A note on quantum Liouville theory via the quantum group An approach to strong coupling Liouville theory

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Nuclear Physics B 492 [ PM] (1997) 717-742

A note on quantum Liouville theory via the quantum group An approach to strong coupling Liouville theory Takashi Suzuki 1 Department of Electronics, Hiroshima Institute of Technology, Saeki, Hiroshima 731-51, Japan Received 13 March 1995; revised 9 December 1996; accepted 15 January 1997

Abstract Quantum Liouville theory is analysed in terms of the infinite dimensional representations of Uusl(2,C) with q a root of unity. Making full use of the characteristic features of the representations, we show that the vertex operators in this Liouville theory are factorized into classical vertex operators and those which are constructed from the finite dimensional representations of UqSl(2, C). We further show explicitly that the fusion rules in this model also enjoy such a factorization. Upon the conjecture that the Liouville action effectively decouples into the classical Liouville action and that of a quantum theory, correlation functions and transition amplitudes are discussed, especially an intimate relation between our model and geometric quantization of the moduli space of Riemann surfaces is suggested. The most important result is that our Liouville theory is in the strong coupling region, i.e. the central charge cL satisfies 1 < cL < 25. An interpretation of the quantum space-time is also given within this formulation. © 1997 Elsevier Science B.V. PACS: 04.60; 11.25 Keywords: Quantum gravity; Quantum Liouvilletheory; Strong coupling

1. Introduction and setup Liouville theory, classically, has a deep connection to the geometry of Riemann surfaces. Indeed, the solution to the Liouville equation yields a Poincar6 metric on

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the upper-half-plane or the Poincar6 unit disk on which Riemann surfaces are uniformized [1]. In the light of this fact it is natural to expect that quantum Liouville theory gives some insight into the quantum geometry of surfaces. Physically speaking, this problem is nothing more than the quantum gravity of 2D space-time. The appearance of Liouville theory as a theory of quantum gravity, the so-called Liouville gravity, was first recognized by Polyakov in the study of non-critical string theory [2]. In the string theory embedded in the D-dimensional target space, the partition function is a function Z[g] of the surface metric g, and is invariant under the group of diffeomorphisms acting on the metric g, while it transforms covariantly under local rescalings of the surface metric as

Z[e'~'~l=exp

4--~ L ( ~ : ~ )

Z[~].

(1.1)

Here SL(q5 : ~) is the Liouville action with the background metric ~, and is written as

SL(4) • P ' ) = / d 2 z V/~ { ~°bcga~0bq0 +Ae't'~:'~}+ R ~ ( Z , Z) } ,

(1.2)

and C is the central charge of the total system, the string coordinates and the reparameterization ghosts. The metric on the Riemann surface X parameterized by the complex coordinates (z, Z) is given by ds 2 = e¢~:'~}~z~ dz dZ. We denote by A the cosmological constant. R~ is the Gaussian curvature measured with the background metric ~ and is given by R~ = -2~Z~azOe log x/~. Up to now, a number of papers (there are too many papers to cite here, see e.g. Refs. [3-7] for a review) have looked at the quantum Liouville gravity and revealed many remarkable results. One of the important features of the quantum Liouville theory is that it implicitly possesses the quantum group structure of Uqsl(2, C). Precisely, vertex operators in the theory can be expressed in terms of the highest weight representations of Uqsl(2, C). We should now remember that, as in the classical algebra sl(2, C), there are two kinds of highest weight representations of Uqsl(2, C), i.e. of finite dimension and of infinite dimension, and they are completely different from each other. Owing to this fact, one can expect that there are two entirely different versions of the quantum Liouville theory. In one version either finite or infinite dimensional representations are well defined and we will show that which type of representation appears depends on the charges of the vertex operators. The quantum Liouville gravity investigated so far is mainly associated with the finite dimensional representations of Uqsl(2, C) [ 8]. In this case however, we have a strong restriction on the central charge of the Liouville gravity, cL ~> 25, the so-called D = 1 barrier. Such a gravity is often called the Liouville gravity in the weak coupling regime, or for short, the weak coupling Liouville gravity. On the other hand, one expects that the quantum Liouville gravity associated with the infinite dimensional representations is completely different from the previous one and that it can get rid of the barrier, i.e. the central charge may be in the region 1 < cL < 25. The Liouville gravity whose central charge is in this region is often called the strong coupling Liouville gravity. One of the progresses for the strong coupling Liouville gravity

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has been made in Refs. [9,10]. There it is shown that, upon using infinite dimensional representations of Uqsl(2,C), consistent gravity theories can be constructed if and only if the central charge takes the special values 7, 13 and 19. This result makes us confident that the Liouville gravity associated with the infinite dimensional representation is in the strong coupling regime. However, as will be explained in Section 3, the deformation parameter q of the algebra Uqsi(2, C) dealt with in Ref. [ 10] is not a root of unity. It should be emphasized here that the infinite dimensional representations of Uqsl(2, C) when q is a root of unity are drastically different from those with generic q [11 ]. In particular, a new and remarkable feature is that every irreducible infinite dimensional representation necessarily factorizes into a representation of the classical algebra sl(2,R) and a finite dimensional one of Uqsl(2,C). Along this line, it is worthwhile to investigate the Liouville gravity associated with the infinite dimensional representations with q at a root of unity and to observe how the characteristic feature of the representations works in the theory of Liouville gravity. Motivated by this, the aim of this paper is to investigate the quantum Liouville gravity via such representations, hoping that such a theory would lead us to a strong coupling Liouville gravity different from the one in Ref. [ 10]. Hereafter, let us use the notations .'/')root and Tpgene for the infinite dimensional repre' ~-'ln f ' ~lnf sentations of Uqsl(2, C) with q at a root of unity and generic q, respectively, and "/~zi, for the finite dimensional ones. We will see that the fields ~(z, g) of the Liouville theory based on 7~°~t should be expanded around the classical Liouville field ~d(z, g) as • = 9cl • $, ~b representing quantum fluctuations around the classical field (see also Ref. [ 12] ). In other words, the metric with which our quantum Liouville theory is constructed should be chosen as ds 2 = eK~z'~)~e where K is some constant and ~p is the Poincar6 metric, ~e = e ~'~:'~) dz dg, playing the role of the background metric. In view of this, it will turn out that our Liouville theory is effectively composed of two Liouville theories, one is the classical Liouville theory Scl (~Pcl) and the other is the quantum Liouville theory Sq(cb : ~,p) with respect to the field ~b(z, f ) . The sector Sq(d? : ~,e) represents quantum fluctuations around SCl(q~cl) and is the Liouville theory associated with 7~Fin. We will explicitly show that our Liouville gravity is actually in the strongly coupled regime, i.e. 1 < cL < 25. The organization of this paper is as follows. Section 2 looks at the classical Liouville theory in some detail, especially from the viewpoints of the geometric and algebraic structure of the theory. Discussions of the quantum Liouville gravity are given in Section 3. We will review in Section 3.1 the weak coupling Liouville theory briefly, and Sections 3.2-3.5 are devoted to our main concern, i.e. Liouville theory based on 77.[°~t. We will show the decomposition of our theory into the classical and the quantum one, and give a concept of quantum space-time. We further discuss the fusion rules and the transition amplitude. It will turn out that our formulation of the quantum Liouville theory admits a nice interpretation in the context of the geometric quantization of the K~ihler geometry of the moduli space. Here the classical sector of the correlation function, whose appearance is the characteristic feature of our formulation, plays an important

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role, i.e. the Hermitian metric of a line bundle over the moduli space. In Section 4, some discussions are given. Appendix A reviews the infinite dimensional representations of Uqsl(2,C) with q at a root of unity.

2. Classical Liouviile theory In this section, we will introduce some backgrounds of the classical Liouville theory and Riemann surfaces tbr later use, especially the algebraic and geometric aspects of the theory. It is known that the classical Liouville theory describes the hyperbolic geometry of Riemann surfaces. Let X~,N be a Riemann surface with genus g and N branch points {zi} of order kz E N)2. The equation 6St = 0 yields the Liouville equation of motion ~90q~(z, ~) = Ae~(Z'~). 2

(2.1)

Here we have made a specific choice for the background metric, ~ab = 6z~. In that case, the Gaussian curvature is given by Rg = - A , i.e. a constant curvature. According to the Gauss-Bonnet theorem, Zg,u admits a metric g with constant negative curvature, called the Poincar6 metric, for the negative Euler characteristic x(X~,N) < 0. Noticing that, for any metric ~ o n Xg,N, there exists a scaling factor e a such that g = e - a ~ has a constant negative c u r v a t u r e Rg = - 1 , we will set A = 1 hereafter. Note however that such a setting cannot be allowed for the quantum case. Let us explain the connection between the Liouville theory and the Fuchsian uniformization of Riemann surfaces. The uniformization theorem states that every Riemann surface with negative constant curvature is conformally equivalent to the quotient of the unit disk D = {w E CI [wl < 1} by the action of a finitely generated Fuchsian group F, i.e. Xg.N ~ D/F. In terms of the uniformization map, Jz : D --~ £u,N a solution to the Liouville equation (2.1) is written as

e ¢c'(z'') =

]OzJzl(Z)]2

(2.2)

(l - I J ; ~ ( z ) 7 ) 2 Upon rewriting J~71 (z) = w, the coordinate on D, the solution (2.2) gives the Poincar6 metric on D,

ds 2 = dw A dv~ ( I - wff,) 2"

(2.3)

Note that the PSL (2, ~ ) fractional transformations for J~J (z) leave the metric invariant. The classical Liouville theory is a conformally invariant theory. This is due to the fact that the energy-momentum (EM) tensor defined by Tab = 27r/x/~(SSL/8~b) is traceless, i.e.T.C~ = 0. The (2, 0)-component is given by

Tc' l [ --~Oz~oc 1 :: = ~-~ Ozq~c + c~2q~c,J

(2.4)

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and is conserved, i.e. OgTzC~= 0. Here it should be emphasized that the second term in Eq. (2.4) coming from the last term of the action (1.2) is indispensable in order for the EM tensor to be traceless and satisfies the correct transformation law of a projective connection under the holomorphic coordinate change z = f ( ~ ) , 2 = (\ ~d-f-' ~~ j

~(dg)

fm {f,e}S= f,

2 T~z~(dz) 1 2 + _~{f,Z}s(dg)2, 3 {f,,'~2 2 \f'J '

(2.5)

Indeed, using the fact that the metric e ~'(z'e) is a primary field of dimension ( 1, 1 ), or equivalently, it transforms under the coordinate change as

dE 2 e ~c'(z'e) dz dg, e ~'(e2) d~ d~ = ~z

(2.6)

one can check that the EM tensor satisfies the transformation law (2.5). This transformation law of the EM tensor indicates that the central charge co of the classical theory is 12 co = ~-~.

(2.7)

We next discuss the algebraic structure of the classical Liouville theory. To see this structure, let us take the otth power of the metric (2.2) at a special point P. Geometrically it corresponds to a branch point P on the surface, and a is related to the order k of the branch point as 1

o~i -

k -2 23,2

--

(2.8)

One immediately finds that there are two entirely different regions of oe, namely, (I) o~ = - j ~< 0, (j E Z / 2 ) and (II) oe = h > 0. It lies in this fact that there are two kinds of quantum Liouville gravity, i.e. the strong coupling and weak coupling regimes. In order to see this more explicitly, we calculate the otth power of F_q. (2.2), 2j (I)

e_./,c,(z,~)= (

1-IJ~l] 2 )

=Z

.l

J Om( J Z )fffi'm( ~), NJn,

(2.9)

111=--.j

2h

=~

Nrhat~(z ) ah'r(~),

(2.10)

r--0

where N J,, and Nrh are binomial coefficients. Although we can represent the functions ~P,J,( z ) and a h (z) in terms of a free field, we are not interested in the explicit expressions within the latter discussions. The crucial fact is that ~bJ,(z) and Arh (Z) form, respectively, the finite and the infinite dimensional representations, V/d and Vh~j, of sl(2, C). Denoting

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by E+, E_ and H the generators of sl(2,C) satisfying the relations [ E + , E _ ] = 2H and [ H, E+ ] = +E±, these representations are ~c]= {~P~,(z)t E+~(~ = E_~J_j = O,H~, = m~bJm,-j <~m ~ j},

v,7' - { g ( z ) l E_,~ = 0, HA)' = ( h + r)Ar,r =0, 1. . . . ). /2

(2.11)

One can further show that the chiral sector tpi~,(z) and A) (z) satisfy the Poisson-Lie relations of the algebra sl(2,C) [15,161, { O)~]I ( Z] ) @, ~Pi{l2,. ( Z2 ) } ..... "#]"2 \'[rjl j2 'mlm2tvB, Ill; m2 ,]tj 22 \[ Z2 ) ~11~i,1 ( Z l ) ,

( 2.1 2 )

where rj'j: is the sl(2, C) ®sl(2, C) valued classical r-matrix, r = H ® H + E + ® E _ . The same relations hold also for A)(z ). Upon the general philosophy that the quantization of the Poisson-Lie algebra g yields the quantum universal enveloping algebra Uqg endowed with the Yang-Baxter relation, these algebraic structures will be essential later. As a last comment, it is necessary for later discussions to summarize the geometrical aspects of the classical Liouville action. Since our main concern will be on the Npunctured sphere, i.e. a sphere with N branch points of order infinity, we will confine ourselves to the surface •0,N where the punctures are located at {Zl, Z2. . . . . Z N -~ (iX3}. TWO Riemann surfaces of this type are isomorphic if and only if they are related by an element of the group PSL(2, C) - a group of all automorphisms of PJ. Using this freedom we can normalize such Riemann surfaces by setting ZN-2 = O, Z N - I = 1 a n d Z N = ( x ) , i.e..a~0, u ---- (~\{Z..I . . . . . Z N - - 3 , 0 , 1}. Defining the space of punctures as TN = {(Zl . . . . . ZN-3) C C®N-31 zi 4= Zj, for i 4= j}, one sees that a point in TN represents a Riemann surface of the type XO,N. Moreover, if two XO,N'S are connected by an action of the symmetric group Symm(N), they should be regarded as the same. Hence, the moduli space of Riemann surfaces of the type Xo,N is given by

.A40,N = 7-N/Symm( N).

(2.13)

For the surface XO,N, the Liouville field ~ocl(z, g) has the following asymptotics near the punctures:

-4. I-21ogei-21ogllogei],

ej

~

IZ - - Zi],

~c,(z.Z) " " ; . -21oglzl - 21ogl loglzll,

f o r / 4= N, for i = N.

(2.14)

Of course, such asymptotics enjoy the Liouville equation (2.1). Denote by ~ a class of fields on XO,N satisfying the asymptotics (2.14). Due to the asymptotics, the action (1.2) diverges for ~Pd c ~ and a regularized action has been obtained by Zograf and Takhtajan [18] in a reparameterization invariant manner as

XL(~cl) = l i m ~ / d2z (Ozq~c,O~c, + e ~''z'~)) + 27rNlog e +47"r(N

-

2) log I loge I"~, J

(2.15)

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where 2 , -- 2 \ Ui=IN-I{IZ -- Zi] < e} tO {[Z] > l / e ) . The Euler-Lagrange equation 6SL = 0 gives again the Liouville equation (2.1). Owing to the regularization, however, the classical action SL(q~cl) is no longer invariant under the action of Symm(N) on TN. According to Ref. [ 17], one can find, for a real constant k, a l-cocycle f~ of Symm(N) satisfying exp (kSL(~od) o o') ,f~12 = exp ( k s L ( ~ c , ) ) .

(2.16)

Here Symm(N) acts on the trivial bundle TN x C ~ TN as (t,z) ~ (o't,f~,~(t)z), where t C 7-N, Z C C and o- E Symm(N). It follows that the cocycle f~ defines a Hermitian line bundle £k = 7-N X C / S y m m ( N ) over the moduli space .AA0,N and the function exp( Lff~', ~- /.d ) can be interpreted as a Hermitian metric in the line bundle £k. This fact will play an important role in the next section.

3. Quantum Liouville gravity As we have seen in the previous section, the classical Liouville theory describes the hyperbolic geometry of Riemann surfaces 2". Indeed, the classical Liouville field q~cl(z,~) is a function on 27 defining the Poincar6 metric in terms of the Fuchsian uniformization map. On the contrary, in quantum theory the Liouville field ~(z, g) expresses quantum fluctuations of the metric and so does the uniformization map j~l (z). In other words, the coordinates (w, v~) on the unit disk are not ordinary complex numbers but in some sense quantum objects. This is the reason why we regard the quantum Liouville theory as a theory of quantum 2D gravity or quantum geometry of surfaces.

3.1. Introduction of the quantum Liouville action We start by summarizing how the quantum Liouville theory appears in the string theory. The original definition of the string partition function is

Z = . f dr[d~]~[dX]g[d(gh)]eexp

( - S x ( X ; g ) - Sgh(b,c;g) - Ao f v ~ ) , (3.1)

with 7- the Teichmiiller parameter, X = (X ~} the string coordinates embedded in the D-dimensional target space and (gh) stands for the ghost coordinates {b, c} associated with the diffeomorphism invariance. Choosing a metric slice g = e ' ~ gives the following relation for the path integral measures:

[ d~ ]e~,k[dX]e~[ d(gh) ]e~~ = J( qb; ~) [ dcIg]~[dX]k[ d(gh) ]~,

(3.2)

where J(q~;~) is the Jacobian. The contributions to the Jacobian from [dX] and [d(gh)] were obtained by Polyakov [2], and that from [dq~] was postulated [6] so that the partition function (3.1) finally had the following form:

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Z =/dr[dqg]~[dX]~[d(gh)]~exp (-Sx( X; ~) - Sgh( b,c; ~) ) 1

x exp (-fd2,

(

X/~ Ag"bO.q~Gq~ + BR~q5 , +Ae°t')) .

(3.3)

The constants A and B are determined by calculating the responses of the Weyl rescaling ---* e ' ~ and demanding the invariance of Z, one finds A = B = (25 - D ) / 4 8 ~ . Upon replacing ¢/' --* v / 1 2 / ( 2 5 - D)rb, we obtain the Liouville action

u1 sa(

1

f d2ev ~

{ _lgab o~aq~Oqb(I) +QoRk@ + Ae ~'~} ,

(3.4)

2' where Q0 =

2~/-~ - D i2

(3.5)

On the other hand, the constant C plays the role of the renormalized coupling constant. In Eq. (3.4), we have replaced OCC by y, since the field rb has been rescaled and y will be related to the deformation parameter q of quantum group Uqsl(2, C). The energy-momentum tensor of the Liouville sector is obtained as 1 2 2 T= = -~(OzcP) + QoOz~,

Tz~ =

0.

(3.6)

The second equation is derived from the equation of motion. The central charge of the quantum Liouville theory is calculated as

ca

= 1 + 12Q 2.

(3.7)

Note that the total conformal anomaly vanishes,

Cx + ca + Cgh =

D + ( 2 6 - D) - 26 = 0,

(3.8)

which is consistent with the requirement of conformal invariance. For the time being, we will concentrate only on the Liouville sector. The correlation function of N vertices is given by

(V~,, (zI, Z1 )V~2 (Z2, Z 2 ) . . . VecN(ZN, ZN)) N

s,),

(3.9)

i=l where the Liouville vertex operator with charge oti is given by

V~,,( zi, zi) = e ~';z'u'(z'';'')•

(3.10)

Note that, in this definition of the correlation function, the base manifold on which the vertices live is not a manifold with branch points but just the sphere pl. Then the functional integral is performed over the space @ of all smooth metrics qS(z, g) on pl.

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Since puncture corresponds to the branch point of order infinity, all the punctures on X0,N correspond to the vertex operators with charges ai = l / 2 y 2 (see Eq. (2.8)). As in the classical theory, (2.9) and (2.10), there are two distinct regions according to the value of the charge a, i.e. (I) a <~ 0 and (II) a > 0. We will refer these two regions as (I) the weak coupling region and (II) the strong coupling region. 2 In quantum theory however, the difference between them becomes more severe than in the classical theory. There is a big gap known as the D = l barrier between the two regions. The weak coupling region, studied first by KPZ and DDK, has been investigated by many authors. On the other hand, the quantum Liouville theory in the strong coupling region is a long-standing problem. Our concern in this article is on this region and we will investigate it by using quantum group methods. Before doing this, it is instructive to review the Liouville theory in the weak coupling region. 3.2. Brief review o f the weak coupling region

The standard approach to the weak coupling Liouville theory begins with the action (3.4). Until now remarkable progress has been made by many authors. Let us, in this section, observe the quantum group aspects of the weak coupling Liouville theory. The quantum group structure appears through the vertex operators, and has been studied extensively in Refs. [8,9]. A crucial fact is that, in this region, the finite dimensional representations of Uqsl(2,C) appear. To be precise, writing the vertex operator with charge - j as J

e-JYq~(z'£) = Z

j j 7"~7~.j,m AfJmgtm(Z)9 t ( z_) ,

(3.1l)

hi= - - j

it was shown that ap.J ( z ) , m = - j . . . . . j form the 2j + 1 dimensional representation of Uqsl(2, C) with (3.12)

q = e ~riy2/2

and satisfy the braiding-commutation relations, ltt'J' (Zl) @ 1/tJ2 (Z2) /'/11 \

nl2

t

=

t

(RJlj2)mlm2 m I hi2

J2 ~,,;(z2) ® lttJ,m,(Zl),

(3.13)

where R j'j2 is the universal R-matrix of Uqsl(2,C). The factorization property of the quantum vertices is a natural guess from the classical result (2.9). The braidingcommutation relation (3.13) reduces to the Poisson-Lie relation (2.12) in the classical limit, y --, 0. This fact is in agreement with the general concept of quantum groups as the quantization of Poisson-Lie algebras of classical Lie groups. 2 According to the standard convention, the Liouville theories which are defined in the space-time of dimension D ~< 1 and I < D < 25 are called, respectively, the weak coupling and the strong coupling theory. The reason why we use here the terms weak and strong is that it will turn out later that vertices defined in the weak (strong) coupling theory carry negative (positive) charges.

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An important fact of region (I) arises in the relation between Q0 and the renormalized coupling constant % that is Q0 = -1 + __y. 9' 2

(3.14)

Substituting (3.14) into the general expression of the central charge (3.7), one obtains cc=13+6

+

(3.15)

and finds that the central charge always satisfies cL ~> 25 (~< 1) for a real (imaginary) y. Note that y ~ should be real since e r~ is the metric in this model. The very origin that restricts the theory to the weak coupling region is the relation (3.14). This was first obtained in Ref. [20] by requiring that the EM tensor (3.6) satisfies the Virasoro algebra on the cylindrical basis S j × R. Once the Virasoro structure is found, one can apply the Coulomb gas formulation of the minimal CFT to the Liouville theory, although the Liouville field is no longer a free field. Let us explain briefly. Note that the parameter Q0 corresponds to ia0, ce0 being the background charge in the Coulomb gas formulation and the highest weight vector ~ j ( z ) can be related to the primary field of the type

V2i+i.l(z) in Kac's table, where V,,m(Z) =: exp(ia,,m~b(z)) : with the charge ten,,,, = { ( 1 - n) a+ + ( 1 - m) c~_ }/2 and a free field 0~( z ). Upon comparing Eq. (3.14) and the relations o~0 = (ce+ + a _ )/2, ce+a_ = - 2 , one immediately sees that the renormalized coupling constant 9' is related to one of the screening charges, say ce+, as y = ice+. With the above connections at hand, the correspondence between the metric e r~z'~) and the screening operator e ia~da(z) which has conformal dimension 1 is now clear. Thus we obtain the correct dimension (1,1) for the metric e ~z'~) [6]. Furthermore, the relation (3.14) is consistent with the quantum group structure of Uqsl(2, C) when the representation is of finite dimension. Indeed, the braiding commutation relation which is deeply related to the q-6j symbols indicates that the operator e -.ir~ has dimension _ j _ ½y2j(j+ I ). By comparing with the result of CFT that the dimension of the operator is given as - ½ j y ( j y + 2 Q o ) , we then obtain (3.14). Thus in the weak coupling Liouville theory, the relation (3.14) works well and plays an important role, although it is the origin of the severe problem, the D = 1 barrier. ^

3.3. The strong coupling region Now we are at the stage to enter into the mysterious world of the strong coupling Liouville theory. At the beginning, we propose a Liouville action for the strong coupling region, SL(q~'~)=St.+S',

with

s'=afa2zv e ,

(3.16)

where SL is the Liouviile action (3.4). We have introduced the additional cosmological term S' whose necessity will be understood later. Note here that, owing to this term, the

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potential reduces to the classical metric e ~c'(z'~) in the limit • ~ - ~ , while without this term the potential vanishes in the limit. We will work, however, not with this action but with the representations of the quantum group Uqsl(2, C) in the following discussions. Let us turn to the quantum group structure. To do this, we introduce the vertex operator e hyqJ(z'~) denoting the charge h instead of a in this region. By definition the charge h is always positive. As in the weak coupling case, the vertex operator allows the following holomorphic decomposition: ehYq,(zd) = ~~....mit A rh,r--h, lb~z)-Ah'r(~).

(3.17)

r=O It can be shown that the vectors A h ( z ) , r = 0,1 . . . . form an infinite dimensional highest weight representation ~, of Uqsl(2, C) with (3.12). Thus in this region, infinite dimensional representations appear instead of finite ones in the weak coupling region. The question here is the relation between Q0 and the renormalized coupling constant y. If we require cL > 1 while maintaining relation (3.14), the constant y becomes complex. This fact means that, as suggested in Section 3.1, the screening charges become complex and, therefore, the dimensions of the primaries are not real in general. Gervais et al. [ 10] have discussed the strong coupling Liouville theory under this situation. Their solution to the problem is summarized in the "truncation theorem", which guarantees that only in the special central charges CL = 7, 13 and 19, the chiral components of the Liouville vertices tbrm Virasoro Verma modules with real highest weights. Using relation (3.14) together with the central charge (3.7), these central charges CL = 7, 13 and 19 give the values of the constants y, y 2 = - 1 + i v Y , 2i and y2 = 1 +iv/3, respectively. Note here that, with the definition (3.12), the parameter q of the quantum group Uqsl(2, C) is generic. In such a situation the relation (3.14) is actually consistent. As in the weak coupling region, the braiding commutation relation in the case of infinite dimensional representations shows that the dimension of the operator e hr'p is calculated as h - ½ y 2 h ( h - 1). By comparing this dimension with that obtained from the CFT method, that is, -½ h y ( h y - 2Q0), we again obtain (3.14). Now we would like to ask what will happen if we take the parameter q to be a root of unity. With the idea that the infinite dimensional representations T~r°°t and ' ~lnf .~gene lnf are drastically different from each other, it is quite natural to expect that the quantum Liouville theory associated with R[o~t differs greatly from the one based on 7"pgene'~lnf Furthermore, since the q - 6 j symbols for the representations -~root '~tnf are completely different from those for "Tpgene '~Inf we could get rid of the relation (3.14) and, therefore, have any other values of the central charge. In the following discussions, the only essential assumption is that our Liouville theory possesses the structure of ./-proot ' ~Inf " Let q be a pth root of unity,3 i.e. .

'

• ,

q = e ~q'/P,

y2

i.e.

p~

-- = --, 2 p

3 In this paper, as a matter of convention, q such that qP

(3.18)

=

--

1 for 3p C N is also called a root of unity.

728

T. S u z u k i / N u c l e a r P h y s i c s B 4 9 2 [ P M ] ( 1 9 9 7 ) 7 1 7 - 7 4 2

with p , p ' C N coprime with each other. Such representations are reported in Appendix A. Once the parameter q is set to the value (3.18), the infinite dimensional highest weight representations are parameterized by two integers /x and v such that h ---, h~, = ( p - j where sr = v / 2 p ~, and j = ( / z - 1)/2. We denote by Vu,~ the module on the highest weight state of weight hu~ and by AUra(Z) the rth weight vector in the module Vu.~, namely, it corresponds to the rth holomorphic sector in the decomposition of the vertex operator e hJ~(z'e). One immediately sees that the terms A ~kp+g~~-Z~)~'l ~u,,,kp+~g), k = 0, 1, • • . , g =/x . . ... p - - 1, disappear from the vertex operator k #v e h""u'(z'~) because, as shown in Eq. (A.6), Akp+.~ are the elements in A'u.~, the space of zero norm states, and so the vertex operator is given by means of the summation only of ~'kt,+,~ ~ j ' t z), s = 0 . . . . . /x - 1, namely, the elements in the irreducible highest weight module V~,, = Vu.~\P(~, . Therefore we can now use the important theorem in Appendix A, more concretely, the relation (A.18): the irreducible highest weight module v~rr is isomorphic to the tensor product V~'@ "l)i, where V~~ and Vj are the highest weight representation of the classical algebra s l ( 2 , R ) with highest weight ( and the 2j + 1 dimensional representation of Uqsl(2, C), respectively. Owing to the isomorphism, the holomorphic sector factorizes as /zv

Akp+s(Z) ~ A(~(Z) @ ~ i I , ( Z ) ,

(3.19)

and also does the anti-holomorphic sector. Keeping these remarkable facts of -/")root ~ ' I n f in mind, let us look at the N-point correlation function in the Liouville theory associated with -r)root '~Inf '

Z S [ m : txi, vi] := (Vm~, (zl, g,) ... V u ~ ( Z N ,

ZN)) N

=/[dq~]exp(-4@SL(@;~))IXvu~(zi,~.),

(3.20)

i=1

where V~,,~,(zi, ~ ) = e h~'~rev(z,.~), and m = (m, N) stands for the moduli parameters of the surface. We will come back later to the discussions of the correlation function and the transition amplitude as well. Before that, let us enjoy our model with the remarkable nature o f, q~root '~nf • Thanks to the factorizations (3.19) of the holomorphic sectors and the similar factorization of the anti-holomorphic sector, the vertex operators in (3.20) are factorized as OO

V~,~(z,g) =Z ,,

Ji

N(',t ")@ ~ k; ' ,t .~i )"A ( "~ 't- zi

k=O

= e ;'~`~(z''~') @ e -j'~'4'(z''e'l ,

Z

A/'.~, J,,e.,j, (z,). ~ej,m (z,) -.

m= -.i~

(3.21)

where ~'~r(z, g) and ~ ( z , Z) are, respectively, the classical and the quantum Liouville fields. With the remarkable decompositions of the vertex operators at hand, we can now conjecture that the action also separates effectively as SL (q~; 1) ---, scl(q:~cl; 1) + g(q~;,~v).

(3.22)

T. Suzuki~Nuclear Physics B 492 [PM] (1997) 717-742

729

Here the actions Scl (~Ocl;1) and S q (05; ~e ) are the Liouville actions with respect to the fields q~d(z, ~) and 05(z, f ) , respectively, and are explicitly given by

,/

S~l(~od; 1) = ~

s',(05; ~p) =

d2z (Ozq~dOf~ocl + Ae~°c'(z'~)),

f a2z ~

(~05~05+ OR~05(z,~) + Ae~
(3.23)

(3.24)

where /3 is some constant and ~p = e ~'~d dz d~ is chosen as the background metric for the quantum sector. When ~" = 1 in the exponent of ~e, the background metric is just the Poincar6 metric. The reason why we have introduced a new coupling constant/3 is that, at this stage, we have no idea how the coupling constant of the classical sector should be. On the contrary, remembering that (see Appendix A) the finite dimensional representation ~i that appeared at the decomposition has the same deformation parameter q as that of V~, the renormalized coupling constant ~ of the quantum sector should satisfy ~2 2

y2 -

2

+2n,

n = 0,-t-1, +2 . . . .

(3.25)

It should be emphasized that, upon the condition yQo = yQ = 1, the conjecture (3.22) together with (3.23) and (3.24) is exactly true under the substitution y ~ = ~'¢cl + y05 with ~- = y / ~ . Some remarks are now in order. The above observations seem to suggest that, in the quantum Liouville theory via/~[o~t, the Liouville field ~ ( z , Z) should be expanded as y ~ = 7"¢cl + ~05 and, therefore, one can interpret the field 05 as the quantum fluctuation around ~'~Pcl. In other words, the metric ds 2 = e ~(z'~) dz d~ is to be written as ds 2 = e~'4'(:'~)~p and e ~(z'~) represents the quantum fluctuation of the metrics around the classical background metric ~p = e ¢¢~' dz d~. If ~- is set to be 1, the background metric gp becomes the Poincar6 metric. Now we can give a possible interpretation of the quantum 2D manifold according to the above observations: the quantum manifold with metric e ~t'(z'~) should be considered as the total system of 2D classical manifolds with the metric e ¢¢¢' dz d~ and quantum fluctuations around the classical surface. This interpretation matches quite well to the general concept of quantum object. Note also that the action Sq(05) governing the quantum fluctuations is again the quantum Liouville theory associated with ~Fi,. It is worthwhile to give a comment here on the difference between our formulation and the standard weak coupling Liouville theory [3-6]. In the standard approaches whose action is given in (3.4), the theory also has the 7~Fi, structure as shown in Section 3.1. But unlike the quantum sector S q (05; ~,p) of our model, the background metric can be freely chosen together with the shift of the Liouville field @(z, Z). For example, in Refs. [3,4] the metric in the light-cone gauge d s 2 = I d x + + t z d x _ l ~, which is conformally equivalent to the metric in the conformal gauge d s 2 = e ~:b(z'~) dz d~, was chosen. These choices have the flat background metric. On the contrary, in our formulation the background metric is automatically selected as ~p. Thus the appearance of the classical

730

T. Suzuki~Nuclear Physics B 492 [PM] (1997) 717-742

sector in addition to the quantum theory associated with T~Fi n is the characteristic feature of our formulation. We will recognize the importance of the classical sector from the viewpoint of geometric quantization of moduli space in Section 3.5.

3.4. Fusion rules Before going to the discussion of the correlation function, it is interesting to look at the fusion rules in our model. First, we have to investigate the Clebsch-Gordan (CG) decomposition rule for the tensor product of two irreducible infinite dimensional representations of Uqsl(2, C), V~ ® V2

, V3,

(3.26)

where ~ := v~rir~~ ~ V~j ® Vj~. The quantum CG coefficient, known as the q-3j symbol, for the infinite dimensional representations of Uqsl(2, C) when q is not a root of unity is given as follows: [ h i h2 h3 ]vqsl(2'C) =C(q)6r~+r2. r33(hl,h2, h3 ) FI r2 r3 . q x

f[2j-

l][r3-h3l![rl

x ~-~(_)Rq~(r3+h3-') R)0

x

-hl]![r2-h2]![rl+hl[r3+h31]!

l]![r2+h2-

1 ] ! ] I/2

1

[R]![r3-h3-R]![rj-hj-R]![rl+hi-R1

[h3 - h i - r l + R]![h3+ h 2 - r l + R - 1 ] ! '

1]! (3.27)

where C(q) is a factor which is not important for our analysis below and z~(hl,h2,h3) = {[h3 - hi - h2]![h3 - hi + h2 - 1] ![h3 + hi - h2 - 1]! x [ h l q'- h2 -+- h3 - 2 ] ! } 1/2.

(3.28)

The notations hi := h~iu ~ = ( i P - j i and r i = ( ( i -+- k i ) p - mi have been used. O f course

for our case, i.e. q = exp(Trip'/p), the CG coefficient is not necessarily well defined due to the factor [p] = 0. What we have to do is only to find conditions which give finite CG coefficients. Since the calculation is lengthy and our interest is not in the details, we describe here only the quite interesting result; finite CG coefficients exist if and only if

IJl --Jzl -- 1 < J3 ~ m i n ( j t + j 2 , P

--2 --jl

--j2).

(3.29)

It should be noted that, on the modules Vcl and ]2, the coproducts of the operators K = qU and L0 are A(K) = K®K and A(L0) = L0 ® 1 + 1 ®L0, respectively. The coproduct of the Cartan operator yields the conservation law of the highest weights, physically speaking,

T. Suzuki~NuclearPhysicsB 492 [PM](1997)717-742

731

the conservation of the spin or angular momentum along the z-axis. Now, from the above coproducts, we have the conservation laws ((I + kl) + ( ( 2 + k2) = ((3 + k3) and ml + m2 = m3 (modp). Therefore the minimum value of j3 is just [jl - J z l because the difference between ]jr - J 2 l and j3 is always integer. We therefore obtain the following decomposition rule of the tensor product of two infinite dimensional representations of Uqsl(2,C) at a root of unity:

(V;,1 "~,,)@ (V;)® ]")j2)=

~

~'~+¢2~¢3

V;]) @ \

J3=lJ'-.izl

Vh

" (3.30)

The decomposition rules for the tensor products of V~1 and of V) are the same as those for the tensor products of the sl(2,R) and of the finite dimensional representation of Uqsl(2, C), respectively. Applying this decomposition rule to our model, the fusion rule is

()(min{jl+j2,~2--Jl--j2}

[ehlr4']x[eh2r4']= ~1+¢2<~3~[e~'~*'~l]

@\

,

(3.31) namely, exactly the same as the tensor product of the fusion rules of the classical theory and those of the weak coupling theory. This result means that the classical sector and the quantum sector never mix with each other.

3.5. Correlationfunctions and amplitudes Let us return to the discussion of the correlation functions in our model. The original definition was given in Eq. (3.20). What we will see in this section is how the correlation function and amplitude can be written by the decomposition of vertices and actions. Since, as we have seen, the Liouville field • can be expanded around the classical solution, the functional measure is [dqb] = [d¢9] up to a constant. The correlation function (3.20) is expected finally to be factorized into the classical and quantum sector, i.e. up to some constant, it can be written as

ZS[m : {/z,v}]

= ZCl[m :

{(}]Zq[m: {J}l-

(3.32)

Here the quantum sector is given by N

zq[m : {j}] := /[ddp]exp (-~----~Sq(dA;~p))He-J'~(z"e')

(3.33)

i=1

and the classical sector is

(

co

Z cj[m : {(}] = exp \-4---~

(~Ocl)],

(3.34)

T. Suzuki~Nuclear Physics B 492 [PM] (1997) 717-742

732

where co = 12//32 is the central charge of the classical Liouville theory. The action SCl(~,cl) denotes the classical action defined on the surface with N branch points and regularized by subtracting the singularities near the branch points. In particular, in the case when the topology is the N-punctured sphere, i.e. all the vertices carry the charges (i = 1/2/3 2, SCJ(q~cl) is given by Eq. (2.15). Let us discuss this in more detail. As stated above, the action Sq (qS; ~'t,) corresponds to the Liouville action in the conformal gauge ds 2 = eT6(:'~)(~p):.: dz dg with ,~e the classical background. Suppose that the correlation function Zq[m, {j}] admits holomorphic factorization as

N~'J-~l[g~'{J}]g'g[m" { j } l ,

Z q l m "{J}] = Z

(3.35)

l,J

where N l,J is some constant matrix. Here ~ t [ m : {j}] is expected to be a holomorphic section of a line bundle over the moduli space .A.40.N. We will see later that this expectation is acceptable, Now we are at the stage to discuss the transition amplitudes of our Liouville theory. In order to obtain a transition amplitude, we have to integrate the correlation function Z Ira] over the moduli space of the surface metric with the Weil-Peterson metric. Combining (3.34) and (3.35) and integrating over the moduli space, the N-point transition amplitude . A N ( { ( } , { j } ) has the following form:

NI'J j '

,AN({f},{j})=Z l,J

co if:el/ ~ - [ ~ ; {j}]g~g[m" {j}], d ( W P ) exp ( - 48¢r

~.40,N

(3.36) where d ( W P ) stands lbr the Weil-Peterson measure on the moduli space. Let us look here at the geometrical meaning of the classical sector. For the topology of the Npunctured sphere, there are quite remarkable facts shown by Zograf and Takhtajan [ 18] about the connection between the classical Liouville theory and the K~ihler geometry of the moduli space of complex structure. The important facts are as follows. First, the Liouville action evaluated on the classical solution is just the K~ihler potential of the Weil-Peterson symplectic structure, precisely, wwp = i~O-SCl/2. Second, the accessory parameters ci, i = 1 ~,, N are written as -2"rrci = 3-scl/azi. 4 From these facts, one can show that the functions ci are in involution [ 19], i.e. {ci, zj}wp = i~ij, where { , }wp is the Poisson bracket with respect to the Weil-Peterson symplectic 2-form o~wp. Thus the classical Liouville theory can be regarded as the K~ihler geometry of the moduli space of surfaces. Due to the relation, it is natural to expect a deep connection between our quantum Liouville theory and the geometric quantization of the moduli space which corresponds to the phase space of the classical geometry. With this hope, we should try to observe our result from the viewpoint of geometric quantization. Before doing this, it is helpful 4 This relation between the classical Liouville action and the accessory parameter which is associated with every puncture was first conjectured by Polyakov [21 I.

T. Suzuki~Nuclear Physics B 492 [PM] (1997) 717-742

733

to summarize the basic facts about the geometric quantization of a classical theory. Consider a K~ihler manifold 3,4 equipped with a symplectic structure w which is written in terms of the Kahler potential K as oJ = iO'JK. The Kahler manifold plays the role of the phase space of the classical theory. Geometric quantization is performed by building a line bundle /2 --+ AA over the 2N-dimensional manifold A4 with the curvature twoform F = - i w . Let us parameterize A4 by (qi,Pi), i = I . . . . . N, and denote the section a s ~b(q i, Pi). The final manipulation to complete the geometric quantization is to impose on /2 the condition that the section is annihilated by the derivatives of half of the variables, which is called a polarization. In other words, by the choice of a polarization, sections are represented by N variables. Let us, for example, choose the polarization as Op,,p = 0, i = 0 . . . . . N. Then the Hilbert space 7-i is the space of sections with the polarization

= {¢'1 aMP = 0}.

(3.37)

The inner product on 7-/is introduced with the Hermitian metric e - x , i.e. for ~bl, ~b2 7"/, it is (~bl,~0z) = f e - X ~ • ']'2, where the measure is that defined by w. Putting our expression (3.36) together with the fact that, as we have seen in the last part of Section 2, a line bundle with the Hermitian metric exp(~S k--el ) can be constructed on the moduli space .M0.N = (mi, ci), the holomorphic part 7"1 [m] of the quantum sector can be regarded as a holomorphic section of the Hermitian line bundle £co ~ .AA0.N with the polarization ac,~ = 0 and the curvature is ~12,rwwo" On the other hand, the classical correlation function (3.34) corresponds to a Hermitian metric defining an inner product ( , )co on the Hilbert space which is the space of sections 1/" [m]. Hence, at least for the topology of the N-punctured sphere, the amplitude can be written as

A = ~ NI'J(aIQ, qZJ>co. Thus the quantum Liouville theory associated with --/-:root f i t s well with the geometric '~lnf quantization of the moduli space. The factorization property into the classical sector and the quantum sector plays an important role here and is just the special feature appearing only in the Liouville theory associated with T2root • ~lnf "

3.6. Central charge and some discussions Finally we give some discussions about our Liouville theory. What we have understood in the above discussions is that the quantum Liouville theory based on the infinite dimensional representations of Uqsl(2, C) at a root of unity factorizes into the classical Liouville theory and the quantum Liouville theory based on the finite dimensional representations of Uqsl(2, C), and the latter governs quantum fluctuations around the classical surface. The following discussions are about this observation, namely, we start from the action

1 f d2z

L = 4~/32

if

(0~odO~oc,+ e ec') + ~

d2z ~

(V~&

+ QR~5.dp) , (3.38)

T. Suzuki~NuclearPhysics B 492 [PM] (1997) 717-742

734

rather than from the original one (3.16). Here we have set A = 0 for convenience. It is important to note that, since e z'~ is a Riemannian metric, the Liouville field 9,q) should be real, and therefore, ~ b is real as well. Recall the relation (3.25) and note that ~2 can be negative. When "~ is purely imaginary, the quantum filed ~b is also imaginary and the second term in (3.38) has the wrong sign, while if "~ is real, ¢b is a real field and the quantum sector has the correct sign. In the following discussions, we would like to impose another assumption, namely that/3 = 9,, or equivalently, r = 1, in order for the background metric ~p to be the Poincar6 metric. It is easy to calculate the EM tensor from the total action, and one finds T~°t =TCJ(z) + T q ( z ) + Tmix(z),

l(-~ TC'(z) = ~y rq(z)=lim

w-~

(3.39)

&°c'°~c' + 02~c~) ,

[(1 -

~

Ozd~&,d~+QO2d~

)

(z

,j

-

w) 2 '

Tn~X( z ) = Q a~ct&b, r~ °t = O.

(3.40)

The term T mix c o m e s from the curvature term in (3.38), where the classical and quantum Liouville fields ~'cr and ~b interact with each other. The second line (3.40) guarantees that the total system is again conformally invariant. Now we must estimate the central charge to confirm that our model is actually a strong coupling theory. To do this, let us observe how the EM tensor T ( z ) = T~z°t transforms under the change of variable z ~ f ( z ) . Since the EM tensor is a second-rank tensor with central extension, it should satisfy the following transformation law:

( d f ' ~ 2 T ( f ) + ~2 { f , z } s . ¢(z) =

\az}

(3.41)

However, one sees that Eq. (3.39) satisfies (3.41) if and only if Q = 0, and the central charge is given by 12 cc = 1 + 9,2"

(3.42)

The first term 1 arises from the quantum sector Tq(z ), to be precise, from the subtraction of the singularity 1 / ( z - w)2, 5 and the second term 12/y 2 comes from the classical sector. Putting together the central charge (3.42) with Eq. (3.8), one finds 1

9,

1//~---~ v lZ

Q0.

(3.43)

Note that the original coupling constant 9, is real for the dimension D ~< 25, and that the central charge of our Liouville theory can actually be in the strong coupling region, 5 The author thanks N. Ano for a very useful discussion on this.

T. Suzuki~Nuclear Physics B 492 [PM] (1997) 717-742

735

1 < cL < 25 for y > 1/x/~. It is interesting to give some of the allowed dimensions D of the target space in our model. Recalling the assignment 72/2 = p'/p with the integers p,p' being coprime with each other and (3.43), one finds (i) w h e n p ' = l ,

D=1,7,13,19,

(ii) w h e n p ' = 2 ,

D=4,10,16,22,

(iii) w h e n p ' = 3 ,

D=3,5,9,15,17,21,23,

(3.44)

etc. We can obtain the fractional dimensions as well. If another matter whose central charge is fractional couples to the string theory, such a fractional dimension will become crucial. In case (i), the dimensions are completely the same as those given in Ref. [8]. This is a strange coincidence, since we are dealing with the case when q is a root of unity, whereas in Ref. [8], the parameter q is not a root of unity. This central charge (3.42) coincides with that obtained by Takhtajan in Refs. [ 13,14], where a manifestly geometrical approach was used. Along Polyakov's original formulation of the quantum Liouville theory, he started with conformal Ward identities via the functional integral. Using perturbation expansions around the classical solution corresponding to the Poincar6 metric, he obtained the central charge (3.42). It is worth mentioning that in both approaches the quantum Liouville theory is expanded around the classical solution, i.e. the Poincar6 metric. In other words, a quantum Riemann surface is treated as a total of a classical surface endowed with the hyperbolic geometry and quantum corrections around it. Indeed, in Takhtajan's approach, the classical limit recovers the underlying hyperbolic geometry of the Riemann surface. In our case, owing to the condition Q = 0, the total action (3.38) is just the sum of the classical Liouville action and a quantum action without any mixing terms. The quantum field ~b(z, ~) can be interpreted as the (D 4-1 )th component of the string, and we obtain the D 4- 1 dimensional space-time. As suggested previously, the signature the space-time has depends on the sign of 5'2. Next we turn our attention to the conformal dimension. Also, in this discussion, we conjecture Q = 0. The vertex operator of the string dressed by gravity is written as

V(Z,~) =

/

d2z ~

: Ox .=

d2z

: e ~ O x :,

(3.45)

where Ox stands for a pure string vertex operator. Let A0 and dx be the dimensions of the classical Liouville exponential and the string vertex, respectively. Then one finds the following relation between the conformal dimensions: 5,2

A o - ~ + Ax = 1.

(3.46)

The middle term in the left-hand side of (3.46) is the conformal dimension of the exponential e ~ . Since we have chosen r = 1, the background metric ~e is the Poincar6 metric and so A0 = 1. Then, together with (3.25), one finds

5'2

72

=Ax=-~4-2n,

n=O,4-1 . . . .

(3.47)

736

T. Suzuki~Nuclear Physics B 492 [PM] (1997) 717-742

This equation suggests that the matter (string) which couples to the Liouville theory forms a discrete series in the conformal dimension and that the constant 5, can be imaginary. In summary, our Liouville theory can be interpreted as follows. It is the total system of the classical Liouville theory plus the fluctuation around the classical manifold, which becomes the (D + l)th component of the string. The same situation occurs in the weak coupling Liouville theory, although in that case the classical Liouville sector does not appear. In the weak coupling theory when the dimension of the target space is D = 25, the quantum Liouville field is regarded as the time component of the string with the wrong sign and, as a result, the space-time becomes the 26-dimensional Minkowski space. On the contrary, in our case, we have more choices for the dimension D as listed in (3.44), especially for y2 = 6 the dimension of the target space is D = 3 and therefore, with the field ~ as the time component, our space-time is the 4-dimensional Minkowski or Euclidean space according to the sign of the quantum sector Sq.

4. Summary and discussions We have developed a new approach to the quantum Liouville theory via the infinite dimensional representations of Uqsl(2, C) when q = e x p ( ~ r i p ' / p ) . In the Liouville theory we have dealt with, only the vertex operators with positive charges o~ can be defined. The characteristic feature of -/')root '~'lnf is that every irreducible highest weight module Virr~,,,, necessarily factorizes into the highest weight module V~j of the classical algebra sl(2, R) and V/, the 2j + 1 dimensional representation of U,/sI(2, C). Our investigations in this article have been performed by making full use of this feature. Owing to this fact, we observed that the vertex operators with positive charge o~ = hu~ factorized into the classical vertex operator with charge sr and the vertex operator with negative charge - j as in Eq. (3.21), where the relation h/~ = ( p - j was understood. We further conjectured that the Liouville action sc(q~) should also decompose into the classical Liouville action Sd(q~d) and sq(~b;~p) with the classical background ~p = e T~''!-'~t, where e ~'' defines the Poincar6 metric. Since the field &(z,,5) can be interpreted as the Liouville field which measures the quantum fluctuations of the metric around the classical metric e T~', the Liouville theory Sq(~b;~p) governs the theory of quantum fluctuations of quantum Riemann surfaces around the classical surfaces. Namely in our formulation, the quantum Liouville theory S'c (@) describes the quantum 2D spacetime as the total system of the classical space-time and the quantum fluctuations around it. We tbund that the Liouville theory governed by the action Sq(d~; ~p) was associated with the finite dimensional representations 7~vin. We remarked the difference between our Liouville theory and the standard Liouville theory [3-6] which is confined to the weak coupling region. Although in both theories quantum fluctuations of the metric are deeply related to 7-~Fin, our theory contains the classical Liouville theory yielding the underlying K~ihler geometry, whereas the latter does not. It is quite important to emphasize that our quantum Liouville theory is certainly in the strong coupling regime.

T. Suzuki~Nuclear Physics B 492 [PM] (1997) 717-742

737

Another interesting result was that our Liouville theory fitted well with the concepts of geometric quantization of the moduli space of metrics. First of all, we have noted that the classical Liouville theory is in agreement with the K~ihler geometry of the moduli space, where the Liouville action evaluated on the classical solution ~Pcl is nothing more but the K~ihler potential of the Weil-Peterson metric wwp. Second, we have observed that the Hermitian line bundle L:c0 can be built over the moduli space .A40,N with the Hermitian metric e x p ( - 4 - ~ S t ) . The role of the holomorphic sections of the line bundle £~,, is played by the holomorphic part ~'[m; {j}] of the quantum sector Z q [ m : {j}]. Thus, as we have seen in Eq. (3.36), the transition amplitude in our theory agrees with the inner product of two wave functions corresponding to the sections of the line bundle £c,~. Remembering that the quantum Liouville theory can be regarded as a quantum geometry of Riemann surfaces, our observation of our quantum Liouville theory from the viewpoint of geometric quantization of the moduli space seems to be quite natural. At this stage, it is worthwhile to compare our quantum gravity with that formulated in Ref. [22] where c > 1 two-dimensional quantum gravity was treated via geometric quantization of moduli space. Riemann surfaces in his consideration are compact and have genus g > 1. There, the transition amplitude between some initial state ~1 and some final state g'v is given by the inner product (see Eq. (5.7) in Ref. [22] )

<~'~, ~'F> = j f

d(WP)

ZL[ml~[m]~ttF[m],

(4.1)

where Z L [ m ] is the Liouville partition function (let o-(z,#) be the Liouville field). q ' [ m [ is a holomorphic section of a line bundle over the moduli space and can be identified with the conformal block obtained by solving the conformal Ward identity of Polyakov [3]. Therefore qt[m] is considered as the holomorphic sector of the partition function in the weakly coupled Liouville gravity. In that paper, the metric on the Riemann surface is parameterized as ds 2 = e "(z'e) ]dz +1x dZ[ 2. Now it is easy to find intimate relations between the formulation of Verlinde and our Liouville gravity except for only one big difference. In both theories, a quantum Riemann surface is composed of two sectors, one is the Riemann surfaces as a background and the other corresponds to quantum fluctuations around the background surface. In Verlinde's formulation, the quantum fluctuation is parameterized by the Beltrami differential /x. The difference arises in the choice of background surfaces; the background Riemann surface in our theory is just the classical surface with the Poincar6 metric e ~°c'(z'~), while in Verlinde's formulation it is again a quantum surface with the metric e "(~'~). Because o-(z, zT) is not a classical field, the partition function ZL[m] in Eq. (4.1) cannot be written as ZCl[m] in our theory. Note again that the holomorphic sections ~ [ m ] in both theories are related only to the quantum fluctuations. Although we have observed some remarkable features of the quantum Liouville theory associated with ~o~t and obtained a natural concept of quantum 2D space-time within the framework of our formulation, there still remain some important problems to be investigated. I list some of them below. The first and maybe the most important problem is the explicit relation between our quantum Liouville theory and Takhtajan's. These

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T. Suzuki~Nuclear Physics B 492 [PM] (1997) 717-742

two models give the same central charge. Moreover, in both approaches the classical geometry appears as a background, namely the quantum metrics are to be expanded around the classical metric of the Riemann surface, the Poincar6 metric. Inspired by the agreements, it is quite interesting and important to find the relation, although at first glance these two approaches are completely different, i.e. Takhtajan's approach is fully geometric and ours is algebraic. Second, the Virasoro structure of this model is an interesting problem. Finding this structure will allow more explicit discussions of our Liouville theory, especially of the correlation functions. This also maybe shed some light on the problem of c > 1 discrete series of the Virasoro algebra. It is well known that representations of the Virasoro algebra with c > 1 always form continuous series. However, since the highest weight representations ~o~t form a discrete series and our model can have central charges greater than 1, it can be expected to find such a discrete series. These discussions will appear elsewhere [23].

Acknowledgements I wish to thank Profs. L.D. Faddeev and L. Takhtajan for communications on strongly coupled Liouville gravity. It is also a pleasure to acknowledge Profs. M. Ninomiya, R. Sasaki and Dr. N. Ano for discussions and comments.

Appendix A. Infinite dimensional representations with q at a root of unity This appendix gives a brief review of Ref. [ 11 ], where the infinite dimensional highest weight representations of Uqsl(2, C) are examined for the case when the deformation parameter q is a root of unity. Let q = exp rri/p. 6 The most essential feature of which we have made full use in our discussions is stated in the following theorem.

Theorem 1. Every infinite dimensional irreducible highest weight representation, denoted as Vtu'sl(2'c) is necessarily isomorphic to the tensor product of the two highest weight representations as

vyqsl(2,C) ,~ VSl(2,~) (~) gGtsl(2,C) F '

(A.1)

where V~1(2'~) is a representation of the classical algebra s u ( l , 1) ~ sl(2, R) and gFUqSl(2, C) is a finite dimensional representation of Uqsl(2, C). Below we will prove this theorem (see Ref. [ l l ] for detailed discussions). Let X+, X_ and K = qH be the generators of the quantum universal enveloping algebra Uqsl(2, C) satisfying the relations 6 In the previous sections, we used the choice q = exp(cript/p). For the brevity of our discussions, we will choose p' = l in this appendix. The essential parts of our discussion are independent of the choice for pt.

T. Suzuki~Nuclear Physics B 492 [PM] (1997) 717-742

K 2 _ K-2

q_q-I

[X+,X_]-

'

KX+ =q+IX+K.

739

(A.2)

To get infinite dimensional representations, we define the Hermitian conjugations as X~ = -XT:, Kt = K -1. As in the classical case, we represent Uqsl(2,C) with these conjugations by constructing the highest weight module ~1 on the highest weight state A~ such that X_A~ = 0, KA h = qha~, X~_ A0h,r = 0, 1 . . . . }. k), = {a~! I A~ := [r][

(A.3)

Using the Hermitian conjugations and the relations (A.2), the norm of the state A~ is easily calculated as

IIA~II2 = I 2h + r - 1

(A.4) q

with the normalization IIA~II2 = 1. The first problem we come across when q is a pth root of unity is that the norm of the state A~', diverges owing to the factor [p] = 0 in the denominator. Thus, for an arbitrary highest weight h, the highest weight module Vh is not necessarily well defined. The only way to avoid this undesirable situation is to require that there exist two integers /z E {0, ! . . . . . p} and u E 1~1, such that the highest weight is given by

1

h = hu, := ~ ( p u - / . t +

1).

(A.5)

For the highest weight hu,, the factor [2hu, + / z - l] = 0 appearing in the numerator of the right-hand side of the norm (A.4) makes the pth state a finite-norm state, and so the module VI,,~ is well defined. However, due to the zero, there appears a set of zero-norm states, oo

,~tzu = ~ { A k p +h # ,

h

h Akt,+u+ I . . . . . A(k+l)p_t}.

(A.6)

k=0

One can immediately show that, with the relation [kp + x] = ( - - ) k [ x ] , II)(112 = 0 for X C Xu.~. It is interesting to note that the Vh~ actually form a discrete series parameterized by the two integers /x and u. This is a characteristic feature of ~o~t. Indeed this is not the case for the infinite dimensional representations of the classical algebra sl(2, C) and of Uqsl(2,C) with generic q as well. It will turn out that one of the parameters, say u, parameterizes the classical sector Vs1~2'~) and the other, /z, • , ,U~sl(2 C) parametenzes VF'" ' . Let Vu.,, := k),,,~ and A~" :----- A rh~ . The second problem we encounter in the construction of an irreducible highest weight module is that the X t', are nilpotent on the module V~,,, i.e. ( X ~ ) t ' A = 0 for VA C Vu,,,. Therefore one cannot move from one state to another by acting X+ or X_ successively. Moreover, the relation K 2N = 1 on Vu,, indicates that one

740

T. Suzuki~Nuclear Physics B 492 [PM] (1997) 717-742

cannot measure the weight of a state completely. In order to remedy these situations, we should change the definition of Uusi(2, C) by adding new generators L1 . -

(-X-)/~' [p]! '

L_~ :=

X~ [p].

i'

L0 :--

1 [2H+p-lJ p

2

q

(A.7)

to the original ones, X+, X_ and K = qH. The complete highest weight module Vuw is /,1.. constructed by acting X+ and L - i on the highest weight state A0 , which is defined by the relations X_ A~" = Li A~" = O. The third problem, which we do not encounter in the classical case and the case when q is not a root of unity, is that there is an infinite chain of submodules in Vuw. That is to say, Vu,,, is no longer irreducible. Let us observe this characteristic feature briefly. First of all, one sees that the state A~," is a highest weight state because both X_ and LI annihilate this state. Therefore a submodule exists on the state A~ v. Since this state has weight h,, u +/1. = h_u,, the submodule can be regarded as V_u.~. Next one again --UP finds a submodule Vuw+2 on At,_ ~ C V_u.,,. Repeating this procedure, one obtains the following chain of submodules in the original module Vu,,,: Vp.,,, ~ V_#.p D Vua,+2 D V-u.`,+2 ~ . . . D Vtz.,,+2k D V-#a,+2k D . . .

(A.8)

Then, the irreducible highest weight module on the highest weight state A 0U`, , denoted by V~' i,-r .... is obtained by subtracting all the submodules, and it is obtained as

&

u,v'

u.,' := t"kp+., I s = 0 ' l

.... .

/.t - l}.

(A.9)

k---0

It should be noted that all the states subtracted in the procedure are the elements in the set A'u.~ and, therefore, no zero norm state exists in V~., namely,

V~'ir,, = Vu.,\d(uw.

(A. 10)

We have now obtained irreducible highest weight modules. Note that every irreducible infinite dimensional highest weight representation is composed of an infinite number of blocks V ~k), k = 0, 1. . . . . each of which contains a finite number of states. This feature is the very origin of the fact stated in Theorem A.I. To see this is the next task. Now we are the last stage to prove the theorem. To complete the proof, we make some observations. In the following we will denote the level of a state by kp + s where s runs from 0 to ~ - 1 and k = 0, 1 . . . . . instead of r, r = 0, 1 . . . .

Observation 1. On V irr +s xkP ±

[kp+s]!

_ (_)~kik_l>+k.,t,~j X~ k! I s ] ! '

(A.ll)

Observation 2. The sets of generators { X + , X _ , K} and { L I , L - 1 , L0} are mutually commutable on the irreducible highest weight module

irr • g/~./..

T. Suzuki~Nuclear Physics B 492 [PM] (1997) 717-742

741

These observations indicate that there exists a map p : Vu,~ i~ ~ V inf ® ~;, where V is the finite dimensional space composed o f / z states. Further p induces another map /) - Uqsl(2,C) + U inf ®b/, such that p ( O A ) = ~ ( O ) p ( A ) . In the second paper in Ref. [ 11 ], such isomorphisms p and t3 have been obtained. In the following, we shall restrict the module V//.,.v i~r to the unitary irreducible representation. In that case, p( a@+.,) = ( -

-

~

~FJ.,

(A.12)

and

#(x±) =1 ® (±,y±), . b ( L i t ) = ( qzG±l ) ® 1, where sr = v / 2 , j = ( % -

F3(K) = 1 Q/C,

(A.13)

t3(L0) = Go ® 1,

1 ) / 2 and m = - j

(A.14)

+ s.

Observation 3. The actions of {if±,/C} E/.4 on the state g'J, C V are calculated as J i g ~j, = [ j ± m +

1 ] ~/t~,±l ,

K~P'j' = qm~F'/,,

Observation 4. The actions of G,, (n = ± 1 , 0 ) ¢ U i'f on a{ C

G,a{=(2(+k-

l)a{_,,

G_,A

=

(A.I 5) V inf a r e

as follows:

(k+ I )ak+ 1, (A.16)

Observation 3 shows that 3"+ and /C satisfy the relations of Uqsl(2, C), i.e. /.4 = Uqsl(2, C) with the Hermitian conjugations ,J,~ = ff~,/C t =/C -1. Therefore, taking into account that V~'i{, has positive norm, V is a unitary finite dimensional representation of Uqsl(2, C) with highest weight j. We rewrite such a representation as Vj, i.e. dim ~i = 2j + 1. On the other hand, Observation 4 leads us to the following relations among G,, n = 0, i l :

[ G,,, G,,, ] = ( n - m) Gn+,,,,

(A.17)

and Hermit±an conjugations G ~ I = Gq:l, G~o = Go. Therefore we can conclude that U inf is just the classical universal enveloping algebra of su(1, 1) ~ sl(2,]~), i.e. U inf = Usl (2, IR) and V inf is the unitary highest weight representation of sl(2, R) with highest weight (. We denote the representation by Vii. We have now finished the proof of the theorem and obtained the important structure o f "]~o~t as

v~rr~, ~ V~! (~V.. i.

(A.18)

In the theorem, V)'ru.~, V~l and Vj. are denoted by Vy'~l(2'c), V~1(2'~) and wUqsl 'F (2,C) , respectively.

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T. Suzuki/Nuclear Physics B 492 [PM] (1997) 717-742

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H. Poincar6, J. Math. Pure 5(4) (1898) 365. A. Polyakov, Phys. Lett. B 103 ( 1981 ) 207, 211. A. Polyakov, Mod. Phys. Lett. A 2 (1987) 893. V. Knizhnik, A. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819. F. David, Mod. Phys. Lett. A 3 (1988) 1651. J. Distler and H. Kawai, Nucl. Phys. B 321 (1989) 509. See e.g., N. Seiberg, Prog. Theor. Phys. (Suppl.) 102 (1990) 319; P Ginsparg and G. Moore, Lectures on 2D Gravity and 2D String Theory, Lectures given at TASI Summer School, 1992, hep-th/9304011, and references therein. i81 J.-L. Gervais, Phys. Lett. B 243 (1990) 85; Commun. Math. Phys. 130 (1990) 257; 138 (1991) 301; J.-L. Gervais and B. Rostand, Commun. Math. Phys. 143 (1991) 175; E. Cremmer and J.-L. Gervais, Commun. Math. Phys. 144 (1992) 279. [91 E Smirnoff and L. Takhtajan, Towards a quantum Liouville theory with c > I, University of Colorado preprint, 1990. [10l J.-L. Gervais and A. Neveu, Phys. Lett. B 151 (1985) 271; J.-L. Gervais and B. Rostand, Nucl. Phys. B 346 (1990) 473; J.-L. Gervais, Commun. Math. Phys. 138 (1991) 301; J.-L. Gervais and J.-L. Roussel, Nucl. Phys. 426 (1994) 140; J.-L. Gervais and J. Schnittger, Phys. Lett. B 315 (1993) 258; Nucl. Phys. B 431 (1994) 273. [ I 1 I T. Matsuzaki and T. Suzuki, J. Phys. A 26 (1993) 4355; T. Suzuki, J. Math. Phys. 35 (1994) 6857. 1121 T. Suzuki, Phys. Lett. B 338 (1994) 175. 113] L. Takhtajan, Mod. Phys. Lett. A 8 (1993) 3529; A 9 (1994) 2293. [141 L. Takhtajan, Topics of quantum geometry of Riemann surfaces: Two-dimensional quantum gravity, hep-th/9409088. [ 151 J.-L. Gervais and A. Neveu, Nucl. Phys. B 199 (1982) 59; B 209 (1982) 125. 1161 L.D. Faddeev and L, Takhtajan, Lecture Notes in Physics 246 (1986) 166. 1171 P. Zograf, Leningrad Math. J. I (1990) 941. [ 181 P Zograf and L. Takhtajan, Funct. Anal. Appl. 19 (1986) 216; Math. USSR Sbomik 60 (1988) 143. 1191 L. Takhtajan, Uniformization, local index theorem, and the geometry of Riemann surfaces and vector bundles, Proc. Symp. Pure Math. 49 (1989) 581. [201 T. Curtright and C. Thorn, Phys. Rev. Lett. 48 (1982) 1309; E. Braaten, T. Curtright and C. Thorn, Phys. Lett. B 118 (1982) 115; E. Braaten, T. Curtright, G. Ghandour and C. Thorn, Phys. Rev. Lett. 51 (1983) 19; Ann. Phys. 147 (1984) 365. 121 ) A. Polyakov, Lectures at Steklov, Leningrad 1982, unpublished. 1221 H. Verlinde, Nucl. Phys. B 337 (1990) 652. 1231 N. Ano and T. Suzuki, work in progress.