Open and Closed Supermembranes with Winding Bernard De Wita, Kasper Peetersb and Jan C

THU-97-44 NIKHEF 97-045 hep-th/9710215 1

Open and Closed Supermembranes with Winding Bernard de Wita, Kasper Peetersb and Jan C. Plefkab aInstitute for Theoretical Physics, Utrecht University Princetonplein 5, 3508 TA Utrecht, The Netherlands bNIKHEF, P.O. Box 41882, 1009 DB Amsterdam, The Netherlands

Motivated by manifest Lorentz symmetry and a well-deﬁned large-N limit prescription, we study the supersymmetric quantum mechanics proposed as a model for the collective dynamics of D0-branes from the point of view of the 11-dimensional supermembrane. We argue that the continuity of the spectrum persists irrespective of the presence of winding around compact target-space directions and discuss the central charges in the superalgebra arising from winding membrane conﬁgurations. Along the way we comment on the structure of open supermembranes.

M-theory is defined as the strong-coupling limit More recently it was shown that the same of type-IIA string theory and is supposed to cap- quantum-mechanical matrix models based on ture all the relevant degrees of freedom of all U(N) describe the short-distance dynamics of N known string theories, both at the perturbative D0-branes [10,11]. Subsequently there has been a and the nonperturbative level [1–4]. In this de- large number of studies of these models for finite scription the various string-string dualities play a N [12,13] and some of them have been reported at central role. At large distances M-theory is de- this conference. These studies were further mo- scribed by 11-dimensional supergravity. In [5] it tivated by a conjecture according to which the was shown that elementary supermembranes can degrees of freedom captured in M-theory, are in live in a superspace background that is a solution fact described by the U(N) super-matrix mod- of the source-free supergravity field equations in els in the N →∞limit [14]. A further conjec- 11 dimensions. In the light-cone gauge (in a flat ture, also discussed at this conference, is that the target space) it was subsequently shown [6] that finite-N matrix model coincides with M-theory the supermembrane theory takes the form of a su- compactified on a light-like circle [15]. persymmetric quantum-mechanical model, which So it turns out that M-theory, supermembranes coincides with the zero-volume reduction of su- and super-matrix models are intricately related. persymmetric Yang-Mills theory based on 16 su- A direct relation between supermembranes and percharges. For the supermembrane the underly- type-IIA theory was emphasized in particular in ing gauge group is the group of area-preserving [1], based on the relation between d =10ex- diffeomorphisms of the two-dimensional mem- tremal black holes in 10-dimensional supergravity brane surface. This group can be described by and the Kaluza-Klein states of 11-dimensional su- the N →∞limit of SU(N) (the role of the mem- pergravity. From the string point of view these brane topology is subtle, as we will discuss in due states carry Ramond-Ramond charges, just as the course). For the finite groups the phase-space D0-branes [16]. Strings can arise from membranes variables take the form of matrices, associated by a so-called double-dimensional reduction [17]. with the Lie algebra of some group (in this case Similarly supermembranes were employed to pro- U(N)orSU(N)). For this reason, these models vide evidence for the duality of M-theory on 10 are commonly referred to as matrix models. It has R × S1/Z2 and 10-dimensional E8 × E8 het- been discussed that the possible massless ground erotic strings [3]. states of the supermembrane coincide with the Here we choose the supermembrane perspec- physical states of 11-dimensional supergravity1. 9]. According to [9] such states do indeed exist in eleven 1For discussions on the existence of massless states, see [6– dimensions. 2 tive, motivated by its manifest Lorentz invari- this can be generalized to a hypertorus. There- ance and well-defined large-N limit (but not nec- fore models based on (subgroups of) U(∞)can essarily committing ourselves to the view that M- describe an enormous variety of models. theory is a theory of fundamental membranes). Using the SU(N) regularisation it was shown Let us nevertheless first consider the supersym- in [19] that the spectrum of the supermembrane metric matrix models, whose Hamiltonian equals is continuous, with no mass gap. This result 1 is expected when the supermembrane Hamilto- H = Tr 1 P2 + 1 [X a ,Xb]2 +gθTγ [Xa,θ] .(1) nian is viewed as the Hamiltonian for the col- g 2 4 a h i lective dynamics of arbitrarily large numbers of Here, X, P and θ take values in the Lie algebra D-particles. From the supermembrane point of of the gauge group. From the supermembrane view, these instabilities arise because arbitrar- point of view they are vectors and spinors of the ily long, string-like (zero-area) configurations can ‘transverse’ SO(9) rotation group. This Hamilto- form. There are thus no asymptotic membrane nian can alternatively be interpreted as the zero- states, but rather multimembrane configurations volume limit of supersymmetric Yang-Mills the- connected by these infinitely thin strings. The ory with some arbitrary gauge group. For the massless ground states, which correspond to the supermembrane one must choose the (infinite- states of 11-dimensional supergravity, thus ap- dimensional) group of area-preserving diffeomor- pear in a continuum of multimembrane states. phisms (which also plays an important role in The connection with M-theory and finite-N selfdual gravity and N = 2 strings) and put g matrix models is made by compactification of + equal to the light-cone momentum P0 .Thema- the 11-dimensional action on a circle or higher- trix model has 16 supercharges, but additional dimensional tori. However, this compactification charges can be obtained by splitting off an abelian has only recently been studied [20] from the point factor of the U(N) gauge group, of view of the supermembrane. In this talk we discuss the supermembrane with winding, paying + a a b Q =Tr (2P γa +[X ,X ]γab)θ , attention to the extension of the supersymmetric gauge theory, the Lorentz invariance and the Q− = g Trh [θ] . i (2) effect of winding on the mass spectrum. Along For the supermembrane the second charge is as- the way we shall simultaneously develop the the- sociated with the center-of-mass superalgebra (we ory of open supermembranes, which has recently return to the membrane supersymmetry algebra received some attention [21–23]. shortly). The actions of fundamental supermembranes The form of the area-preserving diffeomor- are of the Green-Schwarz type [5]. In flat target phisms that remain as an invariance of the model, space, they read depends in general on the topology of the membrane surface. For spherical and toroidal topolo- L = −g(X, θ) gies, it has been shown that the algebra can be ap- ijk 1 µ ν ¯ ν p− [ 2 ∂iX (∂j X + θγ ∂jθ) proximated by SU(N)inthelarge-Nlimit. This 1 µ ν + θγ¯ ∂iθ θγ¯ ∂jθ] θγ¯ µν ∂kθ, (3) limit is subtle. However, once one assumes an in- 6 finitesimal gauge group, one can describe a large where Xµ denote the 11-dimensional target-space variety of different theories. For instance, the embedding coordinates lying in T d × R1,10−d and gauge group [U(N)]M , which is a subgroup of thus permitting us to have winding on the d- U(N ·M), leads in the limit M →∞to the possi- dimensional torus T d.Moreoverwehavethe bility of describing the collective dynamics of D0- fermionic variables θ, which are 32-component branes on a circle by supersymmetric Yang-Mills Majorana spinors and g =detgij with the in- theories in 1 + 1 dimensions [18]. Hence, it is duced metric possible to extract extra dimensions from a suit- µ µ ν ν able infinite-dimensional gauge group. Obviously gij =(∂iX +θγ¯ ∂iθ)(∂jX +θγ¯ ∂jθ)ηµν . (4) 3

Next to supersymmetry the action (3) exhibits that one space has an excess of fermionic and the an additional local fermionic symmetry called κ- other an excess of bosonic coordinates. More- symmetry. In the case of the open supermem- over, we note that supersymmetry may be fur- brane, κ-symmetry imposes boundary conditions ther broken by e.g. choosing different Dirichlet on the fields. They must ensure that the follow- conditions on non-connected segments of the su- ing integral over the boundary of the membrane permembrane boundary. world volume vanishes, The Dirichlet boundary conditions can be sup- plemented by the following Neumann boundary 1 µ ν ¯ ν ¯ 2dX ∧ (dX + θγ dθ) θγµν δκθ conditions, Z∂M m h 1 µ 1 ¯ µ ¯ ¯ ν ∂⊥ X =0 m=0,1,...,p, + 2 (dX − 3 θγ dθ) ∧ θγµν dθ θγ δκθ P+∂⊥θ =0. (8) 1 µ ν + θγ¯ dθ ∧ θγ¯ dθ θγ¯ µν δκθ . (5) 6 These do not lead to a further breakdown of the i rigid spacetime symmetries. This can be achieved by having a “membrane D- We now continue and follow the light-cone p-brane” at the boundary with p =1,5, or 9, quantization described in [6]. The supermem- whichisdefinedintermsof(p+ 1) Neumann and µ brane Hamiltonian takes the form (10−p) Dirichlet boundary conditions for the X , a 1 √ P Pa together with corresponding boundary conditions H = d2σ w + 1 { Xa,Xb}2 2 P + 2 w 4 on the fermionic coordinates . More explicitly, we 0 Z define projection operators + ¯ a −P0 θγ−γa{X ,θ} . (9) P = 1 1 ± γp+1 γp+2 ···γ10 , (6) ± 2 Here the integral runs over the spatial compo- 1 2 and impose the Dirichlet boundary conditions nents of the worldvolume denoted by σ and σ , while P a(σ)(a=2,...,10) are the momentum M a ∂k X =0,M=p+1,...,10 , conjugates to the transverse X . In this gauge√ the light-cone coordinate X+ =(X1+X0)/ 2is P−θ =0, (7) linearly related to the world-volume time. The + where ∂ ⊥ and ∂k define the world-volume deriva- momentum P is time independent and pro- + tives perpendicular or tangential to the surface portional to the center-of-mass value P0 times swept out by the membrane boundary in the tar- some density w(σ) of the spacesheet, whose get space. Note that the fermionic boundary con- spacesheet integral is normalized to unity. The p − dition implies that P−∂kθ = 0. Furthermore, it center-of-mass momentum P0 is equal to minus implies that spacetime supersymmetry is reduced the Hamiltonian (9) subject to the gauge condi- to only 16 supercharges associated with spinor tion γ+ θ = 0. And finally we made use of the parameters P+,whichischiral with respect to Poisson bracket {A, B} defined by the (p + 1)-dimensional world volume of the D- 1 rs p-brane at the boundary. With respect to this {A(σ),B(σ)} = ∂rA(σ)∂sB(σ). (10) w(σ) reduced supersymmetry, the superspace coordi- √ nates decompose into two parts, one correspond- Note that the coordinatep X− =(X1−X0)/ 2 M ingto(X ,P−θ) and the other corresponding itself does not appear in the Hamiltonian (9). It m to (X , P+θ)wherem=0,1,...,p. While for is defined via the five-brane these superspaces exhibit a some- P · ∂rX P + ∂ X− = − √ − P + θγ¯ ∂ θ, (11) what balanced decomposition in terms of an equal 0 r w 0 − r number of bosonic and fermionic coordinates, the and implies a number of constraints that will be situation for p =1,9 shows heterotic features in important in the following. First of all, the right- 2Here our conclusions concur with those of [23] but not hand side must be closed. If there is no winding with those of [22]. in X−, it must moreover be exact. 4

The equivalence of the large-N limit of SU(N) For open membranes the boundary conditions on quantum mechanics with the closed supermem- the fields (7) lead to an SO(N) group structure, brane model is based on the residual invariance as we shall see in the sequel. of the supermembrane action in the light-cone The presence of the closed but non-exact forms gauge. It is given by the area-preserving diffeo- is crucial for the winding of the embedding coor- morphisms of the membrane surface. These are dinates. More precisely, while the momenta P(σ) defined by transformations of the worldsheet co- and the fermionic coordinates θ(σ) remain sin- ordinates gle valued on the spacesheet, the embedding co-

r r r ordinates, written as one-forms with components σ → σ + ξ (σ) , (12) − ∂rX(σ)and∂rX (σ), are decomposed into closed with forms. Their non-exact contributions are multi- plied by an integer times the length of the com- r ∂r( w(σ) ξ (σ))=0. (13) pact direction. The constraint alluded to above amounts to the condition that the right-hand side p We wish to rewrite this condition in terms of dual of (11) is closed. spacesheet vectors by Under the full group of area-preserving diffeo- r rs morphisms the fields Xa, X− and θ transform w(σ) ξ (σ)= Fs(σ). (14) according to Inp the language of differential forms the condition rs rs a a − − (12) may then be simply recast as dF =0.The δX = √ ξr ∂sX ,δX=√ ξr ∂sX , w w trivial solutions are the exact forms F =dξ,orin rs components δθa = √ ξ ∂ θ, (18) w r s Fs = ∂sξ(σ), (15) where the time-dependent reparametrization ξr for any globally defined function ξ(σ). The non- consists of closed exact and non-exact parts. Ac- trivial solutions are the closed forms which are not cordingly there is a gauge field ωr, which is there- exact. On a Riemann surface of genus g there are fore closed as well, transforming as precisely 2g linearly independent non-exact closed st δω = ∂ ξ + ∂ √ ξ ω , (19) forms, whose integrals along the homology cycles r 0 r r w s t are normalized to unity3.Incomponentswewrite and corresponding covariant derivatives Fs=φ(λ)s,λ=1,...,2g. (16) rs a a a D0X = ∂0X − √ ωr ∂sX , The commutator of two infinitesimal area- w preserving diffeomorphisms is determined by the rs D θ = ∂ θ − √ ω ∂ θ, (20) product rule 0 0 w r s st and similarly for D X−. ξ(3) = ∂ √ ξ(2)ξ(1) , (17) 0 r r w s t The action corresponding to the following La- grangian density is then gauge invariant under (1,2) (3) where both ξr are closed vectors. Because ξr the transformations (18) and (19), is exact, the exact vectors generate an invariant + √ 1 2 ¯ subgroup of the area-preserving diffeomorphisms, L = P0 w 2 (D0X) + θγ−D0θ which can be approximated by SU(N)inthe h−1 (P+)−2{Xa,Xb}2 (21) large-N limit in the case of closed membranes. 4 0 +(P +)−1 θγ¯ γ Xa,θ +D X− , 3In the mathematical literature the globally defined exact 0 − a{ } 0 forms are called “hamiltonian vector fields”, whereas the i closed but not exact forms which are not globally defined where we draw attention to the last term propor- go under the name “locally hamiltonian vector fields”. tional to X−, which can be dropped in the ab- 5 sence of winding and did not appear in [6]. More- Hamiltonian (9). Observe that in the above gauge over, we note that for the open supermembranes, theoretical construction the space-sheet metric √ (21) is invariant under the transformations (18) wrs enters only through its density w and hence and (19) only if ξk = 0 holds on the boundary. vanishing or singular metric components do not This condition defines a subgroup of the group of pose problems. area-preserving transformations, which is consis- We are now in a position to study the full 11- tent with the Dirichlet conditions (7). Observe dimensional supersymmetry algebra of the wind- that ∂k and ∂⊥ will now refer to the spacesheet ing supermembrane. For this we decompose derivatives tangential and perpendicular to the the supersymmetry charge Q associated with membrane boundary4. the transformations (22), into two 16-component The action corresponding to (21) is also invari- spinors, ant under the supersymmetry transformations + − ± 1 Q = Q + Q , where Q = 2 γ± γ∓ Q, (23) δXa = −2¯γa θ, 1 a to obtain δθ = 2 γ+ (D0X γa + γ−) 1 + −1 a b + 2 a √ a b +4 (P0 ) {X ,X }γ+ γab , Q = d σ 2 P γa + w { X ,X }γab θ, + −1 Z δωr = −2(P0 ) ∂¯ rθ. (22) √ Q− =2P+ d2σ wγ θ. (24) The supersymmetry variation of X− is not rele- 0 − Z vant and may be set to zero. For the open case The canonical Dirac brackets are derived by the one finds that the boundary conditions ω =0 k standard methods and read and = P+ must be fulfilled in order for (22) to be a symmetry of the action. In that case the the- a b 0 ab 2 0 ( X (σ),P (σ ))DB = δ δ (σ − σ ) , ory takes the form of a gauge theory coupled to ( θ (σ), θ¯ (σ0)) = matter. The pure gauge theory is associated with α β DB 1 + −1 −1/2 2 0 the Dirichlet and the matter with the Neumann 4 (P0 ) w (γ+)αβ δ (σ − σ ) . (25) (bosonic and fermionic) coordinates. In the presence of winding the results given in [6] In the case of a ‘membrane D-9-brane’ one now yield the supersymmetry algebra sees that the degrees of freedom on the ‘end-of- the world’ 9-brane precisely match those of 10- + ¯+ ( Qα , Qβ )DB =2(γ+)αβ H dimensional heterotic strings. On the bound- 2 √ a − ary we are left with eight propagating bosons −2(γa γ+)αβ d σ w { X ,X }, m 10 X (with m =2,...,9), as X is constant Z + ¯− a on the boundary due to (7), paired with the 8- (Qα,Qβ )DB = −(γa γ+ γ−)αβ P0 dimensional chiral spinors θ (subject to γ+θ = 1 2 √ a b − (γab γ+γ−)αβ d σ w { X ,X }, P−θ = 0), i.e., the scenario of Ho˘rava-Witten [3]. 2 Z The full equivalence with the membrane Hamil- − ¯− + (Qα,Qβ )DB = −2(γ−)αβ P0 , (26) tonian is now established by choosing the ωr =0 gauge and passing to the Hamiltonian formalism. where use has been made of the defining equation The field equations for ωr then lead to the mem- (11) for X−. brane constraint (11) (up to exact contributions), The new feature of this supersymmetry alge- partially defining X−. Moreover the Hamiltonian bra is the emergence of the central charges in the corresponding to the gauge theory Lagrangian of first two anticommutators, which are generated (21) is nothing but the light-cone supermembrane through the winding contributions. They repre- 4Consistency of the Neumann boundary conditions (8) sent topological quantities obtained by integrat- with the area-preserving diffeomorphisms (18) further im- ing the winding densities k poses ∂⊥ξ = 0 on the boundary, where indices are raised a rs a − accordingto(14). z (σ)= ∂rX ∂sX (27) 6 and 2 rs fλBC = d σ φ(λ) r ∂sYB YC , zab(σ)=rs ∂ Xa ∂ Xb (28) Z r s 2 rs fλλ0C = d σ φ(λ) r φ(λ0) s YC . (30) over the space-sheet. It is gratifying to observe Z the manifest Lorentz invariance of (26). Here we Note that with Y0 =1,wehavefAB0 = fλB0 =0. should point out that, in adopting the light-cone The raising and lowering of the A indices is per- gauge, we assumed that there was no winding for formed with the invariant metric the coordinate X+. In [28] the corresponding al- √ η = d2σ wY (σ)Y (σ) (31) gebra for the matrix regularization was studied. AB A B Z The result obtained in [28] coincides with ours in and there is no need to introduce a metric for the the large-N limit, in which an additional longi- λ indices. tudinal five-brane charge vanishes, provided that By plugging the mode expansions (29) into the one identifies the longitudinal two-brane charge Hamiltonian (9) one obtains the decomposition with the central charge in the first line of (26). 1 This requires the definition of X− in the ma- H = 2 P0 · P0 trix regularization, a topic that was dealt with 1 0 aλ bλ0 λ00 λ000 + 4 fλλ0 fλ00λ000 0 X X Xa Xb in [24]. We observe that the discrepancy noted + 1 PA P f θ¯C γ γ θB XaA in [28] between the matrix calculation and cer- 2 · A − ABC − a ¯C B aλ tain surface terms derived in [6], seems to have −fλBC θ γ− γa θ X no consequences for the supersymmetry algebra. 1 E aA bB C D +4 fAB fCDE X X Xa Xb A possible reason for this could be that certain +f E f XaλXbB XC XD Schwinger terms have not been treated correctly λB CDE a b 1 E aλ bB λ0 D in the matrix computation, as was claimed in a +2 fλB fλ0DE X X Xa Xb recent paper [25]. 1 E aλ bB C λ0 + fλB fCλ0E X X X X The form of the algebra is another indica- 2 a b 1 E aλ bλ0 C D tion of the consistency of the supermembrane- +2 fλλ0 fCDE X X Xa Xb supergravity system. E aλ bλ0 λ00 D +fλλ0 fλ00DE X X Xa Xb In order to define a matrix approximation one 1 E aλ bλ0 λ00 λ000 + f 0 f 00 000 X X X X , (32) introduces a complete orthonormal basis of func- 4 λλ λ λ E a b tions YA(σ) for the globally defined ξ(σ) of (15). where here and henceforth we spell out the zero- One may then write down the following mode ex- mode dependence explicitly, i.e. the range of val- pansions for the phase space variables of the su- ues for A no longer includes A = 0. Note that for permembrane, the toroidal supermembrane fλλ0A =0andthus the last three terms in (32) vanish. The second λ A ∂rX(σ)=X φ(λ)r+ X ∂rYA(σ), term in the first line represents the winding num- A X ber squared. In the matrix formulation, the wind- √ A P(σ)= wP Ya(σ), ing number takes the form of a trace over a com- A mutator. We have scaled the Hamiltonian by a X + θ(σ)= θAY (σ), (29) factor of P0 and the fermionic variables by a fac- A + −1/2 tor (P0 ) . Supercharges will be rescaled as A + X well, such as to eliminate explicit factors of P0 . introducing winding modes for the transverse Xa. The constraint equation (11) is translated into − rs A similar expansion exists for X . One then nat- mode language by contracting it with φ(λ) s rs urally introduces the structure constants of the and ∂sYC respectively and integrating the re- group of area-preserving diffeomorphism by [24] sult over the spacesheet to obtain the two constraints 2 rs fABC = d σ ∂rYA ∂sYB YC , λ0 − λ0 + ϕλ = fλλ00 ( X · P0 + X P0 ) Z 7

λ0 C +fλλ0C X · P defining a supersymmetric quantum-mechanical B C ¯C B model. +fλBC ( X · P + θ γ− θ )=0, At this stage it would be desirable to present ϕ =f ( XB · PC + θ¯C γ θB ) A ABC − a matrix model regularization of the superme- λ C +fAλC X · P =0, (33) mbrane with winding contributions, generalizing taking also possible winding in the X− direction the matrix approximation to the exact subgroup into account. Note that even for the non-winding of area-preserving diffeomorphisms [26,6], at least aλ for toroidal geometries. However, this program case X = 0 there remain the extra ϕλ constraints. These have so far not played any role seems to fail due to the fact that the finite-N ap- in the matrix formulation. proximation to the structure constants fλBC vi- The zero-mode contributions completely de- olates the Jacobi identity, as was already noticed couple in the Hamiltonian and the supercharges. in [24]. We thus perform a split in Q+ treating zero Let us nevertheless discuss an example of this modes and fluctuations separately to obtain the regularization in the case of non-winding, open mode expansions, supermembranes in some detail. Consider a spacesheet topology of an annulus. Its set of ba- − 0 + + + Q =2γ−θ ,Q=Q(0) + Q , (34) sis functions is easily obtained by starting from the well-known torus functions Y (σ) [6] labeled where m b by a two-dimensional vector m =(m1,m2)with + a aλ bλ0 Q(0) = 2 P0 γa + fλλ00 X X γab θ0 , m1,m2 integer numbers and

i(m1σ1+m2σ2) + a aA bB Ym(σ1,σ2)=e . (42) Q = 2 PC γa + fABC X X γab aλ bB Consider now the involution m → m where +2 fλBC X X γab b m =(m , m ). One then deﬁnes the new basis aλ bλ0 C 1 − 2 +f 0 X X γ θ . (35) λλ C ab functions [27,23] e Upon introducing the supermembrane mass op- e± Cm = Ym ± Ym, (43) erator by where σ1 ∈ [0, 2π]andσ2 ∈[0,π]. It turns out 2 1 aλ bλ0 2 =2 P P (f 0 X X ) , (36) + − M H− 0 · 0 − 2 λλ 0 that Cm obeyse Neumann and Cm Dirichlet condi- + − the supersymmetry algebra (26) then takes the tions on the boundaries, i.e., ∂2Cm| = ∂1Cm| =0. form These basis functions possess the algebra − − − − + ¯ + 2 {Cm,Cn }=i(m×n)Cm+n −i(m×n)C , {Qα , Qβ } =(γ+)αβ M (37) m+n + + − − aλ −λ0 + λ0 {C ,C }=i(m×n)C +i(m×n)C , −2(γ γ ) f 0 X (X P +X ·P ), m n m+n b b a + αβ λλ 0 0 0 e m+n + ¯+ + − + + e {Q , Q } = (38) {Cm,Cn }=i(m×n)Cm+n −i(m×n)C . (0) α (0) β e m+n 1 aλ bλ0 2 e(44) (γ ) P · P + (f 0 X X ) + αβ 0 0 2 λλ 0 e aλ λ0 Note that the C− form a closed subalgebra.e +2 (γa γ+)αβ fλλ00 X X ·P0, m The matrix regularization now comes about by + ¯− a {Q , Q } = −(γa γ+ γ−)αβ P0 (39) 2 (0) α β replacing Ym with the (N − 1) adjoint SU(N) 0 1 aλ bλ matrices Tm of [6]. The corresponding operation − 2 (γab γ+ γ−)αβ fλλ00 X X , to m → m is matrix transposition, i.e. T = T T. + ¯− + ¯ + m m {Qα,Qβ }={Q(0) α , Qβ } =0. (40) Hence we ﬁnd that, in the matrix picture, the − And the mass operator commutes with all the su- antisymmetricb (N(N − 1)/2) Cm matrices form b b e + persymmetry charges, the adjoint and the symmetric (N(N +1)/2) Cm transform as the symmetric rank-two representa- + 2 + 2 − 2 [ Q , M ]=[Q(0), M ]=[Q ,M ]=0, (41) tion of SO(N).

b 8

Finally we turn to the question of the mass indicates that the energy spectrum must be con- spectrum for membrane states with winding. The tinuous. mass spectrum of the supermembrane without Without winding it is clear that the valley con- winding is continuous. This was proven in the figurations are supersymmetric, so that one con- SU(N) regularization [19]. Whether or not non- cludes that the spectrum is continuous. With trivial zero-mass states exist, is not known (for winding the latter aspect is somewhat more sub- some discussion on these questions, we refer the tle. However, we note that, when the wind- reader to [7]). Those would coincide with the ing density is concentrated in one part of the states of 11-dimensional supergravity. It is often spacesheet, then valleys can emerge elsewhere argued that the winding may remove the conti- corresponding to stringlike configurations with nuity of the spectrum (see, for instance, [29]). supersymmetry. Hence, as a space-sheet local There is no question that winding may increase field theory, supersymmetry can be broken in one the energy of the membrane states. A membrane region where the winding is concentrated and un- winding around more than one compact dimen- broken in another. In the latter region stringlike sion gives rise to a nonzero central charge in the configurations can form, which, at least semiclas- supersymmetry algebra. This central charge sets sically, will not be suppressed by quantum cor- a lower limit on the membrane mass. However, rections. this should not be interpreted as an indication We must stress that we are describing only that the spectrum becomes discrete. The possible the generic features of the spectrum. Our ar- continuity of the spectrum hinges on two features. guments by no means preclude the existence of First the system should possess continuous valleys mass gaps. To prove or disprove the existence of of classically degenerate states. Qualitatively one discrete states is extremely difficult. While the recognizes immediately that this feature is not di- contribution of the bosonic part of the Hamilto- rectly affected by the winding. A classical mem- nian increases by concentrating the winding den- brane with winding can still have stringlike con- sity on part of the spacesheet, the matrix ele- figurations of arbitrary length, without increasing ments in the fermionic directions will also grow its area. Hence the classical instability still per- large, making it difficult to estimate the eigen- sists. values. At this moment the only rigorous result The second feature is supersymmetry. Gener- is the BPS bound that follows from the super- ically the classical valley structure is lifted by symmetry algebra. Obviously, the state of lowest quantum-mechanical corrections, so that the mass for given winding numbers, is always a BPS wave function cannot escape to infinity. This state, which is invariant under some residual su- phenomenon can be understood on the basis of persymmetry. the uncertainty principle. Because, at large distances, the valleys become increasingly narrow, REFERENCES the wave function will be squeezed more and more which tends to induce an increasing spread in its 1. P.K. Townsend, Phys. Lett. B350 (1995) 184, momentum. This results in an increase of the hep-th/9501068, Phys. Lett. B373 (1996) 68, kinetic energy. Another way to understand this hep-th/9512062. is by noting that the transverse oscillations per- 2. E. Witten, Nucl. Phys. B443 (1995) 85, hep- pendicular to the valleys give rise to a zero-point th/9503124. energy, which acts as an effective potential barrier 3. P. Ho˘rava and E. Witten, Nucl. Phys. B460 that confines the wave function. When the valley (1996) 506, hep-th/9510209, Nucl. Phys. configurations are supersymmetric the contribu- B475 (96) 94, hep-th/9603142. tions from the bosonic and the fermionic trans- 4. P.K. Townsend, Four Lectures on M-theory, verse oscillations cancel each other, so that the lectures given at the 1996 ICTP Summer wave function will not be confined and can extend School in High Energy Physics and Cosmol- arbitrarily far into the valley. This phenomenon ogy, Trieste, hep-th/9612121. 9

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