a A b XA (θ γ θD ) − X b (θA γ a θD ) ϕD 2  i − XAa (θB γ b θE ) − XAb (θB γ a θE ) dABC (cDC E ϕD + cλC E ϕλ ) 4 i + (θD γ abd θD )XdE ϕE 2 i − (θD γ abd θE )XdC dA DE (cB AC ϕB + cλ AC ϕλ ) 4  1 1 CAB 1 AB C λAB + − ( f ϕC + f ϕλ ) + fC ϕ (PAa PBb − PAb PBa ) ωA ωC 2ωA ωB 1 ~D · X ~ F (cHC F ϕH + cλC F ϕλ ) − dAIC fI DE (XAa XEb − XAb XEa )X 2 ~D · X ~ F ϕF . −f DAE (XAa XEb − XAb XEa )X (31) 3

In this calculation we have used the relation:   (M −a )(0) , (M −b )(0)

DB

7

=

i θ0 γ ab θ0 M2 . 4(P0+ )2

2. The second equation (30): 1 a BD θαC XD d E ϕE + (γ ad θC )α XdD (cE CD ϕE + cλ CD ϕλ ). 2

(32)

In the physical subspace where ϕA , ϕλ ≈ 0 both of the equations vanish. This completes the proof of the Lorentz invariance of the dWHN model.

4

Discussion: Lorentz invariance of M(atrix) Theory

Although our computation is rather tedious, it has a merit that it can be carried out quite systematically. Therefore we are eager to speculate that such an analysis may be applicable to prove the Lorentz invariance of the M(atrix) theory in the large N limit. Indeed there are well known correspondence between SU(N) in the large N limit and the APD. For the simpler part, the translation table is given as follows,

n

APD Y A (σ) {X, Y } R 2 q d σ w(σ)X(σ) o

h

Y A , Y B = f AB C Y C

h

SU(N) TA [X, Y ] Tr X

T A, T B

Y A Y B = dAB C Y C

i

T A , T B = f AB C T C i

+

= dAB C T C

When we compute the anti-commutator of SO(9) supercharges to study the appearance of p-branes of various dimensions, these two theory were essentially the same [2][5] except for the vanishing of the five brane charges in the supermembrane approach4 . In such computation the corresponding generators of the M(atrix) theory are available after the use of above dictionary. It means the computation does not depend on particular geometry of the world volume. For the calculation of the Lorentz invariance, on the other hand, we need to introduce the third tensor cABC which depends explicitly on the metric of the parameter space and the direct translation becomes more involved. Of course, when the geometry of the parameter space is fixed (say the Riemann surface of genus g), we already know the non-commutative analogue of the surface which can be embedded in the large N limit of SU(N)[6]. In such a situation, the construction of the corresponding Lorentz generators in M(atrix) theory becomes possible. For that purpose, it is convenient to indicate an identity for the tensor cABC (see the appendix E for the proof), cABC =





ωB − ωC − 1 dABC . ωA

4

(33)

Some discrepancy observed in [5] can be removed when we carefully keep the Schwinger term in the matrix computation. See appendix F for detail.

8

This formula shows that once the notion of the Laplacian is generalized to the M(atrix) theory we can construct the tensor c. Because we take the orthogonal basis Y I as the eigenfunction of the Laplacian, the basis of SU(N) should be also taken from eigenvectors that diagonalize such an operator. If the Laplacian thus defined have definite “classical” limit in N → ∞, the relations among the tensors f , d, c will also hold in this limit. This means we will recover the Lorentz invariance. We have now the second type of correspondence. APD ∆ ωA ∆YA = −ωA YA

SU(N) e ∆ eA ω e e A TA ∆TA = −ω

Let us illustrate the idea in a concrete example, namely the case of the (non-commutative) torus. In the APD case, the base of the mode expansion is nothing but the Fourier expansion, YA = exp(i(A1 σ1 + A2 σ2 )), ~ 2. ∆YA = −ωA YA , ωA = |A|

(34)

On the SU(N) side the corresponding basis is 1

A2 1 TA = Nz 2 A1 A2 ΩA 2 Ω1 , Ω1 Ω2 = zΩ2 Ω1 , z = e2πi/N .

(35)

The analogue of the Laplacian in this theory may be picked up by using the adjoint action of Ω as,  N2  −1 −1 Ad(Ω ) + Ad(Ω ) + Ad(Ω ) + Ad(Ω ) − 4 , 1 2 1 2 4π 2 e A TA , = −ω N2 = − 2 (z A1 + z A2 + z −A1 + z −A2 − 4). 4π

e ≡ ∆

e ∆T A eA ω



(36)

e A = ωA . In the large N limit, we restore the relation in the continuous limit limN →∞ ω Obviously, in such a situation, one may define the matrix model analogue of the Lorentz generator in such a way that it gives the correct commutation relation. The message here is that we need to specify the Laplacian in the M(atrix) theory to define the Lorentz generators. It should be encoded in the eleven dimensional definition of M theory and have to be tightly restricted since otherwise the various identities discussed in appendix A will be violated and so is the Lorentz symmetry. The situation reminds us of the fact that the consistent string background is depicted by the conformal invariance which is closely related to the Lorentz symmetry in the light cone gauge. One might say that the background dependence of M(atrix) theory appear here as the choice of the Laplacian and the constraint on it comes from the Lorentz invariance as the tensor identities.

9

Acknowledgements: We would like to thank M. Ninomiya, M. Anazawa, A. Ishikawa, K. Sugiyama for invaluable discussions, comments, and encouragements. We are also obliged to S. Watamura for enjoyable conversation on the non-commutative geometry. Note Added: After we submitted this paper, we are notified that the Lorentz algebra was also computed in the unpublished work [7]. We also added appendix F to clarify our argument in section 4. We thank H. Nicolai for the information and the comments.

A

APD Identities

In this section we present several identities satisfied by the APD tensors. Let us recall the definitions of three tensors in (14) and (15): Z

fABC =

Z

dABC =

q

d2 σ w(σ)YA (σ){YB (σ), YC (σ)}, q

(37)

d2 σ w(σ)YA (σ)YB (σ)YC (σ),

cABC = −2

Z

q

d2 σ w(σ)

(38)

w rs (σ) ∂r YA (σ)YB (σ)∂s YC (σ). ωA

(39)

From the above definitions it is evident that fABC is totally antisymmetric and dABC is totally symmetric. The basis functions YA satisfy the completeness relations, X A

X A

1 δ (2) (σ, ρ) − 1, Y A (σ)YA (ρ) = q w(σ)

(40)





1  r ǫrt ǫsu D YA (σ)D s Y A (ρ) + q ∂t YA (σ) q ∂u Y A (ρ) ωA w(σ) w(ρ) =

w rs (σ)

q

w(σ)

δ (2) (σ, ρ) −

X

Φ(λ)r (σ)Φ(λ)s (ρ). (41)

λ

By using (40) and integrating by parts, the authors of [4] showed that the APD tensors satisfy several identities: f[AB E fC]DE = 0,

dABC f A [DE f B F ]G =

d2 σ w(σ)

q

(42)

d2 σ w(σ)YC (σ){Y[D (σ), YE (σ)}{YF ] (σ), YG (σ)} = 0,

Z

q 1 d2 σ w(σ) {YA (σ) , YB (σ)YC (σ)YD (σ)} = 0, 3 Z Z Z q q q 2 2 = d σ w(σ)YA Y[B YC] YD − d σ w(σ)YA Y[B d2 ρ w(ρ)YC] YD

fA(B E dCD)E = dEA[B dC]D E

q

1 ∆YA (σ)YB (σ)YC (σ) = −2dABC , ωA Z q w rs (σ) ∂r YD (σ)∂s Y [A (σ){Y C (σ), Y B](σ) } = 0, = 2 d2 σ w(σ) ω D Z

cABC + cACB = 2 cDE [A f BC]E

Z

= −ηA[B ηC]D .

10

The first identity is nothing but Jacobi identity. In the last step to derive the third and fourth ones, they used the fact that the parameter space of the APD gauge group is two dimensional (Schouten’s identity). The others are straightforward. They also found from (41) that X fAB E cECD = cEAB f E CD − 2fBD E dACE + cλAB f λ CD , (43) λ

where fλAB =

Z

q

d2 σ w(σ)Φ(λ)t (σ)∂t YB (σ)YA (σ), Z

cλAB = −2 d2 σǫrs Φr(λ) (σ)∂s YB (σ)YA (σ).

(44)

Besides these identities we have derived −cABC dC EF + 2cAC(E dC F )B = 4η A(E η F )B , 1 F E [AB]C (cC c + c[A|F |C cB]CE ) 4    1 1 1 1 1 EF CAB EF λAB =− + − fC f + fλ f fC AB f CEF , 2 ωA ωB ωC 2ωA ωB AC DEB A)B DEB D(C|E| AC BD dE (c − 2d ) − 2c dE = 4η η , CG DH I[EF B]A D[EF B]AC d Hd If f G = −f f .

(45)

(46) (47) (48)

They play very important roles in our computation in the following sections. Here we give their brief derivation. Using (40) and integrating by parts, we obtain c

ABC

dC

EF

= −2

Z

q

D r Y A (σ) B Y (σ)∂r (Y E (σ)Y F (σ)), ωA Z q D r Y A (σ) B − 2 d2 σ w(σ) Y (σ)∂r Y E (σ)Y F (σ). ωA

d2 σ w(σ)

cACE dC EF = 2η AB η BF

(49) (50)

The combination of these relations gives (45). In order to derive (46) we rewrite the first term in the l.h.s. of (46) by using (41): 1 F E ABC 1 Z 2 q d σ w(σ)D s Y A (σ)∂s Y E (σ)Y F (σ)Y B (σ) cC c = 4 ωA 1 1 EF λAB − f F E C f BAC − fλ f . ωA ωC ωA

(51)

The completeness relation (40) and integration by parts enable us to rewrite the second term in the l.h.s. of (46): 1 Z 2 q 1 AF BCE d σ w(σ)D s Y B (σ)∂s Y E (σ)Y F (σ)Y A (σ) c Cc = 4 ωA 1 Z 2 q − d σ w(σ)D r Y A (σ)∂r YF (σ)D s Y B (σ)∂s Y E (σ) . (52) ωA ωB

11

From (51) and (52) we obtain 1 F E ABC (cC c + cAF C cBCE ) 4   1 1 EF CAB EF λAB = − fC f + fλ f ωA ωC ! Z q D s Y A (σ) B D s Y B (σ) A E F 2 + d σ w(σ)∂s Y (σ)Y (σ) Y (σ) + Y (σ) ωA ωB 1 − H ABF E , (53) ωA ωB where H ABF E is defined by H

ABF E

=

Z

q

d2 σ w(σ)D r Y A (σ)∂r Y F (σ)D s Y B (σ)∂s Y E (σ).

(54)

When we antisymmetrize the indices A and B in (53), the second term in the r.h.s of (53) vanishes and the third term turns out to be 1 H [AB]F E = − fC AB f CEF . (55) 2 The derivation of this equation requires the identity δtr δus − δts δur = ǫrs ǫtu .

(56)

The combination of (53) and (55) gives (46). Next we derive (47). Eq.(40) and integration by parts give the following relation: dE AC (cDEB − 2dDEB ) − 2cDCE dE AB = 4η AC η BD  r  Z q D YD D r YD C − d2 σ w(σ) ∂r Y B Y A Y C + Y D Y B Y A Y C − Y ∂r (Y A Y B ) ωD ωD Z q r D D Y = 4η AC η BD − 2 d2 σ w(σ) Y B (∂r Y C Y A − ∂r Y A Y C ). (57) ωD From this relation we find the identity (47). Finally we prove the identity (48) . By using (40) and integrating by parts we obtain dCG H dDH I f IEF f BAC =

Z

q

d2 σ w(σ)Y C Y D {Y E , Y F }{Y B , Y A } − f DEF f BAC .

(58)

The first term in the r.h.s. vanishes by antisymmetrizing indices B, E and F .

B

Some Identities of SO(9) Clifford Algebra

In this section we review some properties of SO(9) gamma matrices γ a αβ (a = 1, . . . , 9 ; α, β = 1, . . . , 16). We can take γ a αβ as real and symmetric matrices, i.e. (γ a αβ )∗ = γ a αβ , γ a αβ = γ a βα . 12

(59)

From these gamma matrices we can construct an orthogonal complete basis of 16 × 16 real matrices (or bilinear products of real spinors) : n

Iαβ , γ a αβ , γ ab αβ , γ abc αβ , γ abcd αβ

where

o

,

γ a1 ···ak = γ [a1 γ a2 · · · γ ak ] .

(60) (61)

I, γ a and γ abcd are symmetric, and γ ab and γ abc are antisymmetric with respect to the spinorial indices. The SO(9) gamma matrices satisfy several identities, such as a b1 ···bk

γ γ



ab1 ···bk

+

k X

ˇ

(−)l−1 δ abl γ b1 ···bl ···bk ,

(62)

l=1

(γ b )αβ (γab )γδ + (γ b )γδ (γab )αβ + (γ b )αδ (γab )γβ + (γ b )γβ (γab )αδ − 2Iδβ (γa )γα + 2Iαγ (γa )βδ = 0 .

(63)

β γ ǫ θB θC] ) to (63) we derive By multiplying (γ a )δǫ (θ[A α (γd θ[A )α (θB γ d θC] ) = θ[A (θB θC] ) .

(64)

These identities of gamma matrices are useful in carrying out our calculation.

C

˜ −a, M ˜ −b )DB (M

In this section we show that (29) holds modulo the first class constraints ϕA and ϕλ . First we substitute (18) and (21) into the first and second terms in the r.h.s. of (27). The result is ˜ ab − i Q ˜ + γ ab Q ˜ + = C1 + C2 + C3 + C4 + C5 + C6 , − M2 M 4

(65)

where each CI (I = 1, . . . 6) is composed of the same type of polynomials in X, P and θ:

5

C1 = −P~A2 (−PDa X bD + PDb X aD ), (66) i i C2 = − PAa PBb (θA θB ) − {PAa (θA γ bd θB ) − PAb (θA γ ad θB )}PdB 2 2 i ~2 i i + PA (θB γ ab θB ) − P~A · P~B (θA γ ab θB ) − PdA PeB (θA γ abde θB ), (67) 4 4 4 i C3 = − fABC XBa XCb (θA γ · PD θD ) 2 i ABC a b a bD b f (XC PD − XCb PDa )(θA γ · XB θD ) + if ABC (XD P − XD P aD )(θA γ · XB θC ) + 2 5

Note that we have expanded the products of gamma matrices in the complete set (60) by using the identities such as (62).

13

− − − + C4 = C5 = + +

+ + + + + C6 =

i ABG a ~ G · P~H f {XA (θB γ b θH ) − XAb (θB γ a θH )}X 2 i ABC a f {XA (θB γ bde θD ) − XAb (θB γ ade θD )}XdC PeD 2 i i ABC a ~ G · P~H ) f {PD (θA γ bde θD ) − PDb (θA γ ade θD )}XdB XeC + f ABG (θA γ abd θH )XdB (X 4 2 i ABC f (θA γ abdef θD )XdB XeC Pf D , (68) 4 1 (fABC XdB XeC )2 (PDa X bD − PDb X aD ), (69) 2 i (θA γ ab θA )(fBCD XcB XdD )2 8 i B cdabef C D E F G (θ γ θ )Xc Xd Xe Xf fBDE fCF G 16 i B {θ (−δ ca γ dbef + δ cd γ daef − δ ce γ dabf + δ cf γ dabe + δ da γ cbef − δ db γ caef 16 +δ de γ cabf − δ df γ cabe − δ ae γ cdbf + δ be γ cdaf + δ af γ cdbe − δ bf γ cdae )θC } ×XcD XdE XeF XFG fBDE fCF G i B cd C (θ γ θ )(X aE X bD − X aD X bE )XcF XdG fBDE fCF G 8 i B cd C (θ γ θ )(X aF X bD − X bF X aD )XcE XdG fBDE fCF G 4 i B ab C ~ D ~ G ~ E ~ F (θ γ θ )(X · X )(X · X )fBDE fCF G 8 i ~E ·X ~ G )fBDE fCF G {(θB γ ae θC )X bD − (θB γ be θC )X aD }XeF (X 2 i B C ~D ·X ~ G )fBDE fCF G , (θ θ )(X aE X bF − X aF X bE )(X (70) 4 1 fABC (θA γ · X B θC )(θD γ ab θD ). (71) 4

˜ −a , M ˜ −b )DB . By using the definition of the generator M ˜ −a in (21) Next we calculate (M and the Dirac brackets (16), we find ˜ −a , M ˜ −b )DB = B1 + B2 + B3 + B4 + B5 + B6 , (M

(72)

with B1 , B2 , ..., B6 being given by: 1 DEB AC (c dE − 2cDCE dEAB − 2dACE dEBD )(XBa PDb − XBb PDa )P~A · P~C 4 1 AF BCE ~ E · P~F (P a P b − P b P a ), (c C c − cCF E cBAC )X + A B A B 4 i i = [− dAEC dCBF + {2cCEF (cABC − cBAC ) 4 16 ACE AEC +(c −c )(cBF C − cBCF )}]PAa PBb (θE θF ) i i + [ {−cABC dCEF + (cACE − cAEC )dCBF } − dAEC dBF C ]× 8 4

B1 =

B2

14

(73)

×{PAa (θE γ bd θF ) − PAb (θE γ ad θF )}PdB

+ − B3 =

+

+

+

+

+ − B4 =

+

B5 = +

+

+ +

i ABC EF (d dC − dAEC dCBF )P~A · P~B (θE γ ab θF ) 4 i AEC BF d dC PdA PeB (θE γ abde θF ), (74) 4 i − (2dADF dBEG f CF G + dACG dDBF f EF G − dACG dDGF f F BE ) × 4 a b ×(θB γ · PA θC )(XD XEb − XD XEa ) i {2dCF G dEBF f DAG + 4dDEF dBF G f ACG − dDBF f AF G (cECG − cEGC ) 8 −2dDBF f CF G cEGA − dGDF fFAB (cECG − cEGC ) − 2dGBF fFAC cEGD } × a b b ×(θB γ · XA θC )(XD P E − XD PEa ) i E CDF ABG (d d f + cF CA dDBG f EF G + dBDF dGCE f AF G − dGBF dGCE fFDA ) × 4 FG ~ A · P~C ×{XBa (θD γ b θE ) − XBb (θD γ a θE )}X i (−dEF G dCDF f ABG + dAF G dCEG f DBF − dFAD dGCE f F GB + dFAC dGDE f BGF ) × 4 ×{XAa (θD γ bed θE ) − XAb (θD γ ade θE )}PdC XeB i {(2dCDF − cCF D + cCDF )dEF G f ABG + 2cCF A dDEGf BF G } × 16 ×{PCa (θD γ bed θE ) − PCb (θD γ aed θE )}XdA XeB i F ED BCG A ~ D · P~E ) (c d f F G − 2dCF G dEBF f DAG )(θB γ abd θC )XdA (X 4 i E CDF ABG d d f (θD γ abdef θE )XdA XeB Pf C , (75) 4 FG 1 IC DE F G BHA (d f fC c + 4dAIC fIDE f FHC cBHG − 2dABC dCIH fIGD fHEF ) × 8 H I ~G · X ~ E )(X ~D · X ~F) ×(XAa PBb − XAb PBa )(X 1 AIC DE F H BG (d fI fC cH − 2dCAB dEIH fIDG fHF C ) × 2 ~D · X ~ F )(P~B · X ~ G ), ×(XAa XEb − XAb XEa )(X (76) i CDG EF I (θA γ acdbef θB )XcC XdD XeE Xf F dAGH dBH f I f 16 i {θA (−δ ae γ cdbf + δ af γ cdbe − δ cb γ adef + δ ce γ adbf − δ cf γ adbe + δ db γ acef 16 −δ de γ acbf + δ df γ acbe )θB }dAGH dIBH f CDG f EF I i {(θD γ ebcd θF )XIa − (θD γ eacd θF )XIb }XcG XdH XeE × 8 ×(dCDI dFAB f AGH f BCE + f DEC dICB dAF B f AGH ) i ~D · X ~ E )(X ~C · X ~ F )(dAGH dIBH f CDG f EF I + dGAB dGHI f ECH f DF I ) (θA γ ab θB )(X 8 i ~D · X ~ F )XeE × {(θA γ ae θB )XCb − (θA γ be θB )XCa }(X 4 15

+

+

+

B6 = − + − +

×(dAGH dIBH f CDG f EF I − dGAB dCHI f DGH f EF I F EG DAI +dBGH dIAC f F EG f DHI − dBGH dCH f ) I f i b (θB γ cd θC )XFa XD XEc XGd × 4 ×(fHDE dIBH dJCI f F GJ − fHEB dIF H dJDI f GCJ +fHEB dIF H fJGI dCDJ − dHBF fIEH dJDI f GCJ ) +dHBF fIEH fJGI dCDJ + fHEG dICH fJDI dBF J −fHEGdICH dJF I f DBJ ) i b a ~E · X ~F) × (θA θB )(XCa XD − XCb XD )(X 8 ×(dAGH dIBH f F DG f ECI − 2cGAB dCHI f DEH f F GI ) i ~B · X ~ C )(X a X b ) × (θA θD )(X E F 4 ×(−fGBA dHEG dIF H f CDI + fGBA dHEGfICH dDF I −dHAE fGBH dIF G f CDI + dHAE fIBH fGCI dDF G ), 1 ABC DE GH d fC dA (θG γ ab θH )(θB γ · XD θE ) 4 1 ABC DE GH a d fC dD {XA (θB γc θE )(θG γ bc θH ) − XAb (θB γc θE )(θG γ ac θH )} 8 1 ABC DE GH d d fA dD XE {(θB γ ac θC )(θG γ bdc θH ) − (θB γ bc θC )(θG γ adc θH )} 16 1 DE AC BGH d f c XcC (θD γ abc θE )(θG θH ) 8 A B 1 ABC DE GH a d fC cD {XA (θB γ b θE ) − XAb (θB γ a θE )}(θG θH ). 8

(77)

(78)

In order to see the cancellation between CI and BI (I = 1, ..., 6), it is convenient to rewrite BI by using the identities in the appendices A and B. Eq.(47) reduces B1 to B1 = P~A2 (XBa P bB − XBb P aB ) 1 AF BCE ~ E · P~F (P a P b − P b P a ). + (c C c + cCF E cABC )X A B A B 4

(79)

By using (42),(43) and (45) we rearrange B2 and B3 as i B2 = − [−2PAa PBb (θA θB ) − 2{PAa (θA γ bd θB ) − PAb (θA γ ad θB )}PdB 4 +P~A2 (θB γ ab θB ) − P~A · P~B (θA γ ab θB ) − PdA PeB (θA γ abde θB )] i EF ABC (c c + cAEC cBCF )(PAa PBb − PAb PBa )(θE θF ), (80) + 8 C i ABC B3 = f (θA γ · PD θD )(XBa XCb ) 2 i ABC a bD b f (θA γ · XB θD )(XCa PDb − XCb PDa ) − if ABC (θA γ · XB θC )(XD P − XD P aD ) − 2 16

+

+ + −



i ABG a ~ G · P~H f {XA (θB γ b θH ) − XAb (θB γ a θH )}X 2 i ~ G · P~H + f DGH {XAa (θA γ b θD ) − XAb (θA γ a θD )}X 2 i ~ G · P~H − dABC (cDCE f DGH + cλC E f λGH ){XAa (θB γ b θE ) − XAb (θB γ a θE )}X 4 i ABC a f {XA (θB γ bde θD ) − XAb (θB γ ade θD )}XdC PeD 2 i ABC a f {PD (θA γ bde θD ) − PDb (θA γ ade θD )}XdB XeC 4 i ABG ~ G · P~H ) + i f EGH (θD γ abd θD )XdE (X ~ G · P~H ) f (θA γ abd θH )XdB (X 2 2 i ~ G · P~H ) − dADE (cBAC f BGH + cλAC f λGH )(θD γ abd θE )XdC (X 4 i ABC f (θA γ abdef θD )(XdB XeC PfD ). 4

(81)

B4 is rewritten by using (42), (43) and (47): 1 a bD b (fABC XdB XeC )2 (XD P − XD P aD ) 2 1 AIC DE a b ~D · X ~ F (c F f HBG + c F f λBG )P~B · X ~G d fI (XA XE − XAb XEa )X + HC λC 2 ~D · X ~ F f F BG P~B · X ~ G. +f DAE (XAa XEb − XAb XEa )X

B4 =

(82)

We adopt (42), (43) and (48) to rewrite B5 as B5 = − − − − − −

+

i (θB γ abcdef θC )(XcD XdE XeF Xf G )f BDE f CF G 16

i {θB (−δ ae γ cdbf + δ af γ cdbe − δ cb γ adef + δ ce γ adbf − δ cf γ adbe 16 +δ db γ acef − δ de γ acbf + δ df γ acbe )θC }XcD XdE XeF Xf G f BDE f CF G i {(θD γ ebcd θF )XIa − (θD γ eacd θF )XIb }XcG XdH XeE f DEI f F GH 8 i ~D · X ~ G )(X ~E · X ~ F )f EDA f GF B − i (θA γ ab θA )(f CD XcC XdD )2 (θA γ ab θB )(X I 8 8 i ~D · X ~ F )XeE f CDA f EF B {(θA γ ae θB )XCb − (θA γ be θB )XCa }(X 2 i b a (θB γ cd θC )(XFa XD − XD XFb )XcE XdG f BDE f CF G 4 i a b + (θB γ cd θC )(XD XEb − XD XEa )(XcF XdG )f BDE f CF G 8 i ~E · X ~ F )X a X b (f BDE f CGF + 2f EDGf F BC ) (θB θC )(X D G 2 i ~D · X ~ F )(cHCF f HBG + cλC F f λBG )(θB θG ). (83) + dAKC fKDE (XAa XEb − XAb XEa )(X 4 17

Finally, rearrangement of B6 requires (42), (43) and (63). The result is 1 B6 = − fABC (θA γ · X B θC )(θD γ ab θD ) 4 1 DE d (θD γ abd θE )XdC {cBAC f BGH (θG θH ) + cλAC f λGH (θG θH )} − 8 A 1 + (θD γ abd θD )XdE f EGH (θG θH ) 4 1 ABC a d {XA (θB γ b θE ) − XAb (θB γ a θE )}{cDCE f DGH (θG θH ) + cλC E f λGH (θG θH )} − 8 1 a A b {X (θ γ θD ) − XAb (θA γ a θD )}f DGH (θG θH ). (84) + 4 A We are now in a position to verify the cancellation of (27) modulo the first class constraints (ϕA , ϕλ ). For this purpose we separate (27) into three parts, namely, C = (1) C ≡ (2) C ≡ (3) C ≡

C (1) + C (2) + C (3) , B3 + C3 + B6 + C6 , B1 + C1 + B2 + C2 , B4 + C4 + B5 + C5 .

After simple calculation we find i {X a (θA γ b θD ) − XAb (θA γ a θD )}ϕD C (1) = 2 A i − {XAa (θB γ b θE ) − XAb (θB γ a θE )}dABC (cDCE ϕD + cλC E ϕλ ) 4 i i (θD γ abd θD )XdE ϕE − (θD γ abd θE )XdC dADE (cBAC ϕB + cλAC ϕλ ), + 2 4 1 ~ E · P~F − i θE θF )(P a P b − P b P a ), (cAFC cBCE + cCF E cABC )(X C (2) = A B A B 4 2 1 ~D · X ~ F (c F ϕH + c F ϕλ ) C (3) = − dAIC fIDE (XAa XEb − XAb XEa )X HC λC 2 ~D · X ~ F ϕF . −f DAE (XAa XEb − XAb XEa )X

(85)

(86) (87)

(88)

We see that C (1) and C (3) are already written in the form of linear combinations of ϕA and ϕλ . In order to show that C (2) is also linear in the constraints, we have to apply (46). The result is 1 1 1 f AB ϕC }(PAa PBb − PAb PBa ). (89) C (2) = {− ( f CAB ϕC + f λAB ϕλ ) + ωA ωC 2ωA ωB C Thus we have proved that all the terms in (27) sum up to give (31).

D

˜ +, M ˜ −a )DB (Q

˜ +, M ˜ −a )DB = 0 holds modulo the first class In this section we show that the relation (Q constraints. By using relations (16), we can write down the result in the following way, ˜ +, M ˜ −a )DB = D1 + D2 + D3 + D4 , (Q (90) 18

where 1 1 (D1 )α = − (γ a θA )α P~B · P~C dABC − (γ · PA θB )α PCa (cCBA + cCAB ) 2 4 1 (91) + (γ · PA γ da θC )α PdB dABC , 2 1 ~B · X ~ C )(X ~D · X ~ E )dA F G f BDF f CEG (D2 )α = − (γ a θA )α (X 4 1 + (γ · XA γ · XB γ ade θC )α XdD XeE dC F G f ABG f EDF 8 1 ~D · X ~ E )(dC F G f ADF f BEG + dC F G f BDF f AEG) − (γ · XA θB )α XCa (X 2 1 − (γ · XA γ · XB γ · XC θD )α XEa (dE F G f ABF f CDG + dDE F f AB G f CF G ), (92) 4 1 a (D3 )α = (γ · PA γ · XB θC )α XD (dCD F f BAF ) 2 1 a AD d F f BCF − (γ · XB γ · PA θC )α XD 2 1 ~ C · P~D )cEDC f BA E + (γ ab θA )α XdB (X 2 i 1h − (γ · PA γ ade θB )α XdC XeD + (γ · XC γ · XD γ ab θB )α PdA dAB E f CDE 4 1 + (γ · XA γ · XB θC )α PDa 8 ×(−2cDEB f AC E + 2cDEA f BC E + cDEC f AB E − cDCE f AB E ), (93) i a (γ θA )α (θB γ · XC θD )dAB E f CDE (D4 )α = 2 i + (γ ad θB )α XdA (θC θD )cECD f AB E 4 i ih + −(γd θA )α (θB γ ade θC ) + (γ ed θA )α (θB γ a d θC ) XeD dBC E f DAE 4 i a BD + (γd θA )α (θB γ d θC )XD d E f ACE . (94) 2 After decomposing the products of gamma matrices into the complete basis (60), the identities (42) and (43) lead us to find that D1 and D2 vanish and ~ B · P~A )X a dCD F f BAF + 1 (γ ab θA )α XdB (X ~ C · P~D )(cE AB f ECD +cλ AB f λCD ). (95) (D3 )α = θαC (X D 2 The third term in the r.h.s of (94) is rewritten as i ih (γd )αβ (γ ed )γδ + (γ ed )αβ (γd )γδ θβA (θB γ a )γ θδC XeD dBC E f DAE . (96) 4 The identity (63) and similar calculations enable us to rewrite this term as i ih a BC (γd θC )α (θB γ d θA ) + 2(θC θA )θαB XD d E f DAE 6 i + [(γ a θC )α (θA γ e θB ) + (θB θA )(γ ea θC )α ] XeD dBC E f DAE . (97) 2 19

After combining with this term the other two terms in (94) and using (43), (D4 )α =

i − (γ ad θC )α XdD (θB θA )(cE CD f EBA + cλ CD f λBA ) 4 i a BD d E f CAE − [(θB θA )θαC + (θC θA )θαB ] XD 3 i ih a BD − (γd θ[A )α (θB γ d θC] ) XD d E f CAE . 2

(98)

By using (64) in the third term of the r.h.s. of the above equation, we finally obtain i i (D4 )α = − (γ ad θC )α XdD (θB θA )(cE CD f EBA + cλ CD f λBA ) − (θC θA )θαB dBD E f CAE . (99) 4 2 From (95) and (99) we conclude that ˜ +, M ˜ −a )DB = θαC X a dBD E ϕE + 1 (γ ad θC )α XdD (cE CD ϕE + cλ CD ϕλ ). (Q α D 2

E

(100)

cABC in terms of dABC

In this section we derive the equation (33) which describes cABC in terms of the invariant tensor dABC . Let us recall one of the relations in (42) cABC + cACB = −2dABC .

(101)

By using the definitions of cABC and dABC and performing integration by parts, we find another relation: Z

1 YA (YB ∆YC − YC ∆YB ) ωA ωB − ωC dABC . = 2 ωA

cABC − cACB = 2

d2 σ

(102)

Combining these two relations we can express cABC in terms of dABC : cABC = (

ωB − ωC − 1)dABC . ωA

(103)

This completes the proof of (33).

F

Supersymmetry algebra in M(atrix) theory

The authors of [5] observed some discrepancies in the supercharge algebra between dWHN model and M(atrix) theory. In this section, we would like to indicate that there are some missing terms in the M(atrix) computation and the alleged discrepancy can be removed. 20

We define the supercharge Q and the Dirac brackets between the canonical variables X, P , θ of M(atrix) theory in the same way as [5]. By using their defining relations, we obtain {Qα , Qβ }DB = 4RHδαβ



h



+2RTr −i{P a , [Xa , Xb ]} − i [Xb , θα ] , θα′ 



+2RTr X [a X b X c X d] (γabcd )αβ   3 −4iRTr [Xb , θγ ab θ] (γa )αβ 8   1 +4iRTr [X[a , (θγbcd] θ)] (γ abcd )αβ . 48 

i

(γ b )αβ

(104) 

This result coincides with that of dWHN [2] except that Tr X [a X b X c X d] term is automatically vanishing in the supermembrane calculation. The last two θ-bilinear terms in the r.h.s. of the above equation were absent in [5]. These two terms are originated from the second term in the r.h.s. of the following relation,  √  ′ ′ j [P a i j , Qα ]DB = −i R 2[Xd , θα ]i (γ ab )αα′ + Tr[{D(j i ) , θα } , Xd ](γ ad )αα′ ,

(105)

where D(j i ) is matrix valued quantity whose (k, p)-component is defined as p

D(j i )k = δk j δi p .

(106)

By keeping this term, we have recovered the θ-bilinear terms. Here we should make a remark. The recoverd θ-bilinear terms cannot be observed even if we consider topologically nontrivial configurations such as winding sectors of {X a }. This is because these terms are originated from the second term of (105) which is expected to vanish in the configurations with a well-defined supercharge Qα . In this respect, it is proper to say that there is no discrepancy between the result of dWHN [2] and that of [5], at least practically.

References [1] T. Banks, W. Fischler, S.H. Shenker and L. Susskind, Phys. Rev. D55 (1997), 5112-5128, “M theory as a Matrix Model: A Conjecture”, hep-th/9610043. [2] B. de Wit, J. Hoppe and H. Nicolai, Nucl. Phys. B 305 [FS23] (1988), 545-581. [3] J. Polchinski and P. Pouliot, “Membrane Scattering with M-Momentum Transfer”, hepth/9704029. [4] B. de Wit, U. Marquard and H. Nicolai, Commun. Math. Phys. 128 (1990), 39-62. [5] T. Banks, N. Seiberg and S. Shenker, “Branes from Matrices”, hep-th/9612157.

21

[6] J. Hoppe, MIT Ph.D. Thesis (1982), Elem. Part. Res. J. 80 (Kyoto, 1989), 145; S. Klimek and A. Lesniewski, Commun. Math. Phys. 146 (1992), 103-122; Lett. Math. Phys. 24 (1992), 125; M. Bordemann, E. Meinrenken and M. Schlichenmaier, Commun. Math. Phys. 165 (1994), 281-296. [7] S. Melosch, Diploma Thesis (1990), unpublished.

22