Critical dimensions for noncritical strings

Volume 243, number 1,2 PHYSICS LETTERS B 21 June 1990

Critical dimensions for noncritical strings

Jean-Loup Gervais Laboratoire de Physique Th~orique de I'F.cole Normale Sup~rieure ', 24, rue Lhomond, F- 75231 Paris Cedex 05, France

Received 4 April 1990

It is shown that, for C= 7, 13, and 19, the Virasoro chiral conformal family satisfies a unitary truncation theorem: there exist a physical subspace Y(phys with positive real Virasoro highest weights, and two subsets ~p+hys (respectively Jd~-hy~ ) ofchiral operators with positive (respectively negative) real Virasoro conformal weights, such that, when ~ffhys acts on '~vhy~; (1) it is closed by fusion and braiding, (2) it only gives states that also belong to ~hy~. Thus the continuation to C ~ < 25 of weakly coupled 2D gravity, which in general leads to complex meaningless expressions, gives consistent theories for the above values. This result shows that consistent non critical strings exist for dimensions D= 26-(Tv~ = 19, 13, 7, at least order by order in the expansion over genuses.

1. Introduction

Long ago, Neveu and I were a l ready working on 2D gravity [ 1,2 ]. Many o f our results have been rediscovered and improved since. However , apar t f rom a recent article [ 3 ], no real progress have been made on the strongly coupled regime 1 < Cgrav < 25, after we put forward the special cases of central charges Cgrav = 7, 13, and 19. I just proved that 2D gravity is indeed consistent at these values. This letter contains a pre l iminary account of the der iva t ion which is based on the quan tum group structure o f the Virasoro conformal family unravel led by Babelon [ 4 ] and myself [ 5 ]. Detai ls will be given elsewhere [ 6 ].

In the present algebraic approach to conformal theories [ 1 -3 ,5 -7 ], the basic tools are opera tor differential equat ions for chiral fields. In the Virasoro case (2D gravity and min imal models ) we have a quan tum SchriSdin- ger equat ion (equivalent to the decoupl ing o f Virasoro null vectors) which is der ived by explicit computa t ion for general values of the central charge C. Since one deals with chiral operators , one may consider them on the unit circle. In this way, and since one is working operatorial ly, the discussion is not restr icted to the case of genus zero. By mapping this circle on a closed circuit o f a higher genus surface, and taking the appropr ia te trace, one may in principle consider any fixed genus. Before coming to strongly coupled gravity (section 4) , it is useful to review the progress that led to the uni tary t runcat ion theorem. Firs t recall the basic theorem of ref. [ 8 ]. On the unit circle, z = e x p ( i a ) , and for generic C, it is possible to define two equivalent free fields:

~ j ( a ) = q ~ ) + p ~ ) r r + i Y~ e x p ( - i n a ) p ° ) , j = l , 2 , (1.1) n~O //

so that, if one lets p j ( a ) = Oj(a) , one has

[p , (a l ) ,p , ( t r2 ) ] = [p2(tr ,) , p2(cr2)] =2n i~ ' (am - a 2 ) , P~) = _p~2) , (1 .2a)

N ~ , ) (p2 + p , / v / ~ ) = N cz)(p~ +p,z /V/~) . (1 .2b)

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where one defines two different normal orderings, noted N (~) and N t2), that are with respect to the modes ofp~ and P2 respectively, y is an arbitrary coupling constant. It should be stressed that the two fields act within the same Hilbert space, so that (1.2a), (1.2b) define a complicated non linear transformation between free fields. Eq. (1.2b) gives two equivalent forms of a deformed U( 1 ) Sugawara stress-energy tensor with central charge C= 1 + 3/y. The solutions of the quantum Schr6dinger equation are given by [ 1,5,7 ]

~ = ~ U U ) ( e x p ( h ~ j ) ) , ~ = ~ U ¢ / ) ( e x p ( ~ x / / ~ j ) ) , j = l , 2 , (1.3)

h=~n[C-13-x/ (C-25)(C-1)] , h=~n[C-13+x/(C-25)(C-1)], (1.4)

where d: and ~ are normalization constants. Since there are two possible quantum modifications h and h, there are four solutions. By operator product ~,j, j = 1, 2, and ~ , j = 1, 2, generate two infinite families of chiral fields

~ . , ( 1 / 2 ) which are denoted ~ ) , -J<<.m<~J, and ~t~ J), - J < r h < J , respectively, with ~tk~(~ = ~ , Wl/2 ~--~,/2, and ~-~(~ = ~l, ~}~2/2) = ~t2. An easy computation shows that

= 4 - 4 \ N / 3 + " ( 1 . 5 )

~,~s), ~J ) , are of the type ( 1, 2J) and (2J, 1 ), respectively, in the BPZ classification [ 9 ]. Next it is simpler [ 5 ] to define the rescaled variables

~=iP~') 2N/~' tg=iP~ ' )2N/~ ' t~=t~-hn' ~=~-~'n (1.6)

The Hilbert space in which the operators ~u and ~ live, is a direct sum [ 1-3,5,6 ] of Fock spaces ~(~o) spanned by the harmonic excitations of highest-weight Virasoro states noted I m, 0). They are eigenstates of the quasi momentum tzr, and satisfy L, I~, 0) =0, n> 0; [Lo -A(tv) ] I~r, 0) =0. The corresponding highest weights A(m) may be rewritten as

+ . (1.7)

The commutation relations (1.2a) are to be supplemented by the zero mode ones:

[q61), p~ 1) ] = [q62),p~ 2) ] = i .

The fields ~t and ~t shift the quasi momentum p6t ) = _p~2) by a fixed amount. For an arbitrary c-number function lone has

~ ) f ( o y ) = f ( ~ + 2m) ~,~), ~(mJ)f(~7) =f(~7+ 2thg/h) ~,~;'). ( 1.8 )

The fields ~ and ~t together with their products live in Hilbert spaces ~l of the form

+ o o

~(~o)--- ~ ~(~o+n+hn/h). (1.9) n , t ] = - -oo

~r ° is a constant which is arbitrary so far. The SL (2, C) invariant vacuum corresponds to ~o = 1 + n/h [ 5 ], but this choice is not appropriate for our purpose.

2. T h e quantum group structure

The operators q/and ~ are closed under OPE and braiding. Each family obeys a quantum group symmetry of

t Mathematically they are not really Hilbert spaces since their metrics are not positive definite.

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the SL(2)q type. However, the fusion coefficients and R-matr ix elements depend upon to and thus do not com- mute with the ~'s and ~'s. Thus their explicit form is unusual. One may exhibit the quan tum groups structure by changing basis to new families. Following my recent work [ 5 ], let us introduce

~)(tr)_-__ ~ IJ, t o ) ~ ) ( c r ) , - J ~M<.J , (2.1) - - J , ~ m ~ J

m 2J . 'J, to)M=N/( j+M)exp( lhm/2)

( J - M V , + M x Y, exp[iht(to+m)] \ ½ ( J - M + m - s ) ] \ ½ ( J + M + m + s ) ] ' (2.2)

[ ~ ( J - - M + m - - t ) ] in teger

LPJ! LnJ! = f i LrJ LrJ_= sin h --- LQJ! LP- QJ!' ,=, '

The last equation introduces q-deformed factorials and binomial coefficients. The other fields ~ ) are defined in exactly the same way replacing h by/~ everywhere. The symbols are the same with hats, e.g.

^ ^ r~ s in(hr) LnJ !-= 11 [ r j , [ r j - and so on (2.4) r=l s i n h ' "

The quan tum group structure of the exchange algebra is exhibited by introducing group theoretic states I J, M>, - J ~ < M ~ < J and operators J_+, J3 such that

J+ IJ, M>=x/LJ-T-MJLJ+_M+ lJlJ, M+ 1> , J3IJ, M>=MIJ, M> . (2.5)

These operators satisfy the SL (2)q commuta t ion relation

[J+ , J - l =LZJ3J • (2.6)

In ref. [ 5 ] it was shown that the ~ fields obey the exchange algebra, say for a > a ' ,

~ ) ( a ) ~ d ; ) (cr') = Z (J, J ' ) a ~ ' ~ ' ) ( a ' ) ~ J ) (~) , (2.7) -- J<~N<~J;--J' <~N' <~J'

(j, j, N'N )MM,=( <J, MI®<J' ,M'I)R(IJ, N>®[J',N'> ) , (2.8)

R = e x p ( - 2 i h J 3 ® J 3 )

X ( l + .~'=, [l-exp(2ih)]"exp[½ihn(n-1)]exp(-ihnJ3)(J+)"®exp(ihnJ3)(J_)")Lnj! . (2.9)

R coincides with the universal R-matr ix of SL (2) q with q = exp ( - ½ ih ). Moreover, it was shown in ref. [ 5 ] that the fusion rules to leading order in the singularity ,2 are given by

~ ) ( a ) ~ ; ) (tr ' ) ~ [ d ( a - d ) ] , d ( J + S ' ) - - z l t J ) - - d ( J ' ) ~ ( J , M; J', M' " ~ ' r ( S + J ' ) ( a ) (2.10) I"aMWM'

A(J) = - J - h J ( J + 1 ) , (2.11) 7r

2(J ,M;J ' ,M')=N/ \ j + M I \ J , + M , I \ j + j , + M + M ,] exp[ ih (M'J -MJ ' ) l . (2.12)

#2 It is really leading for C> 1 only, otherwise one just picks up the corresponding singularity. I call it leading for short.

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Since we are working on the unit circle, the divergent factor involves d ( a - a ' ) -= 1 - exp [ - i ( a - a ' ) ] instead of z - z', with z = exp (ia), z' = exp (ia ') . Consider the Clebsch-Gordan coefficients of SL (2) q [ 10 ]. For generic h, they are of the same type as the undeformed coefficients and are noted (J~, M~; ./2, M2 I Jr ,arE; "/3, MI +M2), describing the coupling of the spins J1 and J2 to give a spin -/3. Looking at ref. [ 10 ] one sees that

2 ( J , M ; J ' , M ' ) = ( J , M ; J ' , M ' I J , J ' ; J + J ' , M + M ' ) . (2.13)

This fact, together with ( 2.7 )- (2.9) shows that the structure of the ~ conformal family is dictated by the quantum group structure where the fields transform as

J+_~t)(a) =~/[.JTMILJ+M+ lJ~)+l ( a ) , J 3 ~ ) ( a ) = M ~ ) ( a ) . (2.14)

Indeed, compute

[J_+ ®exp (ihJ3) +exp( - ihJ3) ®J+ ] ( ~ ) ( a ) ~ ; ) ( a ' ) )

= exp (ihM')x/LJ-T-MJlJ+M+ 1/(~)+, (a)~ f f ) (a ' ) )

+exp( - ihM)~/IJ ' T- M' ]LJ' + M' + 13( 4~d~( a)~d: ~ , (a ')) . (2.15)

Let a~a' . This gives

[J+ ®exp (ihJ3) +exp( - ihJ3) ®J+ ] ( ~ ) ( a ) ~ ; ) (a ' ) )

[ d( a - a ' ) ]d(J+ J')-d(J)-d(S')~/EJ + J' :g M T M'J IJ + J' +_M + M' + 1 ~ + ~ ) + , ( a ) . (2.16)

After fusion one has a new representation of the quantum group with

J+_ =-J+ ®exp(ihJ3)+exp(- ihJ3)®J+ , J 3 = J 3 ® l + l®J3 . (2.17)

These last equations coincide with the standard coproduct which thus comes from the short-distance operator product expansion. The complete fusion rules of these fields are

JI +./2 ~ : ) ( a ) ~ t ~ ) ( a ' ) ~ ~ { [ d ( a - a ' ) ] ~J)-~J')-~J~)

J = IJI --./2 I

)< ( Jl, M~ ; arE, ME I Jl , J2; J, M~ + Me ) [ ~ + M2 (a) + descendants ] } . (2.18 )

Indeed one obtains by the same argument as in (2 .15) ,

x/IJT-g~ T-M2][.J+MI +M2 + l.J(J~, M~ ;J2, M2 IJ~, J2; J, g~ +342)

= exp(ihM2)x/lJl ~MIJLJ+M~ + 1.J(J~, M~ + 1; J2, Me IJl, J2; J, M1 +M2 + 1 )

+ exp( - h M 1 ) x/[.J2 -T- M2J [.J2 ±M2 + 1.J(J~, MI ;J2, M2 --- l I J , , ,]2; J, M~ +M2 ± 1 ) , (2.19)

which coincides with the recurrence relation of the q-Clebsch-Gordan coefficients (see ref. [6] for details). Obviously the same structure holds for the hatted fields. One replaces h by h everywhere. Moreover the hatted

and unhatted fields have simple braiding and fusions [5]. The most general (2J, 2J) field ~ ~ ~ ) ~ ) has weight

dK,~ (J, J; C ) = ~ 4 ( C - 1)-~4 [ ( J + J + 1)x/~-- 1 - ( j _ j ) ~ ] 2 , (2.20)

in agreement with Kac's formula.

3. The particularity of the case C larger than one. Negative spins

The structure recalled above is directly handy for C < 1. For minimal models, q and c) are roots of unity. The

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theory thus truncates, as is well known. With the above quantum group structure this truncation becomes strictly equivalent to the corresponding group theoretical truncation. For C > 1 the situation is more complicated, since (2.20) gives negative or complex weights for positive J and J which is the only case considered so far. Our next task is thus to continue the discussion to negative spins. In ref. [ 5 ] the general conformal boots t rap structure is derived f rom recurrence relations that follow from the braiding and fusion of t~ 2~ with q/~). + (,~ These are estab- lished f rom the quan tum differential equation satisfied by the former operator, and only use the fact that the latter is a pr imary field with weight 3 ( J ) . Thus the discussion has a natural extension to negative J as one may verify at every step. Mathematically, this is similar to the extension of the binomial coefficient to negative powers. Thus we shall make use of q-F-functions ~3 F, which are such that

F(a+ l )=La_[F(a) , F (N+ I )=[_NJ! f o r N i n t e g e r . (3.1)

The solution is [ 11 ]

1 - - exp( - 2ih/t) ) F(a)=-exp[½iha(a -1 ) ] (2 i s inh ) '-a f i 1 - e x p [ - 2 i h ( / ~ + a - 1 ) ] '

,u~l (3.2)

where convergence is ensured by giving a small imaginary part to h if necessary. We also need q-hypergeometric functions [ 11 ]:

F(a+ v) F ( a , b ; c ; z ) - ~ LaJ~mbJ" z" LaJ, = - (3.3) .=o LcJ .LvJ ! ' r ( a )

One may show that eq. (2.2) is equivalent to

IJ, t~)m= II j+. . lexp{ih[½m+(to+m)(J-M+m)]} F ( a , b ; c ; e x p [ - 2 i h ( t ~ + m ) ] ) , (3.4)

where a = M - J (respectively a = - M - J ) , b= - m - J (respectively b = m - J ) , c = 1 + M - m (respectively c = 1 - M + m ) for M > m (respectively M < m ). The continuation to negative J is a direct consequence of the Rodgers identity [ 11 ]

F(a, b; c; exp( - 2ihu) ) = (2i sin h) . . . . b exp [ ihu (a+b-c ) ]

x F ( u - ½ ( a + b - c - 1 ) ) F ( c - a , c -b ; c; exp( - 2ihu) ) (3.5)

Eqs. (3.4) , ( 3 . 5 ) g i v e a relationship of the fo rm

F ( ~ + m + J + 1 ) I J, ar)~oc F ( ~ + r n - J ) l - J - 1, ~ r ) ~ . (3.6)

The crucial point o f (3.6) is to show that ~u¢~ s - 1 ) and ~ t - s - 1 ) are to be considered for - J~< m ~<J, and - J ~ M<~J, respectively. In the weak coupling regime of 2D gravity, and letting C = Cgrav > 25, this continuation is instrumental in order to define positive powers of the metric tensor, as is for instance needed to study the quan tum cosmological term. This problem is currently under investigation [ 6,12 ].

.s Since factorials F functions and so on are all q deformed the index q is suppressed.

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4. Solving the reality problem of strongly coupled gravity. The unitary truncation theorem

Coming at last to the core of our subject, we consider the region 1 < C< 25, which is relevant to the strong coupling regime of 2D gravity if one identifies Cwith Cg .... In this case, h and/; are complex. We replace (1.4) by

h = ~ n [ C - 1 3 - i x / ( 2 5 - C ) ( C - 1 ) ] , I ~ = ~ n [ C - 1 3 + i ~ / ( 2 5 - C ) ( C - 1 ) ] . (4.1)

In the weak coupling regime of gravity, the solution of the conformal bootstrap we just outlined arises in a natural way from the chiral decomposition of the 2D metric tensor in the conformal gauge, that is by solving Liouville's equation [ 7 ]. It is thus legitimate to study the strong coupling regime by continuing this chiral structure below C= 25. Complex numbers appear all over the place. However, in a way that is reminiscent of the truncations that give the minimal unitary models, for C= 7, 13, 19, there is a consistent truncation of the above general family down to a unitary theory involving operators with real Virasoro conformal weights only. The proof of this theorem is our next and main topic [ 6 ]. The truncated family is as follows:

(a) The physical Hilbert space. It is given by [ 2,3,6 ]

I --s 1 --s

~ h y s ~ ~ °~--(~)'~)) ~- ~ ( ~ ~ ( ~ r , n ) , ( 4 . 2 ) r = O r = O n = - - ~

tOr, n-tiT,+n(l - h)-(2-~s +n)(1- h) . (4 .3)

The integer s is such that the special values correspond to

C = 1 + 6 ( s + 2 ) s = 0 , + l , h+[~=sn. (4.4)

A (07~,,) is positive and in ~phys the representation of the Virasoro algebra is unitary. The torus partition function corresponds to compactification on a circle with radius R = ~ (see ref. [3] ).

(b) The restricted set of con formal weights. The truncated family only involves operators of the type (2J, 2J) noted Zk ~) and (2( - J - 1 ), 2J) noted Zt+ J~. Their Virasoro conformal weights [2,3,6] which are respectively given by

A - ( J ) = - ~ ( C - 1 ) J ( J + I ) , A + ( J ) = l + ~ ( 2 5 - C ) J ( J + l ) , (4.5)

are real. zJ - (J) in negative for all J (except for J = - ½ where it becomes equal to d + ( - ½ ) = ~ (s+ 2) ). d + (J) is always positive, and is larger than one i f J ~ - ½.

(c) The truncated families, eg~hys is the set of operators Z ~s), j~> O, of the form

J J = EI+)M~;_M , (4.6) X(-'J) 2 l(S) ff'(J'S) )~(+J) J) J--J--l)

c( _ )M%M,--M , -~ Z M~ --J M= --J

where E/~))M are suitably chosen coefficients [ 6 ].

The unitary truncation theorem. For C= 1 + 6 (s + 2), s = 0, + 1, and when it acts on ~hys; the set + dphys (respectively ~¢~-ny~) of operators Z~+ s) (respectively yt_J)) is closed by fusion and braiding, and only gives states that belong to ~hys.

Proof Conditions (4.3) and (4.4) are instrumental since they allow us to relate hatted and unhatted quan- tities. In particular, for Ninteger, one has

[NJ = e x p [ - i ( N - 1 )sn]lNJ, ~r, ,-hoTr., =n[r+n(2 - s ) ] . (4.7)

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The truncation was originally observed [2,3 ] in terms of ~, and @ fields, for the braiding of Z(_ 1/2) with itself. The expression ofz~/2) using ~ fields was written in ref. [ 3 ]. From this one may derive the general formula for Z~ ) recursively in three equivalent ways. First one may deduce Z ~/+ 1/2) from X(_. J ) b y fusion with X~ 1/2) to leading order in the singularity, using (2.10)-(2.13); and this establishes (4.6) for ~¢~-hys. Second, one may derive the same formula by imposing closure of the braiding between Z~ ) and X (1/2) using (2 .7)-(2 .9) . Third one may derive an equivalent expression by imposing the same braiding or fusion conditions in terms of the ~u fields, and using the formulae given in ref. [ 5 ]. - ..~s) In this way one ~shows that z - has an equivalent expression of the form Z(_ s) = EL= _jA(m J) ( ~ ) ~(_J) ~ (m J) , where the commutation "(J) relation between a m and the ~u and ~ fields are of the type ( 1.8 ). In this form, and contrary to (4.6), the m-shift properties are manifest, due to eq. ( 1.8 ). This immediately demonstrates the second assertion of the theorem for X if). The general closure under fusion and braiding follows recursively from the above calculations since the former is associative, and since the order between the latter and the former should be irrelevant. One may also check this directly. The closure by fusion comes out neatly from the orthogonality properties of the Clebsch-Gordan coefficients which appear in eq. (2.18 ). Indeed, it follows from this equation that Z ~l ) and Z if2) have an OPE of the form

dl +d2 ( J l ) O" (,]2) )~- ( ))~- (a' )~ MIE, M2 J,Y=[E--j21OlJy{[d((7--CT' )

× (Ji, M, ; .12, M2 [J) ( J , , - M I ; J2, - M 2 [ J ) [~'I~)q-M2,--MI--M2({~) + .-.]} , (4.8)

where a condensed notation is used for CG coefficients. The hatted ones have ~ as quantum parameter following the general convention. If (4.3), (4.4) do not hold, the fight-hand side involves operators with J ¢ J which have complex Virasoro weights and the closure fails. On the other hand if (4.3), (4.4) do hold, it follows [ 6 ] from (4.7) that

( J l , - M I ; J2, - M 2 [ J ) --- + (Jl, M1 ; J2, M2 [ J ) . (4.9)

The otjj are such [ 6 ] that the sum over M~, ME, for fixed Mt + ME, reduces to

( J1, MI ; J2, M2 I J) ( J, , MI ; J2, M2 [ J) oc 6 j j , (4.10) MI ,M2

due to the orthogonality [ 10 ] of the CG coefficients. Thus only operators with J= , l appear on the right-hand side of (4.8). One may check that they are indeed linear combinations of the X~ ) fields. Finally, in view of the symmetry in J , - - , - J - 1 mentioned above (see eq. (3.6)) similar derivations show that Z~+J) also satisfies the theorem. Details will be given in ref. [ 6 ]. QED

5. Final remarks

A ^ In the present game, one repeatedly encounters products of the type [_NJ × LNJ with N integer, which are equal

to the absolute value squared of[.NJ. One may write

~ = exp(flN) - ( - 1 )sN exp(--fiN) e x p ( f l ) _ ( _ l ) ~ e x p ( _ f l ) , f l = ½ n ~ 2. (5.1)

Apart from the ( - 1 ) factors, this is very similar to/NJ after replacing h by - ifl. Thus it seems that a hyperbolic quantum group deformation is at work. Indeed one can check that if one considers operators J±, ~3 such that

J+_ I J, M ) = (LJT-MJLJ~M'JLJ+M+ l J~ J+M+ lJ ) ' /4lJ, M + 1 ) , (5.2)

where M is the eigenvalue of J3; the operators satisfy

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[J+, J - l = ( - 1)s~J-M~,/L2AAL2~ J • (5.3)

Let us turn, at last to physics. First, taking D free fields as worldsheet matter [ 1,2 ], one sees that one may construct consistent string emission vertices if D = 26 - Cgrav = 19, 13, 7. The mass squared of the emitted string ground state is m2= 2 ( A - 1 ), where A is the conformal weight of the 2D-gravity-dressing operator [ 1,2 ]. Since an infinite number oftachyons is unacceptable, this selects the + d~phy s family with positive weights d +. Bilal and I have already unravelled striking properties of the associated Liouville strings [ 13 ]. Clearly, + dphys is also se- lected if we consider the associated conformal theories by themselves, in order to avoid correlation functions that grow at very large distance. Second, one may play the game of fractal gravity, since eq. (4.5) shows that A-(J , C )+A ÷ (J, 2 6 - C ) = 1, and since the set of values 7, 13, 19 is left invariant by C ~ 2 6 - C . Comparing with the string case, one sees that ~¢~-hys describes the matter. It is a challenge to derive these models from the matrix approach to 2D gravity.

Finally, the truncation theorem holds for any integer s so that it applies to C= 1 (s= - 2 ) , and C = 25 ( s= 2), as well as for C< 1 (s < - 2 ) and C> 25 (s> 2). This will be discussed elsewhere [6 ].

Acknowledgement

I am indebted to Bruno Rostand for his collaboration at an early stage of this work.

References

[ 1 ] J.-L. Gervais and A. Neveu, Nucl. Phys. B 238 (1984) 125; Nucl. Phys. B 238 (1984) 396. [2] J.-L. Gervais and A. Neveu, Phys. Lett. B 151 (1985) 271. [ 3 ] J.-L. Gervais and B. Rostand, Two dimensional gravity and its W 3 extensions: strongly coupled unitary theories, LPTENS preprint

89/23, submitted to Nucl. Phys. B. [4] O. Babelon, Phys. Lett. B 215 (1988) 523. [ 5 ] J.-L. Gervais, The quantum group structure of 2D gravity and minimal models, LPTENS preprint 89/14 (1989), Commun. Math.

Phys., to be published. [6] J.-L. Gervais, to be published. [7] For a review see A. Bilal and J.-L. Gervais, in: Infinite dimensional Lie algebras and groups, Marseille 1988 Meeting, ed. V. Kac

(World Scientific, Singapore). [ 8 ] J.-L. Gervais and A. Neveu, Nucl. Phys. B 224 ( 1982 ) 329. [ 9 ] A. Belavin, A. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 ( 1984 ) 333.

[ 10] For a review see A. Kirillov and N. Reshetikhin, in: Infinite dimensional Lie algebras and groups, Marseille 1988 Meeting, ed. V. Kac (World Scientific, Singapore).

[ 11 ] See e.g.G. Andrews, Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics, Vol. 66 (AMS, Providence, RI).

[ 12 ] E. Cremmer and J.-L. Gervais, to be published. [ 13 ] A. Bilal and J.-L. Gervais, Phys. Lett. B 187 ( 1987 ) 39; Nucl. Phys. B 284 ( 1987 ) 397; B 293 (1987) 1; B 295 [ FS21 ] ( 1988 ) 277;

for reviews see J.-L. Gervais, Liouville superstrings, in: Perspectives in string theory, Proc. Niels Bohr/Nordi ta Meeting (1987) (World Scientific, Singapore); DST Workshop on Particle physics-superstring theory, Proc. I.I.T. Kanpur Meeting (1987) (World Scientific, Singapore).

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Critical dimensions for noncritical strings