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On the anti-de Sitter quantum (super) strings
Volume 261, number 1,2
PHYSICSLETTERSB
23 May 1991
On the anti-de Sitter quantum (super) strings E.S. Fradkin and V.Ya. Linetsky Theoretical Department, P.N. Lebedev Physical Institute, Leninsky Prospect 53, SU-I 17 924 Moscow, USSR
Received 8 March 1991
Closed bosonic and NSR quantum strings in AdSDare considered as exactly solvablemodels. There may exist anomaly-free AdSDstrings in D # 26 (D:# 10) with an appropriate critical cosmologicalconstant fixed by the anomaly-cancellationcondition. All anomaly-freepairs (D, A) are exactlycalculated. Covariant AdS4and AdS6GS-typesuperstringsare discussed.
1. Conventional (super)string theory [ 1 ] may be regarded as an interacting theory of infinitely many particles (string excitations with increasingly high spins, their masses growing with their spins. Interactions are non-local and non-analytical in the string mass parameter, the string tension T = 1 / 2 n a ' . This non-analyticity does not allow one to pass to the massless limit T - , 0 ( a ' ~ o o ) in interacting string theory. If such a limit existed, then all the massive higher spin particles might become massless and an enormously rich infinite-dimensional higher spin gauge symmetry might restore. That symmetry might play a role of the fundamental hidden gauge symmetry behind strings. However such a massless limit T ~ 0 does not exist in conventional string theory as a continuous limiting process. The formal obstruction is a known uncompatibility of the gauge symmetry of massless higher spin particles and gravity. In recent years it was discovered that there nevertheless may exist a consistent interaction among massless higher spins and gravity, but the latter must be either anti-de Sitter gravity with a non-zero negative cosmological constant [ 2-5 ], or Weyl conformal gravity [ 6-8 ]. It is based on the discovery of new classes of non-abelian infinite-dimensional gauge symmetries generalizing local supersymmetries. These arc the anti-de Sitter higher spin symmetry [ 2,4 ] and the superconformal higher spin symmetry [6,8] which generalize AdS supersymmetry and conformal supersymmetry to the case of all higher spins (they mix particles with all spins from zero to infinity). The fundamental property of the gravitational in26
teraction of massless higher spin particles discovered in ref. [ 3 ] is its non-analyticity in the cosmological constant A, which does not permit one to pass to the flat limit A ~ 0 in the interacting theory. These two-analyticities, in T in strings and in A in the AdS higher spin gauge theory, suggested us to conjecture [ 8 ] a deep interrelation between these two theories. Moreover, wc have conjectured [8] that there might bc some sort of phase transition in string theory at high energies when the cosmological constant of the Planck order is induced and an infinitedimensional AdS higher spin gauge symmetry is restored. Conformal higher spin theory then has been regarded as a third ultra-high energy asymptotic phase when all the mass parameters become unessential, all higher spin excitations become equally important and the largest symmetry, superconformal higher spin symmetry, is restored. (This "three-phase scenario" is proposed in ref. [ 8 ] and further discussed in ref. [9].) To approach closely to the investigation of possible interrelations between string theory and AdS higher spin theory it is obviously of importance to try to investigate (super)strings on the anti-de Sitter background. It will be the subject of the present paper. We will employ 2D exactly solvable conformal quantum field theory along with the generalized BFV quantization approach. 2. The basic Weyl-invariant closed bosonic string action in background fields is of the form (see ref.
[10]) Elsevier Science Publishers B.V. (North-Holland)
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A= ~
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dzda[x/~g'~PO,~XuOaX~Gu~
+ EaPOc~XUO#XVAuv] ,
( 1)
where Gu~(X ) and Au~(X ) are the background spacetime metric and antisymmetric tensor field respectively. The space-time is supposed to be D-dimensional, rather than 26-dimensional, by a cause which will be clear later. When G~,~is taken to be the fiat Minkowski metric t/u~= ( - +...+ ) and Au~=O, it becomes the usual string action which is equivalent to the Nambu-Goto one after solving the equations of motion for the 1 + 1 metric g~a. In the generalized BFV quantization approach [ 11 ] one introduces ghosts and anti-ghosts to the first-class Virasoro constraints [ 12 ], which reflect the reparametrization invariance of the action, constructs the BFV generator of the generalized BRST transformations which should be nilpotent, £2z=0,
(2)
to guarantee the BRST invariance at the quantum level, and imposes the BRST invariance condition g21phys) = 0
(3)
on the physical states in an enlarged Hilbert space. In the case of the free closed string on the fiat background the nilpotence condition (2) gives the anomaly-free condition for the space-time dimension D - 2 6 = 0 , where D is the contribution to the Virasoro anomaly from D free scalar fields X u, while - 26 is the contribution from the ghost sector. The condition (3) requires the physical states with the fixed physical ghost number to satisfy the Virasoro constraints
L. Iphys) =/S~ Iphys) = 0 ,
n>0,
(Lo -/5o) [phys) = (Lo + / S o - 2) Iphys) = 0 ,
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the 26-dimensional flat background (usually an expansion in the Riemann normal coordinates). Then the nilpotence condition (2) gives some restrictions (effective equations of motion) on the background fields perturbatively order by order in a ' (see e.g. ref. [ 13 ] ). So (in the absence of Au~) in the one-loop approximation one obtains the Einstein equations for the background metric Ru~ = 0 as a requirment of the one-loop nilpotence of g2. Equivalently, one may calculate perturbatively the effective action for the background fields [ 10 ], or the ~function of the theory and then require it to vanish in order to guarantee the tracelessness of the energymomentum tensor at the quantum level. However, the above-mentioned approaches are perturbative ones which act near the 26-dimensional flat Minkowski vacuum. If there were some non-perturbative string vacua with D ~ 2 6 and with a nonzero background curvature R # 0, one would not find them by means of those perturbative approaches. The goal of the present paper is to demonstrate that in quantum string theory there really may exist nonperturbative vacua with D # 2 6 (and D ¢ 10 in the spinning string case) and a constant R ¢ 0 which nevertheless satisfy the nilpotence condition (2), i.e., which are anomaly-free. 3. We will consider string propagation on D-dimensional anti-de Sitter space with the square ,~,2 of the inverse radius measured in the string tension units ( T = (2not') -s = 1 ). Natural covariant derivatives on AdSo satisfy the SO ( D - 1,2) commutation relations
Mu,, =X~,V,,--X,,VF,,
[Vu, V~] =22Muu,
(6)
and the curvature tensor defined by [Vu, V~IXp =Ru~p "X,,
(7)
(4)
has the form
(5)
Ru~p~=A Z(g~pguo-guag~,,) ,
(8)
R=-D(D-1)22
(9)
where the intercept is also fixed by the nilpotence condition (2). This general scheme can be in principle applied to the more general curved background ( 1 ). However, there appear complicated problems of defining the creation-anihilation operators and Wick normal ordering. To avoid these problems it is customary to employ some perturbative expansion procedure near
.
We consider only the anti-de Sitter case R < 0, since in the de Sitter one R > 0 there are no particle-like unitary irreducible lowest-weight (positive energy) representations of SO(D, 1 ) and, consequently, it does not lead to any physically interesting unitary quantum theory. 27
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AdSo can be represented as a coset symmetric space of the form
of the string tension and the cosmological constant ~ x=-~.
S O ( D - 1, 2) AdSz) - S O ( D - 1, 1 ) '
(10)
where SO ( D - 1, 2) is the anti-de Sitter group, while S O ( D - 1, 1 ) is its Lorentz subgroup. Consequently the metric Gu,, in ( 1 ) is chosen to be the invariant metric on the coset space (10). There also exists an invariant closed three-form on AdSn which may be obtained by restriction from the one on the group manifold SO ( D - 1, 2 ), and we take the second term in ( 1 ) to be the Wess-Zumino term on the coset space. Therefore, we are considering the 2D coset model ( 10 ) with the closed string boundary conditions (amodel on AdSo with the background anti-symmetric tensor) subject to the Virasoro constraints (5), where L, (/S,) are holomorphic (anti-holomorphic) parts of the coset model energy-momentum tensor built up along the lines of the GKO coset generalization [ 14] of the Sugawara construction [ 15] for the WessZumino model [ 16 ]. The affine Kac-Moody algebra SO ( D - 1, 2) of our AdSo coset model is of the form (we display all expressions only for left-movers and similar ones for right-movers are supposed to be satisfied) [ 17 ] --xX'l
~Van+rn--~l
~Van+m~'l
~'~n+m
- r f f " M ~ , . ) - ~ n( rlUPrl~a-rlu°rl"Q6. _,~ , ( l l a ) [ M ~ , p~] =i(q~ppu+,. _q~op~,+.,),
( 1 lb)
[P~,P~m] =i2 . 2M.+,,, ~,~ + Tn rlu~.,_,,,
( 1 lc)
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T
(13)
It is negative in the AdS case (it is also required by unitarity for KM algebras for non-compact groups [18]). The energy-momentum tensor has the usual GKO form ( T = 1 ) 1
T(z) = 211 - ( D - 1 )221 ~P~'(z)PU(z)°°
4
'
1 - ( D - 1)22
[pu, P~m]= Tn rlU"8.,_m,
(12)
i.e. the pu become Fourier modes of the ordinary Fubini-Veneziano fields. From the KM commutation relations one can readily recognize the coset model level to be the ratio 28
2_ 2)22 )
×:Mu.(z)MU"(z)°o ,
(14)
which at 2 = 0 becomes as in conventional string theory T(z) = ½~P~,(z)P~'(z) ~ .
(15)
In (14) it is taken into account that for S O ( D - 1, 2) and S O ( D - 1, 1 ) the dual Coxeter numbers are ( D - 1 ) and ( D - 2 ), respectively. Now, having the explicit expression for the energy-momentum tension at our hands, we can write down an exact BFV operator generating BRST transformations for the quantum string in AdSo: -q= ~ i
{c(z)[T(z)-a]
+ ~b(z)c(z)Ozc(z) ~}.
(16)
The nilpotence condition for this operator yields the following anomaly-free condition for string propagation on the AdSo background: C(D, 22) = 2 6 ,
where we have restored the manifest dependence on the string tension 7". In the fiat limit 2--+0 the latter commutation relations become
1-(
a=l,
(17)
where D ( O + 1) C(D,'~2) = 2 1 1 _ ( O _ 1 ) 2 2 ]
D ( D - 1) - 2[1_ (D_2);t2]
(18) ~ Actually it is the overal normalization ofthe string action (1). Taking Gu~ to be the AdS metric and making X u dimensionless (XU-~2 -~Xa) to get rid of the constant 22 inside the string lagrangian, one obtains that the only coupling constant of the theory is the action overal normalization T / 2 2 .
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is a central charge of the AdSn coset model energymomentum tensor (14). The ghost sector contribution obviously does not depend on the background, being the Virasoro group theoretical factor. In the fiat limit, which can be regarded as a coset model Poincar6/Lorentz, it gives correctly C(D, 0) = D = 2 6 . In this way we have obtained that string propagation on AdSD with D defined is anomaly-free only for the following critical values of the background curvature scalar R = - D ( D - 1 )22 (given in string tension units) R+=A+ ~
,
(19)
D(D 2- 107D+ 156) A= 104(D-2) '
(20a)
B= D 2 ( D - 1 ) ( 9 - 2 6 ) 26(9-2)
(20b)
Therefore it can be said that the anomaly-free condition in quantum closed string theory actually fixes the cosmological constant for given D to be as in (19), (20), rather than the very space-time dimension. However, it is possible to have a flat background if and only if D = 26. Surprisingly, we have obtained just two exact solutions R_+ (D) for any D which satisfy the anomalycancallation condition. At our first trajectory R+ = R÷ (D), illustrate in fig. 1, R+ < 0 (AdSo) for D < 26,
23 May 1991
R + = 0 at D = 2 6 , and R + > 0 for D > 2 6 (dSn). (As we have already discussed, the de Sitter case R+ > 0 does not lead to a positive energy unitary quantum theory.) At the same time the second trajectory R _ = R _ ( D ) (fig. 2) lies entirely below the x-axis, and thus for any D, both less or more than 26, there is a non-perturbative AdSo vacuum. Moreover, just in D = 2 6 there is a second vacuum with R=R_ (26) = - 3 2 5 / 8 (22-----1 ) apart from the known flat one ( R + ( 2 6 ) = 0 ) . To summarize, we have proved the following theorem:
Theorem 1. A. quantum closed string theory in Ddimensional anti-de Sitter space with a curvature scalar R~<0 measured in string tension units (and with an invariant background anti-symmetric tensor) is anomaly-free if and only if: ( 1 ) D < 26 and R = R + (D), or R = R _ (D). (2) D = 2 6 and R=R_ (26) = 325 or R=R+ -~-, (26) =0. (3) D > 26 and R = R (D) (R = R + (D) > 0 is a de Sitter solution), where R+, R_ are given in (19), (2O). Meanwhile, to guess about the existence of the first trajectory R+ of non-perturbative vacua in principle is relatively not very difficult, having in mind known perturbative results (see below), the second AdS.
,it de
Z Z
---;-
~
~t~
Anti-de
bosonic
sitter
Sitter
string
~ 'r
~
Fig. 1. Critical background curvature scalars for the closed bosonic string (R=R+ (D)) and for the NSR spinning string (R=D- 10) measured in string tension units ( T= 1~27tot' = 1 ). 29
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23 May 1991
020
I II
OOO
w
f* ,t
-Q.20
,/
q iI
--(3.40
3 --0.60
" ~ - - ' ~ 4-
I ' 1 4
'
'
~ I 2"*
. . . .
J . . . . .34
I ~, 4-4-
,
'
E . . . . 54
I 64
,~-"'-t
'
,
74
,
~1 a4
~ '
'
I)
Fig. 2. The second non-perturbative trajectory for critical AdSo cosmological constants [A~---,~2~---~Z 2 (D), R=R_(D)= - D ( D - 1 )22_ (D) ] in closed bosonic string theory.
trajectory R_ is quite a non-perturbative surprise. However, as we shall see in the next section, the situation in the world-sheet supersymmetric case is quite different, with only a single trajectory for critical R. The above anomaly-free condition can be altematively re-interpreted as a vanishing of the exact dilaton//-function in the constant curvature background
flO(D, ).1) = (g(D, 22) - 2 6 = 0 .
(21)
Being essentially non-perturbative, it can nevertheless be developed in powers of a ' . Keeping only the one-loop correction, one gets fl(D, ).2) =D_26_3(2na,R) + O ( a , 2 R 2 ) =0.
(22)
It is readily seen that when R < 0 the critical spacetime dimension is decreased, while for ~R> 0 it is increased. However, the existence of the second purely AdS trajectory is not absolutely transparent from any perturbative point of view. From the above results one can see that for each fixed D the absolute value of the background AdS curvature scalar is bounded from above by its critical value Re( = R ± ( D ) ) . Note that introducing any additional internal conformal field theory ("extra dimensions") may only decrease the magnitude of the AdS curvature so that [R] ~< IRcl, see also below). The presence of some upper bound on the curvature 30
in string theory apparently has deep physical roots. The first known example of such a phenomenon was provided by the Born-Infeld electrodynamics, where the electromagnetic field F,~ has its upper bound. Many years later it was discovered [ 19 ] that the Born-Infeld action naturally appears in open string theory, being the effective action for the electromagnetic field occurring in the open string spectrum. In string theory, which is regarded as a finite theory, the presence of some upper bounds on the important physical parameters is a natural phenomenon, being a consequence of finiteness. The existence of some upper bound on the curvature is also expected in closed quantum string theory as a gravitational counterpart of the Born-Infeld phenomenon (it was conjectured in ref. [18] for strings; in the quantum cosmology context the idea of the upper bound on the curvature was put forward in ref. [ 20 ]. The above non-perturbative results show that the magnitude of the anti-de Sitter cosmological constant is indeed restricted from above in string theory. To conclude this section, the above coset construction is actually a unique possible one to describe exactly solvable strings in AdS, D>~ 4. However the D= 3 case is singled out since the 3D anti-de Sitter group is not simple: SO(2, 2)-~SO(2, 1 ) × S O ( 2 , 1). As a consequence, in principle, there is a number of rele-
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vant generally non-equivalent coset models (with different Virasoro central charges) SO(2, 1 )k, ×SO(2, 1 )k~ SO(2, 1 )k,+k2
(23)
(the diagonal KM subalgebra is factored out), and SO(2, 1)k×SO(2, l)k " S O ( 2 , l)k SO(2, 1 )k
(24)
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type fermions which are commuting with the Fubini-Veneziano bosons Pg, being their worldsheet superpartners. The the energy-momentum tensor the supercurrent for a supersymmetric generalization of the coset model ( l 0) can be built up from the super KM currents according to the recipe of ref. [ 21 ]. They satisfy the N = 1 NSR superconformal algebra with the Virasoro central charge given by Cu=~ (D, 22) =3D[1 + ( D - I )22 ] .
(27)
(one of the factors is factored out). To treat the 3D case in a similar fashion with D > 3 and to deal with the central charge of the general form (18) (where now D = 3 ) we believe the coset model
It follows from the general expression for symmetric spaces [ 21 ]
S0(2, 1)k/2 × S 0 ( 2 , l )k/z k= 1 S0(2, 1 )k , - 2--5
g = D - 1 and k = - 1/22.
4. Now let us turn to the Neveu-Schwarz-Ramond strings with manifest world-sheet supersymmetry. A natural basis is provided by the Kazama and Suzuki supercoset generalization [ 21 ] of the GKO coset construction which is based on the notion of super Kac-Moody algebras [22 ]. First consider the SO ( D - 1, 2 ) super Kac-Moody algebra which is an extension of ( 11 ) by means of a free world-sheet fermion in the adjoint representation of the SO ( D - 1,2 ) current algebra
{ ~Vr~ , ~ } =
-
(q~°rl~"-rlu~rl~P)8,._,,,,
(28)
where the dual Coxeter number for S O ( D - 1, 2) is (25)
to'describe string propagation on ADS3, rather than simply the WZ model (24). To factor out the diagonal subalgebra may be also of importance to achieve unitarity after further imposition of the Virasoro constraints on physical states in the coset model Hilbert space.
{ ~ , ~}=TrlU~8~,_s,
C~/H =3 dim(G/H) ( 1 - g / k ) ,
(26a) (26b)
[pu, 7'~] =,~;~2 wu~, ,+~,
(26c)
[pu, 7,~p] =i(r/u~ ~ + , _ r/u¢~ + ~ )
(26d)
( ~ = - ~ ), and the commutation relations with the Lorentz current M~ ~ are obvious (for the NeveuSchwarz (Ramond) boundary conditions r, se7/+
½(Z)).
In the flat limit the ~vu become the usual NS or R-
Then we can write down the BFV operator t2 for this model. Its nilpotence condition yields an anomaly-free condition for the NSR superstring in AdSo: CN=I(D,,~,2)-.~- I 5 .
(29)
Therefore for the critical background scalar curvature R = - D ( D - l ),,],2 we obtain in the supersymmetric case a perfectly simple answer (fig. l ) D-10
R= 2not------7-.
(30)
One can see that there are anti-de Sitter spinning strings in all D < 10, and de Sitter ones in D > 10. The necessary conditions for unitarity and space-time supersymmetry again require R ~<0 and D ~<10. To generalize, we have obtained the following theorem:
Theorem 2. The quantum closed NSR world-sheet superstring theory in D-dimensional anti-de Sitter space with the background curvature scalar R ~<0 measured in string tension units (and with an invariant anti-symmetric tensor) is anomaly-free if and only i f D ~ 10, and R = D - 10. So again the matter of criticality is a background curvature (cosmological constant), rather than the very space-time dimension. Surprisingly enough, the situation in supersymmetric theory is drastically different as compared to the purely bosonic string. We have obtained only a 31
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single trajectory for the critical curvature and it is simply a line, in comparison with the two essentially non-linear trajectories in the bosonic string. As a consequence, the anti-de Sitter NSR spinning string may exist only in D~< 10. A natural question to ask is what has happened when world-sheet supersymmetry was taken into account? The answer is rather typical for supersymmetric (especially superconformal) theories. Actually supersymmetry has corrected a one-loop contribution (compare (18) and (27), (21) and (30)) and cancelled out all the higher-loop contributions to the central charge (fl-function), i.e., only one-loop contributions survive in the supersymmetric case. Technically, the Kazama-Suzuki expression (28) for the central charge contains only a first order correction (proportional to 1/k which may be interpreted as a coupling constant of the theory) to the free central charge 3 dim ( G / H ) for the system of dim ( G / H ) free bosons and the same number of fermions. In fact, it is dictated by the KS orthogonal decomposition for the N = 1 NSR superconformal algebra
which guarantees an additional local fermionic xsymmetry reducing the fermion degrees of freedom. This Wess-Zumino term can be easily constructed if one considers the supercoset [24] (Super Poincar6/ Lorentz). It turns out the closed three-form generating the Wess-Zumino term exists on such a supermanifold only for D = 3 , 4, 6,10 and N = 1, 2 (with appropriate spinor types). The resulting GreenSchwarz superstring actions really possess a necessary x-symmetry [23 ]. However, quantum mechanics puts a further restriction, D = 10, when working in flat space. Now, having confined ourselves, the anomaly-free AdSD NSR spinning strings may be considered in D < 10 when the critical background curvature is taken into account, it is quite natural to ask whether there exist manifestly space-time supersymmetric GS-type superstrings on the AdSD ( D < 1 0 ) background. Such superstrings may be naturally expected to exist in D = 3, 4, 6. Corresponding GS-type actions with the WZ term may be constructed by considering the coset supermanifolds
TG=TH+ TG/H, GG=GH+GG/H,
OSP(114)
(31)
s o ( 3 , 1) '
which must satisfy the following OPE rules:
GH(z)GG/H(W)~O, Grt(z)Tc/H(w)~O, Tn(z)Gc/H(w)~O, TH(z)Tcm(w)~O.
OSP(214) SO(3, 1) × S O ( 2 ) (32)
It leads to the cancellation of the higher-6rder contributions which are presented in the GKO central charge and to the only linear trajectory for critical curvature. This is one of the remarkable surprises of supersymmetry. 5. Now let us discuss the possibilities to achieve manifestly space-time supersymmetric anti-de Sitter superstrings. As for world-sheet supersymmetric AdS spinning strings, we have seen that they may exist in all D~< 10 with the appropriate cosmological constants. However, space-time supersymmetry may put further restrictions on D already at the classical level. In fiat space-time, classically, a manifestly supersymmetric Green-Schwarz superstring action [23] can be constructed in D = 3, 4, 6,10 with an appropriate choice of spinor types. A key point here is the existence of the corresponding Wess-Zumino term 32
F(4) SO(5, 1) × S U ( 2 )
(D=4),
(D=6).
(33)
The actions actually have the standard form
A=~ogA^*cOSGAB+TIogA^coBAo~CfAsC, as
x
(34)
where instead o f d x ~ - ifffUd0 there stand (pull-backs of) coset projections of the Maurer-Cartan one-superforms on the above supermanifolds, and the WZ term is standardly constructed by the closed threesuperform. In the flat limit ;t~0 it gives the ordinary GS actions. The explicit construction will be given in a subsequent publication. The next important problem which arises here is whether the above GS-type formulation is equivalent to the NSR formulation constructed in the previous section. In fact, it is not a simple problem because both theories are non-linear ones (although exactly
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solvable). A useful tool to establish the equivalence between NSR and GS formulations is the light-cone gauge, where both theories become free ones. Therefore it is of importance to find some special gauge for AdS strings which would be a generalization of the light-cone gauge. In this way, it might be possible to have anomalyfree AdS superstrings of the Green-Schwarz (or, perhaps, heterotic) type directly in four dimensions without any compactification, but with an appropriate critical background cosmological constant. 6. To conclude this paper, a few subtle problems should be mentioned. They are concerned with the Hilbert-space of states of our AdS strings. The first problem of importance is if they are all unitary. Here we have proved the BRST invariance of these models. It usually automatically guarantees unitarity in the physical Hilbert space of states defined by (3). However in each concrete case it requires a careful verification. Our coset models are, in fact, obtained by the two-step projection from the non-compact SO ( D - 1, 2) (super) WZW model. The first is a coset projection which projects out many negative norm states related with the Loretz subalgebra. The second is the BRST projection (3) which should eliminate all the negative norm states generated by the time components po of AdS translations. A problem of further study is whether all the negative norm states are indeed eliminated from the physical Hilbert space with an appropriate choice of the AdS ground state (highest weight state for the Verma module). Perhaps it will give some further restrictions on the D, R and the AdS ground state. Another problem of prime importance is whether higher spin massless particles may occur in the AdS string spectrum. To get an answer requires decomposing the physical Hilbert space of states into a direct sum of irreducible representations of the global AdS algebra. Our hope is that in the D = 4 AdS superstring spectrum higher spin massless excitations might occur, as well as particles corresponding to AdS supergravity, and the Fradkin-Vasiliev AdS4higher spin gauge theory [2-5 ] would be some sort of truncation of this yet hypothetical 4D superstring theory (this hope that is based on the rather considerable success achieved in recent years in constructing higher spin gauge theories in AdS4 [2-5 ] was in part a motiva-
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tion behind the present investigation, as mentioned in the introduction). One aesthetically attractive potential possibility to get massless particles may exist in AdS superstrings. If the ground state for right-movers is a Dirac supersingleton [25,26] SR=Di~)Rac, and correspondingly SL for left-movers, then for the closed superstring we have the ground state SR®SL which, according to the result of Flato and Fronsdal [26 ], is nothing else but an infinite chain of massless higher spin particles, its spectrum is in one-to-one correspondence [27 ] with the spectrum of gauge fields of the Fradkin-Vasiliev higher spin superalgebra [4]. In that case string theory might naturally synthesize singletons and massless higher spin particles in the bounds of a unified natural construction. So far in this paper we have adopted a somewhat peculiar point of view. To achieve quantum strings in D ~ 26 ( D ~ 10) one of the two approaches is customary employed. The first is a compactification of extra dimensions (see e.g. the review in ref. [28 ]. The second approach is to consider non-critical strings in subcritical dimensions. In this paper an alternative is considered, we deal with critical strings in D ~ 2 6 ( D ~ 10) without any compactification, but our critical dimensions may not be equal to 26 (10) as a result of the presence of an appropriate critical backgrond negative curvature (AdS cosmological constant) (i.e., background curvature "feeds on" extra dimensions). However, some compactification may be still necessary to take the curvature away from its critical values calculated above. So for the D = 4 NSR spinning string the central charge of the internal superconformal field theory must be (see eqs. (27), (29)) C~.t=9+3R/2=9(1-2
22 ) .
(35)
For R = - 6 one has Cint h 0, while to get "off-critical" values - 6 < R ~<0 one has to add an internal CFT with the above non-zero central charge. If the internal sector is constructed from super WZW or supercoset models with rational central charges, the antide Sitter radius is quantized in units of a ' . To illustrate this, let us choose the internal sector to be one of the Kazama-Suzuki supercoset G / H models [21 ] with N = 2 supersymmetry and with the central charge (28). Flat 4D space corresponds to Cint= 9. Choos33
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ing the Kac-Moody central charge k to be less than /~ which corresponds to Ci.t= 9, one gets the background curvature of the AdS4 R = - ~ dim(G/H) g(k - -~) .
(36)
One interesting particular case is the KazamaSuzuki model S U ( 4 ) / S U ( 3 ) × U ( I ) with Cint= 9 ( 1 - 4 / k ) . It describes a spontaneous compactification from D= 10 to D = 4 and, simultaneously, inducing a cosmological c o n s t a n t ( 2 2 - 2/k, k is an integer) in the resulting 4D space-time.
References [ 1 ] J. Scherk, Rev. Mod. Phys. 17 (1975) 123; J.H. Schwarz, Phys. Rep. 89 (1982) 223; M.B. Green, Surv. High Energy Phys. 3 ( 1983 ) 127; M.B. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge U.P., Cambridge, 1986). [2] E.S. Fradkin and M.A. Vasiliev, Ann. Phys. (NY) 177 (1987) 63. [3] E.S. Fradkin and M.A. Vasiliev, Phys. Lett. B 189 (1987) 89; Nucl. Phys. B 291 (1987) 141. [4] E.S. Fradkin and M.A. Vasiliev, Intern. J. Mod. Phys. A 3 (1988) 2983. [ 5 ] M.A. Vasiliev, Ann. Phys. (NY) 190 ( 1989 ) 59; Phys. Lett. B 243 (1990) 378; CERN preprint CERN-TH-5871/90. [6] E.S. Fradkin and V.Ya. Linetsky, Ann. Phys. (NY) 198 (1990) 252. [ 7 ] E.S. Fradkin and V.Ya. Linetsky, Phys. Lett. B 231 ( 1989 ) 97. [ 8 ] E.S. Fradkin and V.Ya. Linetsky, Nucl. Phys. B 350 ( 1991 ) 274. [9] E.S. Fradkin and V.Ya. Linetsky, Three phases of a unified theory: string theory, higher spin gauge theory in the antide Sitter universe, conformal higher spin theory?, talk, in: Proc. Capri Conf. dedicated to Professor P.G. Bergmann (Capri, October 1990), to be published.
34
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[ 10] E.S. Fradkin and A.A. Tseytlin, Nucl. Phys. B 261 (1985) 1. [ 11 ] E.S. Fradkin, Proc. X Winter School in Carpacz (Poland), Acta Univ. Wratislaviensis No. 207 (1973) 93; E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. B 55 (1975) 224; CERN report TH2332 (1977); I.A. Batalin and G.A. Vilkovisky, Phys. Let. B 69 (1977) 309; E.S. Fradkin and T.E. Fradkin, Phys. Lett. B 72 (1978) 343; I.A. Batalin and E.S. Fradkin, Riv. Nuovo Cimento 9 No 10 (1986) 1; Ann. Inst. Herd Poincar~ 49, No. 2 (1988) 145. [ 12] K. Fujikawa, Phys. Rev. D 25 (1982) 2584; M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443; S. Hwang, Nucl. Phys. B 231 (1984) 386; Phys. Rev. D 28 (1983) 2614. [13] I.L. Buchbinder, E.S. Fradkin, S.L. Lyakhovich and V.D. Pershin, Intern. J. Mod. Phys. A5 (1990). [ 14 ] P. Goddard, A. Kent and D. Olive, Commun. Math. Phys. 103 (1986) 105. [ 15 ] V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B 247 (1984) 83. [16] E. Witten, Commun. Math. Phys. 92 (1984) 455. [17] L. Dixon, M. Peskin and L. Lykken, Nucl. Phys. B 325 (1989) 329; I. Bars, Nucl. Phys. B 334 (1990) 125. [ 18 ] E.S. Fradkin, The problem of unification of all interactions and self-consistency, lecture given on the occasion of awarding the Dirac Medal 1988 (ICTP, Trieste, July 1989). [ 19 ] E.S. Fradkin and A.A. Tseytlin, Phys. Lett. B 163 ( 1985 ) 123. [20] M.A. Markov, Ann. Phys. (NY) 155 (1984) 333. [21 ] Y. Kazama and H. Suzuki, Nucl. Phys. B 321 (1989) 232. [22] V.G. Kac and I.T. Todorob, Commun. Math. Phys. 102 (1985) 337. [23] M.B. Green and J.H. Schwarz, Phys. Lett. B 136 (1984) 367. [ 24 ] M. Henneaux and L. Mezincescu, Phys. Lett. B 152 ( 1985 ) 340. [25] P.A.M. Dirac, J. Math. Phys. 4 (1963) 901. [26] M. Flato and C. Fronsdal, Lett. Math. Phys. 2 (1978) 421; Phys. Lett. B 97 (1980) 236. [ 27 ] S.E. Konstein and M.A. Vasiliev, Nucl. Phys. B 312 (1989) 402. [28] J.H. Schwarz, Intern. J. Mod. Phys. A 4 (1989) 2653.
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23 May 1991
On the anti-de Sitter quantum (super) strings E.S. Fradkin and V.Ya. Linetsky Theoretical Department, P.N. Lebedev Physical Institute, Leninsky Prospect 53, SU-I 17 924 Moscow, USSR
Received 8 March 1991
Closed bosonic and NSR quantum strings in AdSDare considered as exactly solvablemodels. There may exist anomaly-free AdSDstrings in D # 26 (D:# 10) with an appropriate critical cosmologicalconstant fixed by the anomaly-cancellationcondition. All anomaly-freepairs (D, A) are exactlycalculated. Covariant AdS4and AdS6GS-typesuperstringsare discussed.
1. Conventional (super)string theory [ 1 ] may be regarded as an interacting theory of infinitely many particles (string excitations with increasingly high spins, their masses growing with their spins. Interactions are non-local and non-analytical in the string mass parameter, the string tension T = 1 / 2 n a ' . This non-analyticity does not allow one to pass to the massless limit T - , 0 ( a ' ~ o o ) in interacting string theory. If such a limit existed, then all the massive higher spin particles might become massless and an enormously rich infinite-dimensional higher spin gauge symmetry might restore. That symmetry might play a role of the fundamental hidden gauge symmetry behind strings. However such a massless limit T ~ 0 does not exist in conventional string theory as a continuous limiting process. The formal obstruction is a known uncompatibility of the gauge symmetry of massless higher spin particles and gravity. In recent years it was discovered that there nevertheless may exist a consistent interaction among massless higher spins and gravity, but the latter must be either anti-de Sitter gravity with a non-zero negative cosmological constant [ 2-5 ], or Weyl conformal gravity [ 6-8 ]. It is based on the discovery of new classes of non-abelian infinite-dimensional gauge symmetries generalizing local supersymmetries. These arc the anti-de Sitter higher spin symmetry [ 2,4 ] and the superconformal higher spin symmetry [6,8] which generalize AdS supersymmetry and conformal supersymmetry to the case of all higher spins (they mix particles with all spins from zero to infinity). The fundamental property of the gravitational in26
teraction of massless higher spin particles discovered in ref. [ 3 ] is its non-analyticity in the cosmological constant A, which does not permit one to pass to the flat limit A ~ 0 in the interacting theory. These two-analyticities, in T in strings and in A in the AdS higher spin gauge theory, suggested us to conjecture [ 8 ] a deep interrelation between these two theories. Moreover, wc have conjectured [8] that there might bc some sort of phase transition in string theory at high energies when the cosmological constant of the Planck order is induced and an infinitedimensional AdS higher spin gauge symmetry is restored. Conformal higher spin theory then has been regarded as a third ultra-high energy asymptotic phase when all the mass parameters become unessential, all higher spin excitations become equally important and the largest symmetry, superconformal higher spin symmetry, is restored. (This "three-phase scenario" is proposed in ref. [ 8 ] and further discussed in ref. [9].) To approach closely to the investigation of possible interrelations between string theory and AdS higher spin theory it is obviously of importance to try to investigate (super)strings on the anti-de Sitter background. It will be the subject of the present paper. We will employ 2D exactly solvable conformal quantum field theory along with the generalized BFV quantization approach. 2. The basic Weyl-invariant closed bosonic string action in background fields is of the form (see ref.
[10]) Elsevier Science Publishers B.V. (North-Holland)
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A= ~
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dzda[x/~g'~PO,~XuOaX~Gu~
+ EaPOc~XUO#XVAuv] ,
( 1)
where Gu~(X ) and Au~(X ) are the background spacetime metric and antisymmetric tensor field respectively. The space-time is supposed to be D-dimensional, rather than 26-dimensional, by a cause which will be clear later. When G~,~is taken to be the fiat Minkowski metric t/u~= ( - +...+ ) and Au~=O, it becomes the usual string action which is equivalent to the Nambu-Goto one after solving the equations of motion for the 1 + 1 metric g~a. In the generalized BFV quantization approach [ 11 ] one introduces ghosts and anti-ghosts to the first-class Virasoro constraints [ 12 ], which reflect the reparametrization invariance of the action, constructs the BFV generator of the generalized BRST transformations which should be nilpotent, £2z=0,
(2)
to guarantee the BRST invariance at the quantum level, and imposes the BRST invariance condition g21phys) = 0
(3)
on the physical states in an enlarged Hilbert space. In the case of the free closed string on the fiat background the nilpotence condition (2) gives the anomaly-free condition for the space-time dimension D - 2 6 = 0 , where D is the contribution to the Virasoro anomaly from D free scalar fields X u, while - 26 is the contribution from the ghost sector. The condition (3) requires the physical states with the fixed physical ghost number to satisfy the Virasoro constraints
L. Iphys) =/S~ Iphys) = 0 ,
n>0,
(Lo -/5o) [phys) = (Lo + / S o - 2) Iphys) = 0 ,
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the 26-dimensional flat background (usually an expansion in the Riemann normal coordinates). Then the nilpotence condition (2) gives some restrictions (effective equations of motion) on the background fields perturbatively order by order in a ' (see e.g. ref. [ 13 ] ). So (in the absence of Au~) in the one-loop approximation one obtains the Einstein equations for the background metric Ru~ = 0 as a requirment of the one-loop nilpotence of g2. Equivalently, one may calculate perturbatively the effective action for the background fields [ 10 ], or the ~function of the theory and then require it to vanish in order to guarantee the tracelessness of the energymomentum tensor at the quantum level. However, the above-mentioned approaches are perturbative ones which act near the 26-dimensional flat Minkowski vacuum. If there were some non-perturbative string vacua with D ~ 2 6 and with a nonzero background curvature R # 0, one would not find them by means of those perturbative approaches. The goal of the present paper is to demonstrate that in quantum string theory there really may exist nonperturbative vacua with D # 2 6 (and D ¢ 10 in the spinning string case) and a constant R ¢ 0 which nevertheless satisfy the nilpotence condition (2), i.e., which are anomaly-free. 3. We will consider string propagation on D-dimensional anti-de Sitter space with the square ,~,2 of the inverse radius measured in the string tension units ( T = (2not') -s = 1 ). Natural covariant derivatives on AdSo satisfy the SO ( D - 1,2) commutation relations
Mu,, =X~,V,,--X,,VF,,
[Vu, V~] =22Muu,
(6)
and the curvature tensor defined by [Vu, V~IXp =Ru~p "X,,
(7)
(4)
has the form
(5)
Ru~p~=A Z(g~pguo-guag~,,) ,
(8)
R=-D(D-1)22
(9)
where the intercept is also fixed by the nilpotence condition (2). This general scheme can be in principle applied to the more general curved background ( 1 ). However, there appear complicated problems of defining the creation-anihilation operators and Wick normal ordering. To avoid these problems it is customary to employ some perturbative expansion procedure near
.
We consider only the anti-de Sitter case R < 0, since in the de Sitter one R > 0 there are no particle-like unitary irreducible lowest-weight (positive energy) representations of SO(D, 1 ) and, consequently, it does not lead to any physically interesting unitary quantum theory. 27
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AdSo can be represented as a coset symmetric space of the form
of the string tension and the cosmological constant ~ x=-~.
S O ( D - 1, 2) AdSz) - S O ( D - 1, 1 ) '
(10)
where SO ( D - 1, 2) is the anti-de Sitter group, while S O ( D - 1, 1 ) is its Lorentz subgroup. Consequently the metric Gu,, in ( 1 ) is chosen to be the invariant metric on the coset space (10). There also exists an invariant closed three-form on AdSn which may be obtained by restriction from the one on the group manifold SO ( D - 1, 2 ), and we take the second term in ( 1 ) to be the Wess-Zumino term on the coset space. Therefore, we are considering the 2D coset model ( 10 ) with the closed string boundary conditions (amodel on AdSo with the background anti-symmetric tensor) subject to the Virasoro constraints (5), where L, (/S,) are holomorphic (anti-holomorphic) parts of the coset model energy-momentum tensor built up along the lines of the GKO coset generalization [ 14] of the Sugawara construction [ 15] for the WessZumino model [ 16 ]. The affine Kac-Moody algebra SO ( D - 1, 2) of our AdSo coset model is of the form (we display all expressions only for left-movers and similar ones for right-movers are supposed to be satisfied) [ 17 ] --xX'l
~Van+rn--~l
~Van+m~'l
~'~n+m
- r f f " M ~ , . ) - ~ n( rlUPrl~a-rlu°rl"Q6. _,~ , ( l l a ) [ M ~ , p~] =i(q~ppu+,. _q~op~,+.,),
( 1 lb)
[P~,P~m] =i2 . 2M.+,,, ~,~ + Tn rlu~.,_,,,
( 1 lc)
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T
(13)
It is negative in the AdS case (it is also required by unitarity for KM algebras for non-compact groups [18]). The energy-momentum tensor has the usual GKO form ( T = 1 ) 1
T(z) = 211 - ( D - 1 )221 ~P~'(z)PU(z)°°
4
'
1 - ( D - 1)22
[pu, P~m]= Tn rlU"8.,_m,
(12)
i.e. the pu become Fourier modes of the ordinary Fubini-Veneziano fields. From the KM commutation relations one can readily recognize the coset model level to be the ratio 28
2_ 2)22 )
×:Mu.(z)MU"(z)°o ,
(14)
which at 2 = 0 becomes as in conventional string theory T(z) = ½~P~,(z)P~'(z) ~ .
(15)
In (14) it is taken into account that for S O ( D - 1, 2) and S O ( D - 1, 1 ) the dual Coxeter numbers are ( D - 1 ) and ( D - 2 ), respectively. Now, having the explicit expression for the energy-momentum tension at our hands, we can write down an exact BFV operator generating BRST transformations for the quantum string in AdSo: -q= ~ i
{c(z)[T(z)-a]
+ ~b(z)c(z)Ozc(z) ~}.
(16)
The nilpotence condition for this operator yields the following anomaly-free condition for string propagation on the AdSo background: C(D, 22) = 2 6 ,
where we have restored the manifest dependence on the string tension 7". In the fiat limit 2--+0 the latter commutation relations become
1-(
a=l,
(17)
where D ( O + 1) C(D,'~2) = 2 1 1 _ ( O _ 1 ) 2 2 ]
D ( D - 1) - 2[1_ (D_2);t2]
(18) ~ Actually it is the overal normalization ofthe string action (1). Taking Gu~ to be the AdS metric and making X u dimensionless (XU-~2 -~Xa) to get rid of the constant 22 inside the string lagrangian, one obtains that the only coupling constant of the theory is the action overal normalization T / 2 2 .
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is a central charge of the AdSn coset model energymomentum tensor (14). The ghost sector contribution obviously does not depend on the background, being the Virasoro group theoretical factor. In the fiat limit, which can be regarded as a coset model Poincar6/Lorentz, it gives correctly C(D, 0) = D = 2 6 . In this way we have obtained that string propagation on AdSD with D defined is anomaly-free only for the following critical values of the background curvature scalar R = - D ( D - 1 )22 (given in string tension units) R+=A+ ~
,
(19)
D(D 2- 107D+ 156) A= 104(D-2) '
(20a)
B= D 2 ( D - 1 ) ( 9 - 2 6 ) 26(9-2)
(20b)
Therefore it can be said that the anomaly-free condition in quantum closed string theory actually fixes the cosmological constant for given D to be as in (19), (20), rather than the very space-time dimension. However, it is possible to have a flat background if and only if D = 26. Surprisingly, we have obtained just two exact solutions R_+ (D) for any D which satisfy the anomalycancallation condition. At our first trajectory R+ = R÷ (D), illustrate in fig. 1, R+ < 0 (AdSo) for D < 26,
23 May 1991
R + = 0 at D = 2 6 , and R + > 0 for D > 2 6 (dSn). (As we have already discussed, the de Sitter case R+ > 0 does not lead to a positive energy unitary quantum theory.) At the same time the second trajectory R _ = R _ ( D ) (fig. 2) lies entirely below the x-axis, and thus for any D, both less or more than 26, there is a non-perturbative AdSo vacuum. Moreover, just in D = 2 6 there is a second vacuum with R=R_ (26) = - 3 2 5 / 8 (22-----1 ) apart from the known flat one ( R + ( 2 6 ) = 0 ) . To summarize, we have proved the following theorem:
Theorem 1. A. quantum closed string theory in Ddimensional anti-de Sitter space with a curvature scalar R~<0 measured in string tension units (and with an invariant background anti-symmetric tensor) is anomaly-free if and only if: ( 1 ) D < 26 and R = R + (D), or R = R _ (D). (2) D = 2 6 and R=R_ (26) = 325 or R=R+ -~-, (26) =0. (3) D > 26 and R = R (D) (R = R + (D) > 0 is a de Sitter solution), where R+, R_ are given in (19), (2O). Meanwhile, to guess about the existence of the first trajectory R+ of non-perturbative vacua in principle is relatively not very difficult, having in mind known perturbative results (see below), the second AdS.
,it de
Z Z
---;-
~
~t~
Anti-de
bosonic
sitter
Sitter
string
~ 'r
~
Fig. 1. Critical background curvature scalars for the closed bosonic string (R=R+ (D)) and for the NSR spinning string (R=D- 10) measured in string tension units ( T= 1~27tot' = 1 ). 29
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23 May 1991
020
I II
OOO
w
f* ,t
-Q.20
,/
q iI
--(3.40
3 --0.60
" ~ - - ' ~ 4-
I ' 1 4
'
'
~ I 2"*
. . . .
J . . . . .34
I ~, 4-4-
,
'
E . . . . 54
I 64
,~-"'-t
'
,
74
,
~1 a4
~ '
'
I)
Fig. 2. The second non-perturbative trajectory for critical AdSo cosmological constants [A~---,~2~---~Z 2 (D), R=R_(D)= - D ( D - 1 )22_ (D) ] in closed bosonic string theory.
trajectory R_ is quite a non-perturbative surprise. However, as we shall see in the next section, the situation in the world-sheet supersymmetric case is quite different, with only a single trajectory for critical R. The above anomaly-free condition can be altematively re-interpreted as a vanishing of the exact dilaton//-function in the constant curvature background
flO(D, ).1) = (g(D, 22) - 2 6 = 0 .
(21)
Being essentially non-perturbative, it can nevertheless be developed in powers of a ' . Keeping only the one-loop correction, one gets fl(D, ).2) =D_26_3(2na,R) + O ( a , 2 R 2 ) =0.
(22)
It is readily seen that when R < 0 the critical spacetime dimension is decreased, while for ~R> 0 it is increased. However, the existence of the second purely AdS trajectory is not absolutely transparent from any perturbative point of view. From the above results one can see that for each fixed D the absolute value of the background AdS curvature scalar is bounded from above by its critical value Re( = R ± ( D ) ) . Note that introducing any additional internal conformal field theory ("extra dimensions") may only decrease the magnitude of the AdS curvature so that [R] ~< IRcl, see also below). The presence of some upper bound on the curvature 30
in string theory apparently has deep physical roots. The first known example of such a phenomenon was provided by the Born-Infeld electrodynamics, where the electromagnetic field F,~ has its upper bound. Many years later it was discovered [ 19 ] that the Born-Infeld action naturally appears in open string theory, being the effective action for the electromagnetic field occurring in the open string spectrum. In string theory, which is regarded as a finite theory, the presence of some upper bounds on the important physical parameters is a natural phenomenon, being a consequence of finiteness. The existence of some upper bound on the curvature is also expected in closed quantum string theory as a gravitational counterpart of the Born-Infeld phenomenon (it was conjectured in ref. [18] for strings; in the quantum cosmology context the idea of the upper bound on the curvature was put forward in ref. [ 20 ]. The above non-perturbative results show that the magnitude of the anti-de Sitter cosmological constant is indeed restricted from above in string theory. To conclude this section, the above coset construction is actually a unique possible one to describe exactly solvable strings in AdS, D>~ 4. However the D= 3 case is singled out since the 3D anti-de Sitter group is not simple: SO(2, 2)-~SO(2, 1 ) × S O ( 2 , 1). As a consequence, in principle, there is a number of rele-
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vant generally non-equivalent coset models (with different Virasoro central charges) SO(2, 1 )k, ×SO(2, 1 )k~ SO(2, 1 )k,+k2
(23)
(the diagonal KM subalgebra is factored out), and SO(2, 1)k×SO(2, l)k " S O ( 2 , l)k SO(2, 1 )k
(24)
23 May 1991
type fermions which are commuting with the Fubini-Veneziano bosons Pg, being their worldsheet superpartners. The the energy-momentum tensor the supercurrent for a supersymmetric generalization of the coset model ( l 0) can be built up from the super KM currents according to the recipe of ref. [ 21 ]. They satisfy the N = 1 NSR superconformal algebra with the Virasoro central charge given by Cu=~ (D, 22) =3D[1 + ( D - I )22 ] .
(27)
(one of the factors is factored out). To treat the 3D case in a similar fashion with D > 3 and to deal with the central charge of the general form (18) (where now D = 3 ) we believe the coset model
It follows from the general expression for symmetric spaces [ 21 ]
S0(2, 1)k/2 × S 0 ( 2 , l )k/z k= 1 S0(2, 1 )k , - 2--5
g = D - 1 and k = - 1/22.
4. Now let us turn to the Neveu-Schwarz-Ramond strings with manifest world-sheet supersymmetry. A natural basis is provided by the Kazama and Suzuki supercoset generalization [ 21 ] of the GKO coset construction which is based on the notion of super Kac-Moody algebras [22 ]. First consider the SO ( D - 1, 2 ) super Kac-Moody algebra which is an extension of ( 11 ) by means of a free world-sheet fermion in the adjoint representation of the SO ( D - 1,2 ) current algebra
{ ~Vr~ , ~ } =
-
(q~°rl~"-rlu~rl~P)8,._,,,,
(28)
where the dual Coxeter number for S O ( D - 1, 2) is (25)
to'describe string propagation on ADS3, rather than simply the WZ model (24). To factor out the diagonal subalgebra may be also of importance to achieve unitarity after further imposition of the Virasoro constraints on physical states in the coset model Hilbert space.
{ ~ , ~}=TrlU~8~,_s,
C~/H =3 dim(G/H) ( 1 - g / k ) ,
(26a) (26b)
[pu, 7'~] =,~;~2 wu~, ,+~,
(26c)
[pu, 7,~p] =i(r/u~ ~ + , _ r/u¢~ + ~ )
(26d)
( ~ = - ~ ), and the commutation relations with the Lorentz current M~ ~ are obvious (for the NeveuSchwarz (Ramond) boundary conditions r, se7/+
½(Z)).
In the flat limit the ~vu become the usual NS or R-
Then we can write down the BFV operator t2 for this model. Its nilpotence condition yields an anomaly-free condition for the NSR superstring in AdSo: CN=I(D,,~,2)-.~- I 5 .
(29)
Therefore for the critical background scalar curvature R = - D ( D - l ),,],2 we obtain in the supersymmetric case a perfectly simple answer (fig. l ) D-10
R= 2not------7-.
(30)
One can see that there are anti-de Sitter spinning strings in all D < 10, and de Sitter ones in D > 10. The necessary conditions for unitarity and space-time supersymmetry again require R ~<0 and D ~<10. To generalize, we have obtained the following theorem:
Theorem 2. The quantum closed NSR world-sheet superstring theory in D-dimensional anti-de Sitter space with the background curvature scalar R ~<0 measured in string tension units (and with an invariant anti-symmetric tensor) is anomaly-free if and only i f D ~ 10, and R = D - 10. So again the matter of criticality is a background curvature (cosmological constant), rather than the very space-time dimension. Surprisingly enough, the situation in supersymmetric theory is drastically different as compared to the purely bosonic string. We have obtained only a 31
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single trajectory for the critical curvature and it is simply a line, in comparison with the two essentially non-linear trajectories in the bosonic string. As a consequence, the anti-de Sitter NSR spinning string may exist only in D~< 10. A natural question to ask is what has happened when world-sheet supersymmetry was taken into account? The answer is rather typical for supersymmetric (especially superconformal) theories. Actually supersymmetry has corrected a one-loop contribution (compare (18) and (27), (21) and (30)) and cancelled out all the higher-loop contributions to the central charge (fl-function), i.e., only one-loop contributions survive in the supersymmetric case. Technically, the Kazama-Suzuki expression (28) for the central charge contains only a first order correction (proportional to 1/k which may be interpreted as a coupling constant of the theory) to the free central charge 3 dim ( G / H ) for the system of dim ( G / H ) free bosons and the same number of fermions. In fact, it is dictated by the KS orthogonal decomposition for the N = 1 NSR superconformal algebra
which guarantees an additional local fermionic xsymmetry reducing the fermion degrees of freedom. This Wess-Zumino term can be easily constructed if one considers the supercoset [24] (Super Poincar6/ Lorentz). It turns out the closed three-form generating the Wess-Zumino term exists on such a supermanifold only for D = 3 , 4, 6,10 and N = 1, 2 (with appropriate spinor types). The resulting GreenSchwarz superstring actions really possess a necessary x-symmetry [23 ]. However, quantum mechanics puts a further restriction, D = 10, when working in flat space. Now, having confined ourselves, the anomaly-free AdSD NSR spinning strings may be considered in D < 10 when the critical background curvature is taken into account, it is quite natural to ask whether there exist manifestly space-time supersymmetric GS-type superstrings on the AdSD ( D < 1 0 ) background. Such superstrings may be naturally expected to exist in D = 3, 4, 6. Corresponding GS-type actions with the WZ term may be constructed by considering the coset supermanifolds
TG=TH+ TG/H, GG=GH+GG/H,
OSP(114)
(31)
s o ( 3 , 1) '
which must satisfy the following OPE rules:
GH(z)GG/H(W)~O, Grt(z)Tc/H(w)~O, Tn(z)Gc/H(w)~O, TH(z)Tcm(w)~O.
OSP(214) SO(3, 1) × S O ( 2 ) (32)
It leads to the cancellation of the higher-6rder contributions which are presented in the GKO central charge and to the only linear trajectory for critical curvature. This is one of the remarkable surprises of supersymmetry. 5. Now let us discuss the possibilities to achieve manifestly space-time supersymmetric anti-de Sitter superstrings. As for world-sheet supersymmetric AdS spinning strings, we have seen that they may exist in all D~< 10 with the appropriate cosmological constants. However, space-time supersymmetry may put further restrictions on D already at the classical level. In fiat space-time, classically, a manifestly supersymmetric Green-Schwarz superstring action [23] can be constructed in D = 3, 4, 6,10 with an appropriate choice of spinor types. A key point here is the existence of the corresponding Wess-Zumino term 32
F(4) SO(5, 1) × S U ( 2 )
(D=4),
(D=6).
(33)
The actions actually have the standard form
A=~ogA^*cOSGAB+TIogA^coBAo~CfAsC, as
x
(34)
where instead o f d x ~ - ifffUd0 there stand (pull-backs of) coset projections of the Maurer-Cartan one-superforms on the above supermanifolds, and the WZ term is standardly constructed by the closed threesuperform. In the flat limit ;t~0 it gives the ordinary GS actions. The explicit construction will be given in a subsequent publication. The next important problem which arises here is whether the above GS-type formulation is equivalent to the NSR formulation constructed in the previous section. In fact, it is not a simple problem because both theories are non-linear ones (although exactly
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solvable). A useful tool to establish the equivalence between NSR and GS formulations is the light-cone gauge, where both theories become free ones. Therefore it is of importance to find some special gauge for AdS strings which would be a generalization of the light-cone gauge. In this way, it might be possible to have anomalyfree AdS superstrings of the Green-Schwarz (or, perhaps, heterotic) type directly in four dimensions without any compactification, but with an appropriate critical background cosmological constant. 6. To conclude this paper, a few subtle problems should be mentioned. They are concerned with the Hilbert-space of states of our AdS strings. The first problem of importance is if they are all unitary. Here we have proved the BRST invariance of these models. It usually automatically guarantees unitarity in the physical Hilbert space of states defined by (3). However in each concrete case it requires a careful verification. Our coset models are, in fact, obtained by the two-step projection from the non-compact SO ( D - 1, 2) (super) WZW model. The first is a coset projection which projects out many negative norm states related with the Loretz subalgebra. The second is the BRST projection (3) which should eliminate all the negative norm states generated by the time components po of AdS translations. A problem of further study is whether all the negative norm states are indeed eliminated from the physical Hilbert space with an appropriate choice of the AdS ground state (highest weight state for the Verma module). Perhaps it will give some further restrictions on the D, R and the AdS ground state. Another problem of prime importance is whether higher spin massless particles may occur in the AdS string spectrum. To get an answer requires decomposing the physical Hilbert space of states into a direct sum of irreducible representations of the global AdS algebra. Our hope is that in the D = 4 AdS superstring spectrum higher spin massless excitations might occur, as well as particles corresponding to AdS supergravity, and the Fradkin-Vasiliev AdS4higher spin gauge theory [2-5 ] would be some sort of truncation of this yet hypothetical 4D superstring theory (this hope that is based on the rather considerable success achieved in recent years in constructing higher spin gauge theories in AdS4 [2-5 ] was in part a motiva-
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tion behind the present investigation, as mentioned in the introduction). One aesthetically attractive potential possibility to get massless particles may exist in AdS superstrings. If the ground state for right-movers is a Dirac supersingleton [25,26] SR=Di~)Rac, and correspondingly SL for left-movers, then for the closed superstring we have the ground state SR®SL which, according to the result of Flato and Fronsdal [26 ], is nothing else but an infinite chain of massless higher spin particles, its spectrum is in one-to-one correspondence [27 ] with the spectrum of gauge fields of the Fradkin-Vasiliev higher spin superalgebra [4]. In that case string theory might naturally synthesize singletons and massless higher spin particles in the bounds of a unified natural construction. So far in this paper we have adopted a somewhat peculiar point of view. To achieve quantum strings in D ~ 26 ( D ~ 10) one of the two approaches is customary employed. The first is a compactification of extra dimensions (see e.g. the review in ref. [28 ]. The second approach is to consider non-critical strings in subcritical dimensions. In this paper an alternative is considered, we deal with critical strings in D ~ 2 6 ( D ~ 10) without any compactification, but our critical dimensions may not be equal to 26 (10) as a result of the presence of an appropriate critical backgrond negative curvature (AdS cosmological constant) (i.e., background curvature "feeds on" extra dimensions). However, some compactification may be still necessary to take the curvature away from its critical values calculated above. So for the D = 4 NSR spinning string the central charge of the internal superconformal field theory must be (see eqs. (27), (29)) C~.t=9+3R/2=9(1-2
22 ) .
(35)
For R = - 6 one has Cint h 0, while to get "off-critical" values - 6 < R ~<0 one has to add an internal CFT with the above non-zero central charge. If the internal sector is constructed from super WZW or supercoset models with rational central charges, the antide Sitter radius is quantized in units of a ' . To illustrate this, let us choose the internal sector to be one of the Kazama-Suzuki supercoset G / H models [21 ] with N = 2 supersymmetry and with the central charge (28). Flat 4D space corresponds to Cint= 9. Choos33
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ing the Kac-Moody central charge k to be less than /~ which corresponds to Ci.t= 9, one gets the background curvature of the AdS4 R = - ~ dim(G/H) g(k - -~) .
(36)
One interesting particular case is the KazamaSuzuki model S U ( 4 ) / S U ( 3 ) × U ( I ) with Cint= 9 ( 1 - 4 / k ) . It describes a spontaneous compactification from D= 10 to D = 4 and, simultaneously, inducing a cosmological c o n s t a n t ( 2 2 - 2/k, k is an integer) in the resulting 4D space-time.
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