Nuclear Physics B261 (1985) 1-27 © North-Holland Pubhshmg Company

QUANTUM

STRING THEORY

EFFECTIVE ACTION

E S FRADKIN and A A TSEYTLIN Department of Theorettcal Phystcs, P N Levedev Phystcal lnsntute, Lenmsky pr 53, Moscow 117924, USSR Received 8 January 1985 (Final version received 28 May 1985) We present a covanant background field method for quantum stnng dynamics It is based on the effective action F for fields corresponding to &fferent stnng modes A formahsm ts developed for the calculation of F m the a'-->0 hmlt It is shown that m the case of closed Bose stnngs F contains the standard kinetic terms for the scalar, external metric and the antlsymmetnc tensor Our approach makes possible a consistent formulation and solution of a ground state problem (including the problem of space-ttme compactlficatlon) m the stnng theory We suggest a solution to the old "tachyon problem" based on the generatmn of non-trivial vacuum values for the scalar field, metric and antlsymmemc tensor It is shown that a preferred compactlficatlon m the closed Bose stnng theory is to three (anti-de Sitter) space-time dimensions

1. Introduction

String m o d e l s were o r i g i n a l l y d e v e l o p e d for the n e e d s o f the s t r o n g i n t e r a c t i o n theory. T h e n it was r e a l i z e d that a string t h e o r y w h e n p r o p e r l y i n t e r p r e t e d m a y p l a y a m o r e f u n d a m e n t a l role, p r o v i d i n g us with a c o n s i s t e n t t h e o r y o f all i n t e r a c t i o n s i n c l u d i n g q u a n t i z e d g r a v i t a t i o n a l ones [1, 2]. A free ( c l o s e d ) string c a n be d e s c r i b e d in terms o f a n infinite n u m b e r o f its " o s c i l l a t i o n m o d e s " (scalar, s y m m e t r i c 2-tensor, a n t i s y m m e t r i c 2 - t e n s o r , . . . ) . It was o b s e r v e d t h a t the zero string " s i z e " limit (a'--> O) o f scattering a m p l i t u d e s o f different string m o d e s c o i n c i d e s with o n - s h e l l scattering a m p l i t u d e s in a t h e o r y o f fields a s s o c i a t e d with the e l e m e n t a r y string m o d e s [3]. It was s h o w n [3] t h a t the c o r r e s p o n d i n g c o v a r i a n t a c t i o n c o n t a i n s the E i n s t e i n gravitat i o n a l t e r m for the s y m m e t r i c 2 - t e n s o r c o n s i d e r e d as a p e r t u r b a t i o n o f the flat m e t r i c I n a m o r e realistic case o f s u p e r s y m m e t r i c strings in ten d i m e n s i o n s such an a c t i o n c o n t a i n s an N = 2, D = 10 s u p e r g r a v l t y a c t i o n o r N = 1, D = 10 s u p e r g r a v l t y plus N = 1, D = 10 s u p e r - Y a n g - M i l l s a c t i o n in the c l o s e d a n d o p e n string t h e o r y cases c o r r e s p o n d i n g l y [2]. G i v e n t h a t ( c l o s e d ) s u p e r s t r l n g t h e o r y is likely to be finite to all o r d e r s [2] it can b e c o n s i d e r e d as a n interesting c a n d i d a t e for a f u n d a m e n t a l theory. T h e r e is, h o w e v e r , a n u m b e r o f c o n c e p t u a l as well as t e c h n i c a l p r o b l e m s in a ( s u p e r ) string t h e o r y as f o r m u l a t e d t o d a y . T h e a b o v e c o n n e c t i o n b e t w e e n s t n n g a n d c o r r e s p o n d i n g field t h e o r i e s was p r e v i o u s l y e s t a b l i s h e d in a n o n - c o v a r i a n t " o n - s h e l l " w a y (one h a d first to find an ct'-* 0 o n - s h e l l scattering a m p l i t u d e a n d t h e n to guess a c o v a r i a n t a c t i o n f r o m w h i c h it c o u l d be d e r i v e d ) . A l s o , e x p a n s i o n s n e a r a flat 1

2

E S Fradkm, A A Tseythn / Quantum string theory

space-time were used m this procedure. All this made it ddticult to understand how a curved space-time could be built of the "graviton" string modes and thus how a spontaneous compactification from D = 26 or 10 to four space-time dimensions could take place. A lacking formalism is an analog of a background field method known for ordinary field theories, i.e a covarlant "off-shell" effective action F for the infinite number of fields corresponding to string "excitations". Had we such an effective action accumulating a string theory dynamics we could consistently formulate the ground state problem m this theory. If the solution for a ground state metric appeared to be non-trivial this would be a mamfestatlon of a dynamical "condensation" of free string "gravlton modes". I f the corresponding ground state Ddimensional space-time was a product of a four-dimensional one and a compact internal space this would be a solution to the compactlfication problem. If this ground state was stable so that no ghosts and tachyons were present in the expansion of the effective action F near the vacuum values of fields this would be a solution to the unltarity problem ("tachyon problem" of the old Bose string theory). Our aim here is to present such a "background field method" formulation of a quantum string theory starting with a covariant defimtlon of the effective field theory action F in terms of a path integral over "Internal" string variables (sect 2). In this paper we mainly consider only the case of closed Bose strings Generalization to superstrmgs remains an important problem for the future We follow the covariant approach to the stnng theory path integral [4] (see also [5-8]), so that F is given by functional integrals over string coordinates and two-dimensional metrics The action m the exponent which is averaged contains the free string term [9] as well as the infinite number of "source terms" depending on the "external" fields (the arguments of F ) which " p r o b e " different string "excitations". A low-energy approximation for F Is obtained by expanding in ~'-->0 In sect 3 we discuss integration over string coordinates and obtain an "intermediate" effective action W which depends on the external fields and an arbitrary two-dimensional metric. Integration over two-dimensional metrics is studied m sect. 4. We consider the simplest case of the spherical two-dimensional topology ("tree approximation") The resulting effective action for the "lower-lying" fields (scalar, D-dimensional metric and antisymmetric tensor) is then extremized for finding the ground state configurations. We first investigate the D = 26 case using a simplified approach in order to illustrate some general points Then more rigorous treatment of the general D ~< 2 6 case is given and it is shown that there is no ground states with a flat D-dimensional space-time. The theory prefers compactificatlon to a threedimensional anti-de Sitter space-time so that the ground state metric is always curved It seems likely that more realistic four-dimensional compact~ficatlon patterns may exist in the (closed) superstring case. In concluding sect. 5 we dtscuss some points connected with an interpretation and extension of our approach In particular, we present a generalization of the

E S Fradkm, A A Tseythn / Quantum stnng theory

3

effective action to the case of extended objects of arbitrary dimensions (strings, membranes, etc.).

2. Effective action Our starting point is the covariant (closed) Bose string action [9]

1 ,(

Io = 2 ~ a '

1

dax x/g~g

ttv

t

a~o 0 ~

(1)

ernz,

Here [ a ' ] = x u (/.t = 1, 2) are coordinates of a 2-dimensional compact space M 2, is a metric on M 2, ~0' (t = 1 , . . , D) are coordinates of an external space-time M D where strings propagate (external space metric G,j(~o) is assumed to be flat in (1)). We shall use euclidean notation in this paper. The quantum string theory is defined by the path integral [4]

g,~(x)

('")=I [dg~'~]I[dq~']e-CW~)'°t~'g]

(2)

For example, the tree amplitudes for the scattering of the "ground state" (tachyon) scalar stnng modes are given by [4] (see also [6, 10]; cf [11])

GN(c~,,...,C~N)=(kOll d2XkX/-g(-Z~k)6(°)(~bk--~(Xg))>.

(3)

Here ~b~, are the coordinates of N points In M ° and M 2 is assumed to be a closed simply connected manifold As was found in [4, 6, 10] GN reproduce the VlrasoroShapiro amplitudes in the case of D = 26. Introducing a scalar "source" field ~(q~) it is easy to write down the expression for the generating functional corresponding to GN:

F(°)[cl)]=(exp[-f d2xv/-gcp(q~(x))]), G,(4,,,..,~,~)

-8 8m(4,,).-. ~m(4~) ~,:o"

(4)

(5)

The crucial point is to observe that F(°)[q )] is just a "tree" effective action for the scalar field q) which corresponds to the "ground state" mode of the (closed) Bose string. Eq. (5) gives the amplitudes (more correctly, irreducible Green functions) on a "naive" vacuum 4, .. = 0, G o = ~u The true vacuum value of • (and all other fields to be introduced shortly) is to be determined by minimizing the full effective action, thus, hopefully, solving the "tachyon p r o b l e m " (see sect 4) Now it is obvious how to generahze (4) to include fields corresponding to other closed string excitations: we are simply to add all other possible "source" terms

4

E S Fradkm, A A Tseythn / Quantum string theory

which preserve the reparametrlzation invariance

1 it*, +~le 0~ z Ov~j A,j(~p(x))+

VgRC(~(x))

+ x/ggit~g~"0it~p'0~tpJ~~p ko,~ tB,jk,(~(X)) + . - " ] .

(6)

The symmetric tensor H,j is a "source" for the "graviton" modes [12], the antisymmetric tensor A o is a "source" for the anttsymmetrtc 2-tensor modes [13], while higher tensor fields hke Bukl correspond to " s p i n > 2 " massive modes*. C is a "source" for the "dilaton" - a massless scalar of the closed strmg spectrum (R is the curvature scalar of git~)*. Eq. (4) is true m a "first quantlzed" string theory**. To account for processes with a "cubic" interaction of virtual strings (one string splitting into two and two strings recomblning into one) we are to sum over all closed oriented manifolds with arbitrary number of handles n n is thus a number of "loops" m the full "second quantized" stnng theory (note that n is the only topological characteristic of such 2-dimensional manifolds) As a result, we are led to the following expression for the effective field theory action corresponding to the "second quantized" string theory

F[~, Go, Ao,...]= I=Io+Xs

~ e "x [ n~0

[dgit,,] f [d~'] e -~x/~),, J

JM

(7)

.....

= f d2xx/g~(~)+ l---f 1 f d 2 x [ 1~ g +--27ra' +

f

itv

1 its" Oit~p~0~ j G,~(~)+~ze Oit~pl O~pj A,j(~)]

l j d2xV~g itt, g A p O~O~q~O~

k

Op~B,jkl(~)+" ", (Ao-->21ra'A,j) I

(8)

Here X = 2 - 2 n is the Euler number of M~ and Gv=-~,~+2rra'H,~ is an a priori arbitrary metric of the external space-time M D A dimensionless constant or plays * To have c o r r e s p o n d e n c e w i t h the s t a n d a r d d d a t o n e m i s s i o n vertex [14, 15] ( ~ 0 t p ' 0 ~ ' e *p ~) we are also to redefine H~j or the metric G,j m eq (8) G,~ --> G',~ e x p ( 4 C / ( D - 2)), w h e r e G',~ has a flat-space hm~t W e s h a l l n o t d~scuss the d l l a t o n c o n t r i b u t i o n m the effective a c t i o n m the m a r e b o d y o f th~s paper, a s s u m i n g C = Co = const in the v a c u u m a n d a b s o r b i n g C O m tr m e q (7) (see, however, the end o f sect 4) ** A closed s t m p l y c o n n e c t e d surface M ~ can be c o n s i d e r e d as a w o r l d sheet o f a w r t u a l string w h i c h a p p e a r s at s o m e point, p r o p a g a t e s a n d then d~sappears at a n o t h e r p o i n t

E S Fradkm, A A Tseythn / Quantum string theory

5

the role of a coupling constant in the theory*. The choice e ~ for the weight with which different topologies are summed seems to be unique and is distinguished by the fact that tTX can be represented as a local addition to the action (8). In fact,

1

X =4~" f d2x x/-gg.

(9)

Here R = R ~ g ~ and R ~ p = 0 ~ F ~ , - • • is the curvature of M~ It is important to stress that the structure of (6) or (8) is quite simple, we have written down all possible local "source" terms that respect covariance in two dimensions. The only "external" gauge lnvariances of (8) (and hence of F ) are the general covariance in D dimensions (which follows from 2-dimensional covarlance) and the abehan gauge symmetry for the antisymmetric tensor, t~A~j= 0~Aj-0jh~, a~ = 0/0~p ~. Hence the only fields which can be "massless" in /" are the metric G,j and the antisymmetric tensor A o. "Higher-spin" tensor fields B~kl , . . must be massive because of the absence of corresponding gauge symmetries necessary to provide their masslessness in F. In this way we deduce the structure of the free closed string spectrum without use of any a priori knowledge about it To determine F as defined by (7) we are thus to compute the partition function for quantlzed strings propagating on an arbitrarily curved space-time M D and interacting (in addition to gravity Go) with the infinite number of local fields. ~, A,j, B,jkl , The important property of (7) is that it can be represented as an integral over the space-time M D The reason is that the free string theory is insensitive to a position of a string "centre", i.e. the action (1) is lnvarlant under tp'(x) --> ~o'(x) + a', a ' = const Hence a free string partition function contains the corresponding zerom o d e contribution (the volume of R D) as a factor. This translational invariance is broken by the "external" fields in (8) so that a D-dimensional zero mode integral is no longer a trivial one. It is useful to extract this integral over a "string centre" collective coordinate from the very beginning by splitting ~o' into constant and nonconstant parts: ~o'(x) = ~b'+ ~ ' ( x ) ,

I

~b' = const,

FIll= I d% f

hi,

[drl] = dn ,~D)(p,[~, rl])Q[~, rl], Q = det c~P'[r/+ Oa~ a] o~o

(10)

Here P ' = 0 is a "gauge condition" breaking the invariance under ~ -~ n + const to avoid overcountlng and Q is the corresponding "ghost" determinant Using (10) • A s w e w i l l s e e m s e c t 4 ~r ~s n o t s u b j e c t to a r e n o r m a h z a t l o n , should be posmve for convergence of the sum m (7))

~ e ~t h a s a f i x e d v a l u e ( n o t e t h a t cr

6

E S Fradkm, A A Tseythn / Quantum string theory

we get f r = J d % C4B-T~(~)~(~(6), ~,q)(6), ~,~,q)(6), --, Gv(6),~k,m(~b), . . , F o k ( ~ ) , .

,

),

(11)

where ~ depends on all powers of derivatives of fields (multiplied by powers of a'). The gauge invarlances tmply that the derivatives of G o and A,j combine mto k k n the curvature tensors ~ kom = c3jF~m + F . j F , , . - (j ~ m ) and FIj k = 30[,Ajk ], and that all derivatives are covariant with respect to the Christoffel connectmn F~k(G) T h e factor v/-G(&) comes from the expansmn of the covariant measure in (7) 11~ dq~'(x)~/G(~ (x)), G = det G,j* The representation (11) shows that it is F defined by (7) (and not, for example, In F) that is the effective action The lagrangian in (11) ~ - - ~ e " f , , [dgz,] f [dr/'] exp { - ~1I [ t b + */, g,~, ~ ( ~ + T/), . ])

(12)

is expressed in terms of the path integrals over the "internal" string degrees of freedom so that ~ is effectively "non-local" m 4) (this "smearing" may be responsible for the finiteness of ~)** The fields qb, Gv, A,, .. can be considered as some "bound state" excitations of the string degrees of freedom. Thus we get an unusual type of an induced ( g r a v i t y , . . ) theory where the effective fields (metric, ) are not to be further quantized The point is that loop effects of an approximate field theory (valid at energies E .~ (a') -~/2) are automatically accounted for by the string theory loop corrections*** 3. Integration over coordinates

Our aim is to determine the leading terms in a "low-energy" ( a ' ~ 0) expansion of F in (7), (8). According to (12), we are first to expand the fields near ~b' =const and to integrate over "fluctuations" 7/' Integration over 2-dimensional metrics will be carried out in sect. 4 At this first stage the 2-metric g~,~ is considered as an arbitrary background field (in this section we may not also speciahze the value of the Euler number of M 2) The corresponding "first-stage" effective action is given by e-(I/h>wl~.,~.o, l= j [dn] e (I/h)1~+.7.g,.v.'~(~+,~). I

(13)

* The m e a s u r e [dg~,v] is defined as m [4, 8] In p a r t i c u l a r we & v i d e by the v o l u m e o f the full & f f e o m o r p h l s m g r o u p o f M 2 (see also sect 4) ** As ~s c l e a r from (12) all n o n - p o l y n o m l a h t m s m ~ arise as a result of "'virtual c o n t a c t e x c h a n g e s of the m f i m t e n u m b e r of the string m o d e s " , cf [16] ***The m f i m t e set of m o d e s p r o p a g a t i n g m the string t h e o r y l o o p s Is In c o r r e s p o n d e n c e with the set o f " e x t e r n a l " fields, ~, G,j, A w

E S Fradkm, A A Tseythn / Quantum string theory

7

It is useful to start the analysis with the formal case when all the "external" fields except the metric G,j are set equal to zero Then (8), i e. the action for a string propagating m a curved space-time" I~ =

,f

d2x ~/g½g~'O.~'O,,~o'G,j(~)

(14)

exactly coincides with the action for a generalized o--model defined on a curved 2-dimensional space M 2 (the "internal" space of the ~r-model is M D with the metric (;,1; note that the roles of M r' and M 2 are reversed on going from a string picture to a or-model picture) Hence (13) is the partlUon function for the o--model on a curved background g~,~ Being interested in the a ' - ~ 0 expansion of F we may compute W perturbatively in h ( a ' h counts the loop order in (13)). To make perturbation theory manifestly covariant in D dimensions it is convenient to do a local change of the quantum variable ~7' -~ ~r'(~7, d~), introducfng the geodesic normal coordinates with the origin at the point d~' [17, 18]" n ' =- ~ ' - 4,' - ¢ ' -½Ck(~b)¢'¢ k +"

".

(15)

Making sr' dimensionless by the rescaling se'-->(2~ra')~/2~ :', we obtain for (14) (see e g. [19])

Ic =

f

1 i O,,¢3 G,,(qb)-g(2~ra 1 p)~,kA(~b) d2x x/-gg ~ v (~0~,¢

x ¢k1~'¢'0~,¢'0~¢' + O ( a ' 2 ~ z ) } .

(16)

Higher-order terms in (16) have the following structure:

f d2x x/gg~'~(a'¢2)P+~q(~O~Ple~O~,¢O~¢ ,

(17/

so that (16) corresponds to a theory with the infinite number of dimensionless coupling constants (which are powers of the curvature of G,j and its covariant derivatives taken at the point ~b and rescaled by powers of a'). Observing that ~:' transforms as a vector under the point transformations of ~b' we can go to the orthonormal basis introducing ¢~(x)= e~( qS)¢'(x), G v = e, eja a = 1 , . . . , D. Then (16) takes the form 16 =

1 a a d 2 x~/gg p.v {~0~,~ 0~

(~..

I --

a

c

b

d

-g~abca~ ~ O~z~ 0~,~

~ 27ra'~

)

+O(a 13)} (18)

A convenient covariant choice of the "gauge" in (10) is [18]: pa = ~ d2x v ~ : ~ (x) = O (It eliminates the constant zero mode of the scalar laplacian on a compact space

8

E S Fradkm,A A Tseythn/ Quantumstringtheory

M E) Throughout this paper we ignore various local factors (-8(2)(0)) like det (0~'/0~') or Q (as well as various quadratic infinities) which are cancelled by a proper choice of the functional measure (being quadratically divergent they in any case vanish in dimensional regularlzation). For example, the quadratic infinities corresponding to the theory (18) are cancelled by the term ( ~ A 2 ~ d E x In det G(~b + 7/(~, ~b))) coming from the covariant measure [de*] (A-->oo is a cutoff). The classical action (18) is lnvariant under the Weyl transformations g~-> AE(x)g~ as well as under the coordinate transformations of x ". To preserve the latter invariance we use dimensional regularlzation which breaks down the Weyl lnvariance ((18) is Weyl invariant only at d = 2 ) Covarlance and Weyl anomaly considerations (see [20, 21]) suggest that W has the following general structure

W[g' G]=l[3 f Rx/g d2x + y f (Rx/~)x[]~'(Rx/~)x'd2x [3=4a,

.y=al+a2ht~,~+T3ha,2(~ . . ) 2 + . . .

(19) (20)

Here R and ~ = ~ ~ab are the scalar curvatures of g ~ and G,~ respectively, e = d - 2-> 0- and --1 [~x[]xx,=~(2)(x-x'), E]=-O~(x/gg/.*v0~). (21) The first term in (19) is the "topological" ultraviolet mfinity (infrared mfinities are absent because of compactness of M2). The second term m (19) corresponds to the Weyl anomaly ( T~ = (2/x/-g)g~(SW/~g~) = - 4 T R ) . In the one-loop approximation for the two-dimensional theory (13), (18) W in (19) is simply the effective action for D free scalar fields on a curved 2-dimensional background. Hence y = D/96¢r (see e g. [21, 4]) To determine the two-loop coefficient in (20) one may expand the metric near the flat background g ~ = 8 ~ + h ~ and compute the "self-energy" graphs for h~,~ (fig 1) using momentum space representation and the infrared cutoff by the expliot mass term (the contributions of the first two diagrams of fig. 1 mutually cancel) No 1/e 2 or non-local 1/e infinities appear in W= [d2p/(2cr) 2] × h~(p)K~p~(p)hp~(-p) (no counterterms are to be accounted for in computmg W, their use would be equivalent to a non-trivial renormahzation of G,j(q~) m (14) making the effective action F ambiguous)* As a result we reproduce the h ~ term in the expansion of (19) with y(2)=_~/128¢r2 Thus in the

Fig 1

* In order to detect the mfimty m (19) one assumes the background field ~bto be non-constant

E S Fradkm, A A Tseythn / Quantum string theory

9

two-loop approximation 3' =

D 96¢r

hoe' ~ (6) + O(a'2h2~2). 64~r

(22)

(This result being dependent on a ' ~ (i.e. on the square of ratio of a string "radms" to a space-time "radius") disagrees with the expression found in reE [22] where a particular case of M ° = S ° was considered.) Having found Wig, G] we are to integrate e -w/* over the metric g,,~. This will be done in the next section using the methods of refs. [8, 23, 24]. As a result the a ' ~ term in (22) will produce the Einstein-like term in the effective action F. Now let us include in W the contribution of the antisymmetrlc tensor A,~ (the contribution of the scalar field 45 will be discussed in sect. 4) The relevant piece of the action (8)

I~,A = 2--~a, f d2x{½./gg'*%,.~' o.~o'G,,( ~ ) I ~ +~ze 0~.q~JO.~pJ A,s(~o)}

(23)

is exactly the action for the generalized or-model with a generalized "Wess-Zumino" term. In fact the second Kalb-Ramond [13] term in (23) coincides with the so-called Wess-Zumino term [26, 27] for the special choices of M ° and A,j(~) corresponding to the "standard" o-models Expanding ~o' near 4~' according to (10), (15) we find that in addition to (18) the action (23) contains also the terms depending on Fuk = 30t,Ajkl, ~,kl and their covariant derivatives The leading contribution in (20) comes from the following term in (23): 1 f d2xe ,u.v-61 F.bc(¢)~: a O,.~:b 0~: c ,

P,,bc = (2~'°t')'/2Fabc

(24)

Computing the two-loop h~,~ "self-energy" diagrams with two o~'l/2F vertices (fig. 2) we conclude that the total result corresponds to (22) with the following substitution

+ ~ -~F,,kF ug

(25)

(Note a connection of this result with that found previously (on a llnearized level) in the context of the old dual string model [14], and also with recent work [28, 29] on the generalized or-model (23).)

Fig 2

10

E S Fradkm, A A Tseythn / Quantum string theory

One can give an alternative derivation of eqs. (19), (22), (25) using the fact that the Weyl mvariance in two dimensions makes it possible to establish the exact expression for the free scalar Green function [4, 23]. Introducing the conformal coordinates in which g~,~= e208~ (p is not regular on M 2) we have (projecting out the zero-mode contribution and taking X = 2)

,1[

[] 2~' = ~

In ( x - x') 2 +

]

e -°(x)-°(x') .

(26)

Here .4 ~ o o is a covariant cutoff. As follows from (18) the two-loop contribution in W is given by W(2) = v [~ x -1, , ' - V --1 .V. x. .~. -1~',x lim ([~ xx,V - i xp V x' - xl"q ~,~ ~,' x-~x'

(27)

The result of (27) is the same as in (19), (22) (R = - 2 e-Z°v]p). The analysis of this section can be generalized to the case of the Fermi string theory with the classical action mvariant under the N = 1 two-dimensional local supersymmetry [9, 30, 31]. Coupling a free Fermi string to the external metric G,j amounts to constructing a locally supersymmetric generalization of the N = 1 supersymmetric generalized or-model action (see e.g. [32]). T I h e result (of. ref. [33] for the N = 2 case) appears to be a straightforward combination of the actions of refs [9, 30] and [32]. It is possible also to include the coupling to A o using the N = 1 supersymmetrlc extension of the action (23) constructed in [28] (see also [29]) Less clear is how the Fermi string theory (apparently lacking D-dimensional supersymmetry) can be coupled to a sort of ten-dimensional supergravity anticipated as its a ' ~ 0 hmlt (cf. [34]). Such coupling is perfectly possible for the covariant superstring action of ref [35]. Thus generalization of the above treatment to the superstring case remains an interesting problem for the future (m particular, it is important to understand whether a superstnng action with "sources" can be transformed into a sort of supersymmetric ty-model action)

4. Ground state problem (integration over 2-metrics; solution of effective equations)

Given the effectwe field action (7) we can study the problem of a ground state in the closed Bose string theory. Namely, we first have to compute F [ ~ , G,. A , . B,jk~ • .] ( B stands for all higher-spin tensor fields) for arbitrary arguments and then find the effective mean values of fields, i e. the solutions of the effectwe field equations ~F 6 ~ = 0,

6F 8---G= 0,

6F 6A = 0,

6F 6B = 0.

(28)

E S Fradkm, A A Tseythn / Quantum string theory

11

Assuming that the ground state space-time manifold should possess some global isometries (e.g. Poinca~e or de Sitter) we have to look for solutions of (28) with the tensor fields B,~kL equal to zero. As for Ao, it may be non-vanishing in the vacuum with Fabc ~ e - ' a b c (a, b, c = 1, 2, 3) thus a priori distinguishing "compactification" to 3 dimensions [36]. Hence the ground state is determined by the expectation values of the three "lowest m a s s " fields q~, G o and A o. It is easy to guess the general structure of the first several terms in the expansion of F in (7):

r [ @, G, A] = j d ° ab ~f-d{ V( q~) + a'f,( @ )(O,@ ) 2 + a'f2( ~ ) ~ + a'f3( q~)FokF°k + O(a'2)}.

(29)

All the indices are contracted with the help of Go(~b ) To establish the ground state values q~o, Go, Ao one has thus to compute the functions V and fn. A check of the absence of ghosts in the theory will be the physical relative signs of the "kinetic terms" in (29), while the stability of the ground state under small fluctuations will be equivalent to the absence of tachyons in their spectrum. Thus we get a consistent field-theoretic formulation of the "tachyon p r o b l e m " in the Bose string theory It was anticipated previously that the presence of the tachyon in the free string spectrum indicates that the expansion goes near a " w r o n g " vacuum and hence a sort of "spontaneous symmetry breaking" (generation of condensates of some scalar modes) is needed to establish the "true" one (see e.g. [37]). However, an off-shell method for the solution of the ground state problem (analogous to the effective action method for the study of dynamical symmetry breaking in a field theory) was not developed. A circumstance that was also Ignored in the previous approaches is that the flat metric G o = 8 o and A 0 = 0 may not correspond to the true vacuum of the theory. It is of course necessary to have a flat space vacuum if one wishes to apply the theory to hadrons but given that the free spectrum contains the massless "gravlton" and antisymmetric tensor the question about the ground state metric and A,j should be solved dynamically without prejudices about possible relevance of the theory to flat four dimensions. Recalling that the theory has dimensional parameter a ' it is natural to anticipate that being computed for a ground state value of qb F in (29) should contain a cosmological term (which is not ruled out by the symmetries of this theory) of natural value A ~ a '-1 Hence the vacuum space-time (or at least a "factor" of it) is likely to be an (anti) de Sitter space with a characteristic scale of the Planck order. We stress that it is unnatural (m the absence of supersymmetry) to hope that the ground state metric may a p p e a r to be flat. Given that stability criteria in a curved space are in general very different from those In the flat space we conclude that a resolution of the "tachyon p r o b l e m " is essentially connected with the question of the background space-time geometry

E S Fradkm, A A Tseythn / Quantum string theory

12

It seems likely that the ground state problem should have a definite solution already in the "tree" (i.e. first quantlzed string) approximation Hence we shall compute F for the case when the integration in (7) goes over closed 2-dimensional rlemannian manifolds without handles ME2, i.e. with the topology of the sphere S2(x(M 2) 2). Even the treatment of this simplest case is full of technical details which we shall partly omit here (some basic material about computation of a free partition function, i.e. F[0], can be found in refs [23, 24, 8]) Separating the constant part ~' of q~' as in (10), (15) we get (cf. (12)-(15)) :

r[q',G,A]-f d% f [dg..]f [d¢']exp{-Id2x./~(4,,¢) 21c , f d2x[½x/-gg~'~O~,~'tg~JG,j( 4))+. .]} (30) The integral over ~:' goes over the regular non-constantfunctions on M 2 (satisfying S d2x q ~ : ' ( x ) = 0) and

q~(4,, ~) --- q'(6 + ~(@, ~)) = 'P(4,) + (~,q')~¢ + ~2 ~--~!(~,,''" ~, q~)~,~¢"

•

(31)

(9, are covariant derivatives corresponding to G,j). Integrating over ~', we find the "effective action" W (defined in (13)), which depends on ~(~b), 0,~(~b), . . , G,j(~b), ~'jkt(~b), • • •, F,jk(~b),... ( W = ~(~b) S d2x x / g + . . . ) Simple power counting reveals that W contains ultraviolet infinities, i.e. depends on a cutoff A o f the field theory defined on a fixed M 2. For example, ignoring various mixing terms, one finds

W~o(~) =- h~,= l E1n ( 2or'" rt~2~.~. . f dEx ~/~+ n) ||L\ ] lob

•

,

(32)

where 1/e = - I n A Hence W is not "unambiguously calculable" in the Bose string case we consider* It appears likely that W will be calculable in the superstnng case where the "tachyon field" • is absent and infinities in other sectors have a better chance of cancelling The apparent cutoff dependence of W may look hke a serious problem of the Bose string theory. There are two possibilities of overcoming this difficulty. A naive • P o w e r c o u n t i n g i n d i c a t e s t h e p r e s e n c e o f mfimt~es a l s o m o t h e r field s e c t o r s F o r e x a m p l e , in t h e c a s e o f a " h i g h e r - s p i n " field Bvk t w e h a v e t=

II

2,n-~ t

d 2 x ~½(as¢)2 +

+ ~.B,,kj~"+

I

d2xff'ff[B,jkt(Ck)

]~'~,~0~¢~ +

and hence

W~(B)~ e-'-~Bmj R2x/gd2x+

E S Fradkm, A A Tseythn / Quantum string theory

13

one IS to " s u b t r a c t " the infinities by redefining the field (~--> • - a ' In A~Et~-I-- • .) In each o r d e r o f expansion in a ' (a redefinition ~--> ~ + c A 2 l S In any case needed to absorb quadratic infinities). This p r o c e d u r e makes W finite but d e p e n d e n t on an infinite n u m b e r o f arbitrary ( " s u b t r a c t i o n - d e p e n d e n t " ) constants. A n o t h e r more appealing possibility is connected with the existence o f some natural "built-in" m e c h a n i s m leading to a well-defined F as a functtonal o f fields rescaled by powers o f the cutoff so as to make them dimensionless ( 4 --> A 2 t ~ , . . ) The basic observation is that the rescallng o f the cutoff should be equivalent to a rescahng o f a twodimensional metric g ~ But g ~ itself is an integration variable in (7). Hence all the cutoff d e p e n d e n c e (left after the rescahng o f the fields) should be absorbable in a formal redefinition o f the Integration variable, and hence should be absent in F. There o f course m a y be additional infinities c o m i n g from the integral over metrics itself. F o r example, the integral over a " s c a l e " o f g ~ m a y diverge at lower limit However, this integral should necessarily be cutoff at 1/A because a (covariant) short-distance cutoff 1/A -->0 was already introduced in the theory defined on M~ with a fixed metric As a result, this integral will be automatically convergent after the rescaling o f the metric discussed above. That this second a p p r o a c h is sensible can be seen on the example o f a free partition function or a "naive effective potential" F [ ~ =/x 2 = const, G,j = 60, A,j, .. = 0] = (exp ( - / z ~ S dEx x/if)), which as will b e c o m e clear (see also [24, 8]), is a well-defined function o f q~ = ~ A -2. It is interesting to note that if this m e c h a n i s m is indeed operative (as we shall see) no actual renormalizatlon o f tr In (7) IS needed at all so that o- is a fixed coupling constant o f the string theory (eft [8]). Being interested in the lowest-order terms in (29) we can ignore higher-derivative couplings in (31) and carry out explicit (one-loop) integration over so: W = qb(~b) f d2x x/g+½ In det (8,sA + 2 7 r a ' ( ~ , ~ s ~ ) ~ ) + . . . , 1

A=--~gO~,(g~'~q'gO~) ~

v/~

.

(33)

There is no contribution from the linear term because I d2x 4 g ~ ' vanishes for a class o f functions on which the ~:' integral in (30) is defined, namely for the regular functions e x p a n d a b l e In elgenfunctions (with non-zero eigenvalues) o f the Laplace o p e r a t o r on M~* Thus, to the lowest order in a '

W = ~(~b) f d2x x / g - z r ~ ' ( ~ 2 ~ ) ~ f d2x v/-~Dx: + . . . ,

(34)

• This m fact is a subtle point Assuming first that (0,~)~ = J, is non-constant on M2, integrating over and then taking the limit J-->const one finds the additional term in (33) ~ra'(O,~)2~dZx d2x ' x/'g[Z~,x/~, However, the function ~o~ []-lJ needed to compute the gaussmn integral by a shift is not regular on ME For regularity of the hmlt one has to take [~-~ with projectors on non-constant functions so that the final result vanishes

E S Fradkm, A A Tseythn / Quantum string theory

14

w h e r e [7~-~x, = 6<2)(x-x ') I n t r o d u c m g a p r o p e r - t i m e cutoff 1/A2-> 0 w e get (cf.

[4] and (26))

l(,nA~)fdZx4~+lf

f d2x x/~[N-'xx = -4--~

~

d2xd2x'4-~[2x'x,(Rv~L,.

(35)

Hence,

W= Woo(O)+,~ f d~x 4~+¼,~' (lna~)~2~ f d~x 4"ff - ~1 t,'~2~'¢. ~ ~

I

--1 1 t (In a : ) ( ~ - I ' ~ F :) d2x d 2 x ~ v~Gx~,(R,/-ff)~,+~

+ 0(~'~),

~,o~- (D -26)/96~r

(36)

H e r e F 2 = F,jkF 'jk a n d W~(0) = - ~ ( D - 2 6 ) In A 2 ( n o t e t h a t ~ R,g-g d2x = 87r)*. In (36) we a c c o u n t e d for the l e a d i n g g r a v i t a t i o n a l a n d a n t i s y m m e t r i c t e n s o r c o n t r i b u t i o n s a l r e a d y f o u n d m (22), (25). W e a l s o i n c l u d e d the c o n t r i b u t i o n o f the g h o s t o p e r a t o r c o r r e s p o n d i n g to the g a u g e g~.~ = e2°ff~,~ ( w i t h ff~,~ b e i n g a metrxc o n S z) F o r g ~ = e2pg.v **

1 ~~ r;-,-,~2 At. -~,~ ~, f d2x d2x' g-~[];2,(R,/-~)~ + l ,~'(ln a~)(~-l~F~)

[

'

+ ~o-6-~(~-~F2)1

= [](~),

f d~x d2~ ' (R~)~E3;~,(R~)~, /~ = R(ff)

All the cutoff d e p e n d e n c e m (36) can b e a b s o r b e d (g~.~ = A - 2 ~ . ~ ) a n d the s c a l a r field (q0 = A 2 6 ) :

(37) in r e s c a l i n g o f the m e t r i c

* To obtam this value of Woo(0) one has to correctly subtract the contrlbutton of the stx zero modes (corresponding to the conformal Killing vectors) of the ghost operator - V ~2 - ~ R1 g ~ and to define [dg~,~] m (30) by dividing the formal measure by the volume of the dlffeomorphtsm group (see [8]) ** It may seem that S RU ~R should not change under the constant scahng of the metric However, this symbol m (19) should be understood precisely according to (37)

15

E S Fradkln, A A Tseythn / Quantum stnng theory

Expanding e - w in powers of t~' we obtain

F[CI),G,A]~-l f dD4)x/-G-(qb)f [dp]e-Wo x (1-aa'[ ~2cl)f d2xx/-ff+(~-~F2)] RD

Wo=W(O)+4f d2x,/-+7of Rt3-1R,

},

(38)

a ---~ln A z .

Here

g~l, -- (1 -l-X2/4r2) 2'

f RD-'R=-f d2xd2x'~xRxt~:~.~xRx,, f []-lR==-I d2xd2x'x/~[]~,Rx,v~x,. p-integration goes over regular functions on S:. Note that (38) does not contain additional integrations over parameters of "Telchmuller deformations" of the metric [8] because any traceless deformation of a metric on M22 can be generated by a local dlffeomorphlsm. The integrand in (38) is mvariant under the diffeomorphisms that preserve the gauge condition g,~ = e2pff,~. They are generated by the six conformal Killing vectors (three SO(3) rotations and three proper conformal boosts) The corresponding global transformations form the conformal group SO(2, 2) ~ is the (infinite) volume of this group which cancels the analogous factor coming from the integral over p (p is mvarlant under SO(3) but is "shifted" by the conformal boosts) Let us first investigate the case of 70 = 0 or D = 26. The free theory (qb = 0, G o = 8~, Av, . = 0) is then Weyl lnvanant, l.e p is a gauge degree of freedom. The corresponding partition function F[0] then diverges and is to be regulated by introducing a gauge In analogy with philosophy adopted in ordinary gauge theories ~t may seem natural to insert the Weyl gauge (19 = 0) also in the presence of "sources" ("external" fields) even though they formally break the Weyl symmetry In any case, the integral over p is not a well-defined one for 3'0--0 and hence some prescription of how it is to be evaluated is to be adopted. That the gauge-fixing prescription is a reasonable one IS seen from the fact that using it one can prove that the amphtude

E S Fradkm,A A Tseythn / Quantumstring theory

16

(2) r e p r o d u c e s the V i r a s a r o - S h a p i r o (VS) a m p l i t u d e for D - - 2 6 * P u t t i n g / 9 equal to zero m (38) a n d c o m p u t i n g all the integral with the S2 metric ( = ~ . ~ ) we finish w~th the result

r(26)[~,G,A]~__MO f dOqbv/--~e-4,[1 - z 1a l a , ~ 2 - + o t ' b l ( ~ - ~ F z) + . . . ] ,

(39)

where al a n d bl are constants, • -- qb S x/-~d2x a n d M is h n o r m a h z a t i o n mass that can be t a k e n e q u a l to (27m') 1/2. H a v m g c o m p l e t e l y " o m i t t e d " the integral over the metrics, we c a n n o t a p p l y the n a t u r a l " m e c h a n i s m " for e h m l n a t i o n of infinities described above. Thus in (39) we assume that al is a fimte b u t " s u b t r a c t i o n - d e p e n d e n t " constant It m a y a p p e a r that p = 0 is a too stringent condm~on. I n fact, it m a y be destrable to preserve the " s m a l l " c o n f o r m a l g r o u p (see previous footnote) u n d e r which ~(,).gs~ are the p r o p e r c o n f o r m a l Killing vectors), 1 e to fix the Weyl gauge o n l y for the " t r a n s v e r s e " directions. T h e n the (fintte-dimensional) integral over the " p u r e c o n f o r m a l gauge" p's will r e m a i n a n d o u r " m e c h a n i s m " will be applicable, We shall &scuss this more rigorous a p p r o a c h later b u t ~t is mstructlve first to a n a l y z e the g r o u n d state p r o b l e m for the " m o d e l a c t i o n " (39). I n t r o d u c i n g the new d i m e n s i o n l e s s scalar f i e l d / 2 = exp (-½43) we can rewrite (39) as F~26)[ qb, G, A] =

- M

o

f dD qb v/-G[122- a'a,( a,O ) 2

+ a'b,(Yt - ~F2)O: + O(a ,2)].

(40)

H e n c e if we ignore the metric a n d the a n t i s y m m e t r i c tensor b a c k g r o u n d s we get a free scalar field which is a ghost for a~ < 0 a n d a t a c h y o n for al > 0. I f a~ > 0 (as we shall a s s u m e ) all the three " k m e t l c t e r m s " m (40) have the physical signs. C o r r e s p o n d e n c e with the free strmg spectrum** suggests that al = ~, t h o u g h the choice of this p a r t i c u l a r value will not be i m p o r t a n t for the s u b s e q u e n t discussion A naive flat-space v a c u u m /2 = 0 ~s u n s t a b l e due to the t a c h y o n l c n a t u r e of the fluctuation mode. Let us see if the full action (40) admits a stable g r o u n d state ,

)

s

• In the momentum representation GN(pl, PN = (~Ik= 1 J d2XkV~xkeZ'~l'k)Pk) Integration over produces 8(~p) Inserting [I x 8(p(x)) we get .lust the VS amplitude but without the mass-shell condition a'p 2 = 4 This condition is recovered under additional requirement of conformal invariance (duality) It was noted m [10] that using a different prescription for integration over P(Xk) one can obtain in GN the pole factors lqk (4- t~'p~)-~ in front of the VS amplitude However, this prescnpUon is completely ad hoc The only important fact is that the GN with ,O(Xk) integrations included is lnvanantunder the conformal SO(2, 2) transformations for arbitrary p2 = _ m2, while the VS amplitude is conformal invariant only at the point p~ = 4/a' Hence the only effect of the conformally invanant removing of the p-integrations should be restriction of the VS amplitude on the mass shell This justifies the use of the conformal invariance reqmrement m addition to the gauge-fixingprescription • * This correspondence is not obvious in fact The naive vacuum seems to be • = 0 and not/2 = 0 (cf (2), (14)) Note, however, that /2 depends on a "subtracted" value of • and hence a direct interpretation is complicated due to the dependence on the cutoff

E S Fradkm, A A Tseythn / Quantum strmg theory

17

configuration. Suppose we look for maximally symmetric vacua, so that the vacuum value o f / 2 is /20= const # 0. Then the classical field equations corresponding to (40) take the form

1 + a ' b l ( ~ - - 1 F 2 ) = 0, ~lJ

1 mnFjrnn = 0 , -~F,

~ , F 'Jk = 0 .

(41)

The value o f / 2 0 is thus arbitrary. It is easy to see that there are no solutions with F,jg = 0 No solutions exist also if G,j has the euchdean signature (so that F 2 > 0). Hence we deduce that the D-dimensional space M °, where strings are defined, must have at least one "time" direction. Only one "time" is conststent with the absence of ghosts. Solutions with a maximal symmetry are obtained m the case when F,jk ~ e,jk in some two space and one time dimensions. Thus M ° = ~3 × MO-3 where ~3 is an anti-de Sitter three-d~menstonal "space-time" (with s t g n a t u r e - + +), 1 e ~ab = --½f2oGab,

f2o = 1 / a ' b l ,

Fabc = foeab,,

(42)

a, b, c = O, 1, 2.

If Fzj k has all other components equal to zero, M o - 3 is an arbitrary Einstein space with a zero cosmological constant, i e. ~mn = 0 ,

m,n=4 ..... D

(43)

(We assume that G,j is block diagonal) Hence the maximally symmetric case is M D-3 = R °-3. Less symmetric solutions are found if f j j k ~ 8ok for some 3-dimensional subspaces of M o - 3 . Then M °-3 sphts on a product of compact S3 factors"

2

f o-•f

2

, = 1 / b l a ' > O.

(44)

n

The general solution corresponds to MD equal to a product of ~3, of N S3 factors and of some (dim = D - 3 ( N + 1)) Einstein manifold satisfying (43)*. We see that the Bose string model prefers the three space-ttme dimensions wtth the anti-de Sitter metric and the non-vanishing antisymmetrtc tensor field strength. Direct analysts shows that there are no tachyonic modes in the spectrum of fluctuattons of the fields near thetr vacuum values Thus m contrast to the " n a i v e " / 2 = 0 vacuum the true vacuum is stable. Note that the non-zero vacuum value o f / 2 imphes the non-zero constant vacuum value for • ~ - I n / 2 * * . Thus our conclusion is that generation of non-trivial background values for ~, G o and A,j formally solves the "tachyon problem". * In contrast to the Freund-Rubln mechamsm [36], the presence of ~ makes it possible to have flat space-hke subspaces ** Note also that the vacuum value of the action (40) is equal to zero and that the value of O determines the value of the gravitational constant

E S Fradkm,A A Tseythn/ Quantumstnng theory

18

This result (if true also for D < 26) seems to rule out a direct application of the closed Bose string theory to a description of glueballs but is quite natural from the "fundamental" point of view on a string theory as a can&date for a theory of all interactions. In a more realistic superstnng case other antisymmetrlc tensors will be present in (40) (while O will be absent) and hence more reahstic patterns of a ground state compactification may exist. Here we would like to note the following problem that must be carefully analyzed in the approach to compactification based on the effective action like (30), (40). In the derivation of the action (40) we used the t~' ~ 0 approximation (and neglected higher tensor fields). However, the resulting ground state space appears to have a characteristic scale of the order of (~,)w2 Hence we have to study whether the higher-order terms in (40) are m fact irrelevant. It seems likely that it is incorrect to study compactlficatlon in the superstring theory starting only with a D = 10 supergravity action Maybe it is sufficient to include a number of other terms (like C t ' 2 ~ 2 + "" ") but maybe one has to invent a new approximation scheme for F without expanding ~t m powers of t~'. Let us now consider the computation of F for the general case [) <~26 (for D > 26 p is a ghost). It is useful to isolate first the integral over a constant scale of the metric and then to do a loop expansion for the remaining degrees of freedom. This is done systematically by inserting 1=So dA ~(~x/g d E x - A ) in eq. (36), i.e. by extracting the integral over the surface area A [23] (see also [24]). Starting with eq. (38) and using (37) we get oo

I"[~,G,A]~T'-lld°q~V~-GJo

---~e

I d/~ t~(f d2x V~(e2#- 1))

xe-S{ 1-½a'~2(qbA)A-' f d2x x/~ e2p[½ In (A2A)+ 1

+ ¼ '(m - fiF ) [ 1 + In (A 2A) + ~-'-~f d2x #[~o~

S-K-Kln(A2A)+-~[Id2x#(qo~+½Rold2xV~o(e2#-2~-l)].

(45)

Here K = ~ ( D - 2 6 ) and .4=Sd2xx/~ ( g ~ is the same as in (38)) The metric A

A

go~,, = (A/A)g,~,, (Ro is the curvature scalar of go, Ro = 8~r/A, [3o = [](go)) is the stationary point of the action - J R [ - 1 R under the constraint S d2x x/g= A. The integration variable ~ (a regular function on S2) has a vanishing vacuum value (the total metric is now g = e2~;go, cf. (38)) The Liouvllle-type action for ~ in (45) is positive under the constraint j d2x x/~(e 2#- 1) = 0 [38]. All the A-dependences in (45) are shown exphcltly except that coming from the dependence on go~.~~ Aff,~ In view of the Weyl invariance of []o and R o ~ o the

E S Fradkm,A A Tseythn / Quantumstnng theory

19

latter d e p e n d e n c e can be only o f the " a n o m a l o u s " type, f(AEA). Thus we see (in a g r e e m e n t with the suggestion m a d e above) that all the cutoff d e p e n d e n c e in (45) can be eliminated by introducing the new variables ,4 = AEA and ~ = A - 2 4 so that

F expressed m terms of ~, G o and A.j is mdependent of the cutoff, i.e. ~san unambzguously calculable functmnal. This conclusion is u n c h a n g e d if the integral over A formally diverges at A - - 0 because in fact it must be cut off at Am,n~ l / A 2 - > 0 . We shall evaluate F in tile simplest possible a p p r o x i m a t i o n , including the o n e - l o o p contribution from the/~ action but ignoring the ~ d e p e n d e n c e o f the pre-exponential terms. The integral over ~ then is

,~o -= A o - g o ,

1 Ao = - ~ogo0~ (g~%~o0~')

(46)

The o p e r a t o r ,~o defined on S 2 has the s p e c t r u m An = (4~r/A)[n(n + 1 ) - 2 ] , i.e. has one negative m o d e (with elgenfunction ~ = const) and three zero m o d e s (corresponding to the i n v a n a n c e o f the integrand of (45) u n d e r the (proper) c o n f o r m a l transformations). T h e negative m o d e is projected out by the 6-function in (46) while the infinite integral over the (normalized) zero m o d e s is cancelled out by the T "-1 factor in (45). T h e contributton of (46) is thus p r o p o r t i o n a l to (A 2A)~×4(det' ~0) -1/2 ' In det' Ao = - ( B2 - -

4)

In (A 2A) + const,

(47)

where the first factor comes from the normalization o f the negative and zero modes, B 2 = (1/4~r) S d2X x~o(~Ro + Ro) = 7 is the corresponding D e W i t t - S e e l e y coefficient for Ao a n d the prime and ( - 4 ) indicate that the first four m o d e s o f Ao are not included I n t r o d u c i n g ,4 = A2A and tp A-2t~ we can then put the result for F in the form* =

F ~ f dD~b x/-G f d.4 e-~'~+~'nA{l +½a'.,~2c~ x ( 1 - ½ In .4) + ~a'(9~ - ~2F2)(1 - In ,4) + . . . }

(48)

Here u = ~ ( D - 25) if D < 26, and v = 1 if D = 26 (the contribution of det' Ao in (47) is absent in the latter case, while the integral over ~'s e x p a n d a b l e in higher n > 1 elgenfunctions o f Ao is regulated by inserting the Weyl gauge). For the free case ( 4 = const, G,, = ~,,, A,, = 0) eq. (48) agrees with the result of ref [24]. * Note that to absorb the quadratic mfinmes (which we did not mdwate expheltly above) one apparently has to redefine ~ - , • + const A 2 As a result the constant part of t~ appears to be ambiguous Here it may be useful to recall once again that one can get free of this amblgmty by using a dimensionless cutoff or by properly defining the measure of the path integrals revolved

E S Fradkm, A A Tseythn / Quantum string theory

20

The integral over ,4 is also to be e v a l u a t e d in a " s a d d l e - p o i n t " a p p r o x i m a t i o n The " s a d d l e p o i n t " for the " a c t i o n " in (48) is ,4o = ~'/qb

(49)

E x p a n d i n g .4 n e a r .4o a n d integrating o v e r , ~ - . 4 o in the g a u s s i a n a p p r o x i m a t i o n for the " a c t i o n " ( p u t t i n g A = Ao in the p r e - e x p o n e n t i a l terms) we find f r o m (48)

F= cM °

x

f

dDq~ x/-G(~/1.') 8

1 - ½ a ' L , ~q) ----~(1-½1n(~/u))+½a'(~-~F2)(l+ln(~/u))+O(cg

2) ,

(50)

w h e r e 8 = - 1 - 1) 0.e. 8 = 1(19 - D ) for D < 26 a n d 6 = - 2 for D = 26) a n d c = const. I n t r o d u c i n g the d i m e n s i o n l e s s scalar f i e l d / 2 = ( ~ / ) , ) 8 / 2 we can rewrite (50) as

F ~- c M D f dD4) .I-G[/2 2 - a'(0,/2)2(a1 + a2 I n / 2 ) +~ln

,

(51)

where al = - 6 - 2 ( 1 + 8)(3 - 26), a : = - 2 6 - 3 ( 1 + 6). F o r 6 = oc (51) has the s a m e f o r m as the " m o d e l a c t i o n " (40). This suggests that the g r o u n d - s t a t e structure s h o u l d be similar for b o t h actions. A s s u m i n g t h a t the g r o u n d state value o f / 2 i s / 2 0 = const # 0 a n d tgnortng I n / 2 - t e r m s we find that the classical field e q u a t i o n s c o r r e s p o n d i n g to (51) c o i n c i d e with (41)" ~ _~2F2 = _ 4 a , - 1 ,

~ "~o - ~l r. .; . . l~mm "J = 0,

(52)

A g a i n there are no s o l u t i o n s c o r r e s p o n d i n g to the flat D - d i m e n s i o n a l s p a c e - t i m e , while the m a x i m a l l y s y m m e t r i c s o l u t i o n is ( a n t i - d e Sitter)3 x R ° - 3 with F,jk ~ %k* The stabtlity o f the s o l u t i o n s d e p e n d s on the values o f a~ a n d a2 a n d deserves special study. O u r b a s i c c o n c l u s i o n is t h a t (either for D = 26 or for D < 26) it is n e c e s s a r y to account f o r the "condensates" o f the D-d~rnenstonal metrzc and the anttsymmetnc tensor (as well as for the " c o n d e n s a t e " o f the " g r o u n d s t a t e " scalar) in o r d e r to d e t e r m i n e the true ground state o f the c l o s e d Bose string theory. It is i n c o n s i s t e n t ( f r o m the effective field t h e o r y p o i n t o f w e w ) to t a k e D - d i m e n s i o n a l space-Ume to b e flat. A n o n - t r i v i a l q u e s t i o n is w h e t h e r a s o l u t i o n o f (52) a p p r o x i m a t e s a s o l u t i o n o f the full effective e q u a t i o n s (28). This m a y n o t b e the case b e c a u s e (52) i m p l y t h a t a ' ~ = - 2 4 , a ' F 2 = - 9 6 a n d hence all h i g h e r o r d e r ( ~ a ' 2 ~ 2 + "" ") t e r m s in (51) a p r i o r i c a n n o t b e neglected. H o w e v e r , ~t ts p o s s t b l e to g~ve an r e d i r e c t a r g u m e n t that * An interesting property of this solutmn is that it "predmts" not only the dimension of the "etiectlve" space-time (three) but also its non-euchdean s~gnature This raises a hope that both the dlmensmn and the signature of the physmal space-ume may be predmted m a slmdar fashmn m a more reahst~c superstrmg context

21

E S Fradkm, A A Tseythn / Quantum string theory

(adS)3 x R °-3 with 9~-¼F2--0 may still be an exact (tree level) solution of string theory ground state equations (28). Eqs. (28) imply vanishing of expectation values of some string-theory operators constructed of ~o' and g~,~. Particular vacuum backgrounds may be distinguished by some symmetry properties implying the vanishing of these expectation values. For example, if the generalized string action (8) corresponds to a conformal invariant two-dimensional field theory then (at least in the tree approximation and for the critical number of dimensions) the expectation value of any local operator of non-negative dimension vanishes and thus the corresponding background fields solve eqs (28) (cf. [43]). The only known conformally invariant model (with vanishing fl-functlon) of the type (8) is the principal o-model with the Wess-Zumino term having a properly chosen coefficient [26] (supersymmetry offers additional possibilities [19, 28]) The simplest example is (8) with G o being a metric on S3, A 0 having Fok = f % k ( f = const) on S3, • = const and all other fields vanishing. If the scalar curvature of G o is 9~ = 6 / r 2 (r is the radius of S3) then the value o f f ensuring the vanishing of the fl-functlon is f = + 2 / r so lETok= 0 [26, 28]. Note that not only S 3 but also its "analytic continuthat 9~ _11~ 4- ok-ation" obtained by changing the signature from (+ + +) to ( - + +) without violating the relation 9 ~ - I F : = 0 (i.e. (adS)3 with Fok ~ eok ) is admissible as providing the vanishing fl-function. Thus in spite of the fact that higher-order terms in (51) are not naturally suppressed for the above (adS)3 × R D-3 solution it is likely to coincide with an exact (tree-level) ground state solution of the Bose string theory. The action analogous to (51) is found if we include the contribution of the "dilaton" field C in (8) (see ref. [42]). Taking for simplicity • = const we get in the tree approximation (at the critical dimension)

IV'~ f d D t ~ x / ~ e-2C{1 + l a ' [ N + 4(O,C): --1F2k]

(53)

+ O(ol'2)} ,

or after the Weyl rescaling G o --> G o exp ( 4 C / ( D - 2 ) )

4I

F--- --K20l '

-- ---~ K

dD4~x/-dexp D - 2 (0,C)2 _ ~ F2k e-SC / ( D- 2) + O( a')

/¢ ~ g(~,)(D-2)/4,

g = e-".

,

(53')

The second term here has the same form as the bosonic part of the action of D = 10 supergravity. The important advantage of our approach over the previous ones [3, 14] (based on S-matrix considerations) is that we are able to establish the full non-polynomial structure of dilaton couplings. Assuming C = const in the vacuum, the effective equation 8 F / 8 C = 0 implies the vanishing of the "lagrangian" in the curly brackets in (53) (and thus, incidentally, the vanishing of the vacuum value of cosmological constant). Now it is easy to prove the observation made in ref. [43]

E S Fradkm, A A Tseythn / Quantum stringtheory

22

that (in the tree approximation) type I superstnng theory always admits a vacuum solution M ~° = M 4 × ,y6, where M 4 is flat. In fact, because of 8F/8C = 0 the equation 8F/SG,j = 0 looks like (cf. (52), e -2c in (53) plays the role of/22 in (51))

If M 4 is maximally symmetric, all "matter" fields should have vanishing 4dimensional components (the argument breaks down for type IIA theory where one may have Fijkl~ Eukl). Then ~,,:,k,r=0, Z', Jl =0, . . , 3, is always a solution of

8r/~G,,

o

=

One may question how the tachyon problem can be solved in an open Bose string theory. The generalization of (7) to the (oriented) open-string theory (more properly, to the theory of interacting open and closed strings) is

F[~,G,.A,j,CIX,A,.]---~e~XI[dg.~][d~']e-'tr(Pe-1~M),(54) where the summation goes over compact (oriented) 2-manifolds with the topology of a disk with holes (k) and handles (n) (X = I -k-2n); I is given by (8) and I~M = r~M 2

dt { e X ( ~ ( t ) ) + lq~JAj(q~(t))+ • .},

q~- dt '

~o'(t)-= ~o'(x(t)), x"(t) parametrizes the boundary, e2(t) = (g,~X~X~)~M The fields X, A , , . . correspond to the modes of the open-stnng spectrum (scalar tachyon, massless v e c t o r , . . . ) and are assigned to representaUons of an internal symmetry group G (m the oriented string case G = U ( N ) and fields belong to the adjoint representation). P in (54) is the ordering along 0M 2 Ignoring non-trivial backgrounds of the fields of the closed string sector (G,j = 8,j, • = 0) we get for G = U(1) in a tree approximation (M 2= disc)

F - f d°~5 f : df~£ v e-~:r~{l+~ra'kl(f~)[~2+l~r2a'2k2F,jF 'J+ "'} f dD~bf(X){1 + b(X)V]X + rr2a'2F,~F 'J+ ' ' "},

L=AL,

.~ = A - 1 X ,

A-~oe,

L = (the length of 0M 2) = 2 rra,

kl=L~dOK(O,O)=-Lln£+const, k2 =

dO

Fu = O,A~- OjA,, 0<~ 0<~2Ir,

dO' -~K(O, 0 )-~,K(O, O')-K(O, O')O000----~,K(O,0') = 2 ,

where K(O, 0') = K(z, Z')[aM lS the boundary value of the Neumann function on a

E S Fradkm, A A Tseythn / Quantum string theory

23

standard disc with radius a, ,

1

r ( z , z ) = ~--~In I z - z ' l l z - ~ . ' - l l ,

ZIOM ~" a e ~0 .

A stable vacuum may correspond to X = const, F,~- e,j, i.e. to a breakdown of D-dimensional Lorentz symmetry to 0(2) x O ( D - 1 , 1). It may happen, however, that a stable ground state is possible only after the account of loop corrections, and thus of "condensates" of G o and A,r

5. Discussion

The argument which led to the expression (7) for the effective action used a string theory motivation. At the same time, another interpretation of F is possible. One can forget completely about the string theory and consider (7) as a basic definition of the effective action in a fundamental theory of infinite number of fields with the "massless" ones being the metric and the antisymmetrlc tensor. (In the supersymmetric case the massless sector will include also other "lower-spin" Bose and Fermi fields in a number hopefully sufficient for a low-energy correspondence with the "standard model" ) The "internal" variables which are quantlzed in this theory are the coordinates ~p' (and also the "internal" metric gz~). The "classical" space-time coordinates ~b' are the mean values of ~p'(x) (cf. (10)). All "ordinary" fields are not explicitly quantlzed but their quantum dynamics is lmpliotly accounted for due to their dependence on the fluctuating space-time coordinates ~p' (see (8)). A reasonable point of view is that all we need to know from a fundamental "quantum gravity" theory is an effectwe actton F[G,, ~] ( G o is the metric and ~b stands for all other fields) that should satisfy several condmons" (i) it should be mathematically well defined, i e. finite, calculable, etc, (n) it should possess a consistent (unitary) low-energy limit, i.e should contain the Einstein term and kinetic terms for ~ with correct physical signs and no instability should be present (no ghosts and tachyons, i e. positive energy); (iii) it should be a starting point for a study of fundamental problems of Planck scale physics (quantum cosmology, compactificatlon of extra dimensions, black holes, etc). The F in eq. (6) may be viewed as a candidate for such an effective action. It is free of the usual problems of effective actions in ordinary field theories: it is manifestly covariant and "gauge independent" (no guage is needed to be fixed for G,j) and it is free of the ultraviolet infinities. There are also no other problems (hke indefiniteness of the classical action) arising In the naive quantization of the Einstein theory itself It of course remains to be seen whether F as defined by (7) is completely free of pathologies like some special sorts of infinities or inconsistency of the "loop expansion" in (7) In fact, the prescription of summation over topologies may look

E S Fradkm, A A Tseythn / Quantum string theory

24

ad hoc* so that it m a y be desirable to derive it as a perturbation expansion for the q u a n t u m field theory o f string functionals g t [ C ] with the action ~ ~ A ~ + 1/~3 (In analogy with what was d o n e in the light-cone gauge m [39]) A covariant formulation o f such a " s e c o n d quantized" theory (generalizing the Polyakov a p p r o a c h to the "first q u a n t i z e d " string) remains to be developed. A d o p t i n g this point view on the action (7) we are led to the following natural question w h y the action (8) standing In the exponent In (7) is given by a twodimensional integral, i.e. corresponds to one-dimensional extended objects (strings)? We know that historically the string theory was first developed in the context o f a strong interaction theory and only then was it observed that the presence o f "gravlton" in the string excitation spectrum suggests a more f u n d a m e n t a l role o f stnngs N o w d a y s It IS clear that the quark-antlquark pair connected by an o p e n string IS indeed an adequate description o f mesons, following from Q C D but there seems to be no general arguments distinguishing strings as basic objects for the construction o f a f u n d a m e n t a l theory hke (7). In fact, it is straightforward to write d o w n a generalization o f (7) for the case o f a closed " e x t e n d e d " object o f dimension d - 1 (particle, string, m e m b r a n e , . . . ) * *

F[~,G,j,A,,

~. ]=

~

f[dg~,~]f[d~']e-('/h)I,

topologies

+

IM d

f ddx e ~'~ ~'dO~,,~O'L" " " O~,dqg'"A,, ,d(qQ+

(55)

Here /z, v = l , . . , d , t, j = l , , D , M = ( 2 ~ r a ' ) -~/2, A IS a cut off, and dots l t stand for "higher-spin" field terms hke ~ v / g g / x v g A p 0~,~I ,0~q~1~0zq~0oq~"B . . . . ddx, etc The action I (and hence F = ~ d ° ~ b x/GSf(q~(~b), ), cf. (11)), possesses only two kinds o f "external" gauge invarlances: (i) the D - d i m e n s i o n a l covariance which is simply a consequence o f the d-dimensional c o v a n a n c e o f I ; (n) the abelian gauge i n v a n a n c e for the a n t i s y m m e t n c tensor, 8A,, ,~ = 0I,,A,~ ,,1. As a by-product, we deduce that the spectrum of excitations in a theory o f closed d - l ( > 0 ) - d i m e n s i o n a l objects should contain the massless (gauge) fields (represented by the metric G u and the antisymmetric tensor o f rank d) in addition to the infinite n u m b e r o f the masswe fields *** In this way we understand that any theory (either supersymmetric or not) based on extended objects o f an arbitrary dimension should contain (when * One may quesUon why &fferent "loop diagrams" are taken with weights exp (crX) and not with to(X) where w is some unknown funcUon A partial justification is provided by the observation that (rX can be written as a local addmon to the action (8) ** We assume that • contains a constant part ~(1-½d), cf [40] ***The spectrum of a theory of open (d - 1)-&menslonal objects is a "combination" of the spectra of the theories of closed objects of &menslons (d - 1) and (d -2)

E S Fradkm, A A Tseythn / Quantumstring theory

25

interpreted in four dimensions) massless particles only of spins s ~<2. The point is that "higher-spin" fields in (55) lack the corresponding gauge groups needed to ensure their masslessness in F. The distinguished role played by G,j and A,, ,~ is due to the existence of only two covariant objects (g~,~ and e "' "~) on an arbitrary finite-dimensional nemannlan manifold To get massless higher-spin fields we need a new (lnfinite-dimensionalg) geometry with higher symmetric invariants. What is less clear is how to give meaning to the sum over topologies in (55). Lacking topological classification of ( d > 2 ) - d i m e n s i o n a l closed manifolds It is difficult to classify admissible types of interactions of extended objects. At the same time, the inclusion of non-trivial topologies ("loops") is necessary in order to have the effective "duality" between the "source" (external) fields in (55) and virtual states that propagate in loops (this "duality" is absent in the case of particles, i e d = 1). If this duality takes place, we can consistently truncate F to an effective action f for a field theory of finite number of fields defined at energies E ~: M, so that " l o o p " corrections to F reproduce ordinary loop corrections to /~ computed with a cutoff at higher energies. Another problem is that the kinetic part of the action I in (55) lacks Weyl invariance if d > 2 and hence the "first stage" effective action W[g, q~,...] (cf. (13)) will be essentially non-local and difficult to compute. These and other technical problems present for d > 2 distinguish the string case d --2 as a first non-trivial but yet tractable one. Leaving aside these comphcatlons it is possible to guess the low-energy (M--> co) structure of F in (55). As in the string case it IS natural to suppose again that the higher tensor fields B~kt. a r e irrelevant for a study of a ground state problem, so that the 1 / M expansion of F looks like (cf. (39), (48))

F[cI),G,A]~fdDdpv~-G{V(CI))+--~fl(cI))O,cI)Ojcl)G 'j + M1 j 2 ( ¢ ) ~

+~l'2f3(¢)F''M ,~+,F" '~+'+O(1/M4)} .

(56)

In analogy with the string case, it appears likely that the ground state values of fields will be ¢ =const, G o ={block diagonal metric corresponding to a Ddimensional space-time being a product space containing one or several (d + 1)dimensional factors} and F,, ,~+,~ e,, ,~+, (for indices belonging to a (d + 1) subspace). Hence compactification to four dimensions is preferred in the case of membranes We would like to note, however, that the actual compactlfication mechanism may be different from that of ref. [36] due to the presence of the scalar tp (as we already saw in sect. 4). Also, higher-order ((1/M4)9~2+ • .) terms in (56) may be important One of the authors (A.A Ts ) acknowledges useful discussions with S.M Apenko, A.M Semikhatov, I.V. Tyutin, M.A. Vaslliev and B.L. Voronov.

26

E S Fradkm, A A Tseythn / Quantum string theory

Note added

After this paper was written we learned about the preprmt [41] where a "phenomenological" approach to the study of compactification in a Bose string theory is discussed. It seems that the "first principles" approach to a field-theoretic description of a quantum string theory developed in our work provides a consistent framework for investigation of the ground state problem (including space-time compactlficatlon) in this theory and gives answers to the principal questions raised in ref [41]. Note added in proof

Computmg F at the critical dimension we are to discard the integral over the conformal degree of freedom p using a Weyl gauge. The resulting off-shell expression for F a priori depends on the Weyl gauge used. A crucial consistency requirement should be gauge independence of the effective background (a solution of (28)) and o f f computed at the stationary point A sufficient condition for this is the Weyl invariance of the integrand (i.e. a decouphng of 19) for the stationary values of the fields. The on-shell Weyl invariance is also crucial for consistency of the amphtudes generated by F[44] It is remarkable that the effective equations corresponding to (53) do imply (to the leading order) the quantum Weyl mvanance of the stnng action (8). As was noted in ref [45] they also imply the (one-loop) ultraviolet finiteness of the corresponding g-model Our conjecture (which may turn out to be true only in the superstnng case) is that the stationary points of the effective action always correspond to Weyl-lnvarlant (and thus fimte) two-dimensional theories. References [1] J Scherk and J H Schwarz, Phys Lett 57B (1975) 463 [2] J I-I Schwarz, Phys Reports 89 (1982) 223, M B Green, Surveys m High Energy Phys 3 (1983) 127 [3] J Scherk and J H Schwarz, Nucl Phys B81 (1974) 118, T Yoneya, Progr Theor Phys 51 (1974)1907 [4] A M Polyakov, Phys Lett 103B (1981) 207 [5] D Fnedan, Lectures at Copenhagen Workshop (October 1981), prepnnt EFI 82/50 [6] E S Fradkm and A A Tseythn, Ann of Phys 143 (1982) 413 [7] B Durhuus, P Olesen and J L Petersen, Nucl Phys B198 (1982) 157, B Durhuus, Nor&to prepnnt 82/36 [8] O Alvarez, Nucl Phys B216 (1983) 125 [9] L Bnnk, P D1 Vecchla and P S Howe, Phys Lett 65B (1976) 471 [10] R I Nepomechle, Phys Rev D25 (1982) 2706 [11] S Mandelstam, Nucl Phys B64 (1973) 205, J L Gervals and B Saklta, Phys Rev Lett 30 (1973) 706 [12] M Ademollo et al, Nuovo Clm 21A (1974) 77 [13] M Kalb and P Ramond, Phys Rev D9 (1974) 2273 [14] J Scherk and J H Schwarz, Phys Lett 52B (1974) 347 [15] M Ademollo et al, Nucl Phys B94 (1975) 221 [16] M B Green and J H Schwarz, Nucl Phys B243 (1984) 475

E S Fradkm, A A Tseythn / Quantum stnng theory

27

[17] J Honerkamp, Nucl Phys B36 (1972)130, G Ecker and J Honerkamp, Nucl Phys B35 (1971)481 [18] D Frtedan, Nonlinear models m 2+ e dimensions, PhD thesis, LBL-11517 (1980), Phys Rev Lett 45 (1980) 1057 [19] L Alvarez-Gaum6, D Z Freedman and S Mukhl, Ann of Phys 134 (1981) 85 [20] L S Brown and J C Colhns, Ann of Phys 130 (1980) 215, S J Hathrell, Ann of Phys 142 (1982) 34 [21] M J Duff, Nucl Phys B125 (1977)334 [22] C Lovelace, Phys Lett 135B (1984) 75 [23] E Onofn and M A Vlrasoro, Nucl Phys B201 (1982) 159 [24] A B Zamolodchlkov, Phys Lett 117B (1982) 87 [25] E S Fradkln and A A Tseythn, Phys Lett 106B (1981) 63 [26] E Wltten, Comm Math Phys 92 (1984)455 [27] A M Polyakov and P B Wlegman, Phys Lett 131B (1983) 121 [28] T L Curtright and C K Zachos, Phys Rev Lett 53 (1984) 1799 [29] S Gates, C Hull and M Ro/~ek, Nucl Phys B248 (1984) 157 [30] S Deser and B Zumlno, Phys Lett 65B (1976) 369 [31] A M Polyakov, Phys Lett 103B (1981) 211 [32] D Z Freedman and P K Townsend, Nucl Phys B177 (1981) 282 [33] K Yamaglshl, Ann of Phys 150 (1983) 439 [34] F Ghozzl, J Scherk and D I Olive, Nucl Phys B122 (1977) 253 [35] M B Green and J H Schwarz, Phys Lett 136B (1984) 367, Nucl Phys B243 (1984) 285 [36] F G O Freund and M A Rubln, Phys Lett 97B (1980) 223 [37] K Bardakcl and M B Halpern, Nucl Phys B96 (1975) 285, K Bardakcl, Nucl Phys B133 (1978) 297 [38] E Onofri, Comm Math Phys 86 (1982)321 [39] M Kaku and K Kakkawa, Phys Rev D10 (1974)1110, 1823, C Marshall and P Ramond, Nucl Phys B85 (1975) 375 [40] A Sugamoto, Nucl Phys B215 (1983) 381 [41] P G O Freund, P Oh and J T Wheeler, String reduced space compactlfiCatlon, Fermi Institute preprlnt EFI 84/14 Nucl Phys B246 (1984) 371 [42] E S Fradkln and A A Tseythn, Phys Lett, to be published [43] P Candelas, P Horowltz, A Stromlnger and E Wltten, ITP Santa Barbara preprint (1984) [44] H Aoyama, A Dhar and M A Namazle, preprmt SLAC-PUB-3623 (1985), S Wemberg, prepnnt UTTG-5-85 [45] C G Callan, E J Martlnec, M J Perry and D Fnedan, Pnnceton prepnnt (1985)