t'~I)qEVIER

27 October 1994

Physics Letters B 338 (1994) 175-180

PHYSICS LETTERS B

Towards a strong coupling Liouville gravity Takash i Suzuk i 1

Department of Mathematics and Statistics, University of Edinburgh, Edinburgh, UK

Received 7 July 1994 Editor: P.V. Landshoff

Abstract

The possibility of strong coupling quantum Liouville gravity is investigated via infinite dimensional representations of Uqsl(2,C) with q at a root of unity. It is explicitly shown that the vertex operator in this model can be written by a tensor product of a vertex operator of the classical Liouville theory and that of weak coupling quantum Liouville theory. Some discussions about the strong coupling Liouville gravity within this formulation are given.

In the recent developments of 2D quantum Liouville gravity theory [ 1] as a non-critical string theory [2- 6], one of the important features is that it has quantum group structure [7,8]. Precisely, the vertex operators in the theory can be expressed in terms of the rep- resentations of Uqsl(2, C). Upon remembering that there are two kinds of representations of Uqsl(2, C), namely, of finite dimension and of infinite one, and that they are completely different from each other, we can expect that there are two phases in the Liouville gravity. These phases correspond to, so-called, weak coupling regime and strong coupling regime. A num- ber of works have revealed remarkable features of the weak coupling Liouville gravity where finite dimen- sional representations appear. However, in the weak coupling regime, the well-known restriction, that is, the D = 1 barrier has bothered us.

On the other hand, successful work for the strong coupling Liouville gravity has been done by Ger- vais and his collaborators [9]. In these works, they made use of the infinite dimensional representations

of Uqsl(2,C) with generic q and have shown that consistent theories can be built provided the central charge of the Liouville gravity is 7, 13 or 19. However, up to now, no work has been done for the quantum Liouville gravity based on the infinite dimensional representations of Uqsl(2, C) with q at a root of unity. Hereafter we will denote by ~ the infinite dimen- sional representations of Uq~l(2,C) when q is a root of unity. We can expect that the theory with such a q will be drastically different from that with a generic q. Indeed, in [ 10] it has been shown that ~) is com- pletely different from the representations with generic q. It is, therefore, quite interesting and worthwhile to investigate the structure of the quantum Liouville theory via such representations. The importance of the Liouville gravity based on ~ is announced also in Ref. [8]. In this letter we will examine the structure of the strong coupling Liouville gravity when we ap- ply ~ instead of the finite dimensional representations or the infinite dimensional ones with generic q.

The Liouville action to start with is given by

I Present address, YITP-Uji, University of Kyoto, Uji 61 I, Japan.

0370-2693/94/$07.00 (~) 1994 Elsevier Science B.V. All fights reserved SSDI 0 3 7 0 - 2 6 9 3 ( 9 4 ) 0 1 1 4 2 - 7

176 T. Suzuki/Physics Letters B 338 (1994) 175-180

SL(~) 1 = f { ½ ° eo4,e . + Ae 'l'(z'e) }

ao f d2z v'~R(Da'(z, Z) (l)

where we have chosen the metric on a Riemann surface £ as ds 2 = e~'(z'~)~z~dzd~ and y is acoupling constant and A is the cosmological constant. The last term in ( 1 ) is added in order for the theory to be conformal. Indeed, upon the Liouville equation of motion

-OzOe~(z , g) = - A e *(z'~), (2)

the energy-momentum tensor defined by TaLb:- - 2¢ r (6&/6~b) is traceless, T~ = 0, and satisfies the conservation law, OeT~ = 0. In the derivation of the equation of motion, we have made a choice of the background metric as ~,~b = 8z~.

The classical theory of the Liouviile theory, where Q0 = Qel = 1/y, yields uniformization of a Riemann surface £ to the upper-half-plane (conformally equiv- alent to the Poincar6 disk). The equation of motion (2) is solved and the solution is given by means of arbitrary holomorphic and antiholomorphic functions u ( z ) , v ( ~ ) as

eea(zd) = 3zU(Z)Ogv(g) (1 - u ( z ) v ( g ) ) 2' (3)

which, denoting w = u ( z ) and g, = v(~) , gives the Poincar6 metric

ds 2 = dw A d ~ (1 - w~) 2 (4)

on the Poincar6 disk D = {w E C I Iwl < I}. Thus the functions u and v define the inverses of an uniformiza- t i onmap j : D ~ E , j - l ( z ) = w,J - I ( ~ ) = ~.When we quantize this system, the functions u, v and, there- fore, the coordinates w, • on D become operators. In this fact we can glance at the quantum fluctuation of the metric of the manifold, and this is the reason why we regard the quantum Liouville theory as a theory of quantum 2D gravity or a quantum geometry of a surface.

To study an algebraic structure of the classical Li- ouville theory, let us take the hth (h E Z /2 ) power of the metric (3). One immediately finds there are two entirely different cases according to whether (I) h >

0 or (II) h < 0. It lies in this fact that there are two kinds of quantum Liouville gravity, i.e., strong cou- pling and weak coupling regimes. In order to see this more explicitly, we calculate the hth power,

(I)

(u)

e-J~(zd) = \{ ~az'~zl -- U(7.)U(7.)~) ) 2j

J

m=--j

e h~(z'~) = ~(~/tgzU(Z)C~o(~ )2h

oo _~ ~"~ h h h r - N;a,(z)a ' (z) ,

r=0

(5)

(6)

where we used h = - j in (5) and NJm(N h) are nor- realization constants. In the latter discussions, we are not interested in the explicit forms of the functions and A, but the crucial fact is that ~ (z ) and )t h (z) form, respectively, finite and infinite dimensional rep- resentations of~/(2, C). One can further show that the chiral sector ~,J (z ) and a h (z) satisfy the Poisson-Lie relations [ l 1,12].

With the above results of the classical Liouville the- ory in mind, let us turn to the quantum version of the theory. The quantization approach which many authors have followed is the canonical quantization where an equal-time commutation relation between the Liou- ville field ~ ( z , Z) and its canonical conjugate is im- posed. In the following, however, we quantize the Li- ouville theory according to the well-known fact that the quantization of the Poisson-Lie algebra g yields the quantum universal enveloping algebra Uqg. In fact, it has been shown that the quantum Liouville theory has the symmetry UqSl(2, C). We then assume that the quantum version of the hth power of the Liouviile metric, which is called the vertex operator with charge h, also splits into chiral and antichiral sectors as in (5) and (6). The chiral sectors, denoted as ~ ( z ) and Ah(z ), are now the finite and infinite dimensional representations of Uq~l (2, C), respectively. The defor- mation parameter q is given in terms of the coupling constant by q -- exp i¢ry2/2.

The gravity in case (I) is called the weak cou- pling Liouville gravity. This case has been studied in [7,8] from the viewpoint of quantum groups. They

T. Suzuki/Physics Letters B 338 (1994) 175-180

have shown that q ' ~ ( z ) , m = - j ..... j form a finite dimensional representation of Uqsl(2, C) with q = exp i7ry2/2 and satisfy the braiding-commutation re- lations which reduce to the Poisson-Lie relations of sl(2, C) in the classical limit, y ---* 0. In this regime, the central charge of the gravity is given by

cL = 1 + 12Q~ (7)

and the relation between Q0 and the coupling constant y is

1 y = - + (8) Q0 Y ~,

which affirms that the conformal dimension of the metric e *tz'~) is just ( 1, 1 ) [6]. The relation (8) was also obtained in [ 13 ] upon the assumptions that the energy-momentum tensor should satisfy the Virasoro algebra and that Tzz should commute with Tee. In- serting (8) into the central charge (7), we get cL = 13 + 6(or + a - i ) with ot = T2/2 and find the well- known restriction c t >_ 25 (_< 1 ) for real (imaginary) y. If we require the central charge to be I < cL < 25, the coupling constant y becomes a complex num- ber. This is the origin of the difficulty in constructing consistent strong coupling Liouville gravity. The only successful work at present is the work developed by Gervais and his collaborators [9], where the infinite dimensional representations of Uqsl(2, C) have been utilized but the deformation parameter q is not a root of unity. It has been proved that consistent theories can be built when the central charges take the special values cL = 7, 13 or 19.

Contrary to the canonical quantization, Takhtajan has recently performed path-integral quantization of the Liouville gravity in [ 14 ] along Polyakov's original formulation of quantum Liouville theory [2]. Upon a perturbation expansion, it is proved that the central charge of the theory is just (7) with the classical value Qo = Qd = l / y , and so cL is always greater than 1 including the strong coupling region. It is also shown that the dimension of the metric is ( 1, 1 ) as desired. Supported by these illuminating results, let us investi- gate the strong coupling Liouville gravity via ~ , i.e., the infinite dimensional representations with q being a root of unity.

Let us go back to the case (6). The quantum version of (6) is

177

~ D

eh~l,(zd) ~"~Al~h rAh~ z . A h r . - . =2. . , q" r t ) " t z ) . (9) r = 0

The fields Ar h (z ) , r = 0, 1 .... form an infinite dimen- sional lowest weight representation Vh of Uqsl(2, C) with q = exp(zriy2/2). In the following, let q be a pth

t

root of unity, q = exp(Tri~), i.e.,

7 2 p_.~ T = p , (10)

with p, p ' E N, coprime with each other. It is helpful to summarize here the essential features of the repre- sentation with such a q (see Refs. [ 10] for the detailed discussions).

A characteristic feature of the representations is that, upon requiring the representations to be well- defined in the sense that the norm of each vector in the lowest weight module is finite, they are parame- terized by two integers, say It and t,. Thus the repre- sentations are discrete series and the lowest weights are expressed in terms of It, 1: as

h ~ , = ~ P~ + i t - 1 . (11)

We denote by V~,,, = Vh~. the lowest weight module on the lowest weight vector whose weight is h~,,, and by A~ ~ the lth vector in Vu,, where l = 0, 1 ..... One easily sees that the module Vp,, is not irreducible by itself. Indeed, one finds infinite series of submodules in V~,~. This is the essential difference from the infinite dimensional representations with generic q. In order to obtain the irreducible module V~, all submodules should be subtracted correctly. In Ref. [ 10] this task was done and the irreducible module ~'~ was obtained. The remarkable result is that there is an isomorphism p : ~ __., ~ l ( ~ ~j such that

u~ ~__~ A¢ ( ~ ) . j , (12) p : Akp+r

where ( = ~,/2p', j = (i t - I ) / 2 and j + m = r.

Therefore the irreducible lowest weight module V~ is isomorphic to the tensor product of the infinite di- mensional lowest weight module V~ I of the classi- cal sl(2, C) and the finite dimensional highest weight module Uj of Uqsl(2, C),

V~ ~ We' (~) Uj. (13)

178 7". Suzuki / Physics Letters B 338 (1994) 175-180

We will make full use of this result in the following investigations of the strong coupling Liouville gravity.

In order to apply the above results to the strong cou- pling Liouville gravity, we first introduce correlation functions defined by

N P

ZS[m : tzi, vii = / [ d c ~ ] e -sL(~') H eh"'"'l'(z~d')' , 1 i=l

(14)

where m = ( m , ~ ) is the moduli parameter of the surface. In order to obtain physical amplitude, we have to integrate ZS[m] over the moduli space .Ms.N with Weil-Peterson measure d(WP). Putting together (12) and (9) with the replacement h ~ h~,,, we find

eh~,,'t'(zd) ,~ e¢~a(zd) ( ~ e - j ~ ( z d )

O 0

k--O

J ~ A/'mJ xI'Jm ( Z ) xt'J'm ( Z), (15)

m - - j

where ~Pe] (z, Z) and q~(z, Z) are, respectively, the clas- sical and quantum Liouville fields. This result sug- gests that the strong coupling Liouviile gravity based on ~ is decomposed into the classical sector and the quantum sector which is associated with the finite di- mensional representations of UqSl(2, C).

We now conjecture that the action also separates as

SL(aP) ~ ~(~o~) +K2sq(~o) (16)

with

5~'(~Pc,) = ~ d2z (Oz~Pc, az~c, + A,e~"(z'~)),

(17) 1/ + A2e¢(Z'~)). (18)

A new coupling constant fl has been introduced for the classical sector because, at the moment, we have no idea what it should be. On the contrary, since, as described in (13), the finite dimensional representa- tion ~.~j has the same deformation parameter q as Vi~, the coupling constant for the quantum sector should

be the same as the original one, T. The relative con- stant r 2 between the classical and quantum sectors has been introduced and can be renormalized into the coupling constant y. With (15) and (16), the correla- tion function (14) will be factorized into the classical correlation function Zel[m : ~'i] and the quantum one z q [ m : j i] ,

ZS[m : t.ti, Pi] "~ Zd[m : ¢i]Zq[m : ji]. (19)

Let us discuss this result in more detail. We first look at the classical sector briefly. The path-integral in this sector is trivial and yields

ZCl[m : g'i] = exp ~ N ~ 0 c l .] (20)

where c = 12/fl 2 is the central charge of the classi-

cal Liouville gravity, and ff:~(~p) is the regularized ac- tion obtained by subtracting the singularities near the points where vertices are inserted. For example, in the case when the topology is a sphere and all vertices are punctures, i.e., ~'i = 1/fl 2 for Vi, it is [ 15]

S,v(cpd) = d2z dz~pclOeCpc! + Ale ~d(zd)

+ 2~Nlog e + 4¢r(N - 2) log [ log el), (21)

where

E, =E \U~VI{Iz - zil < e} U {Izl > a/e}.

We next proceed to the quantum sector. The action Sq(~p) is just the Liouville action in the conformal gauge ds 2 = e~(Z'~) dzd~. The coupling constant for this action is renormalized to ~ = "r/K and, therefore, the central charge coming from this sector is

Cq ---- 1 + 1202 (22)

where 0 = KQ. Note that the correlation function Z q [ m,ji] admits holomorphic factorization and can be written as

Zq[m: j i ] = ~ - ' ~ N t ' J ~ l [ - ~ : j i ] ~ j [ m : j i ] . (23) I,J

Here ~ t [ m : j i ] is the Virasoro block satisfying the conformai Ward identity obtained by Polyakov [3] and will be regarded as a holomorphic section of a line

T. Suzuki / Physics Letters B

bundle over the moduli space .A4g,N and N H is some constant matrix.

Let us next give a geometrical interpretation of the strong coupling Liouville gravity. Putting (20) and (23) together and integrating ZL[m :/zi, ui] over the moduli space with Weil-Peterson measure, the ampli- tude is written as

" m ( ( i ' j i ) = Z N'J f d(WP) exp ( _ ~ c ~;ct'~/

l,J Ms.I v

x ~ ' / [ ~ ; j i ] Wj[m :ji]. (24)

In order to observe this result from the viewpoint of the geometric quantization, we should notice the fact that the Liouville action evaluated on the classical solution is the K~ihler potential of the Weil-Peterson

metric, i.e., o a ~ t = itowp/2. This remarkable rela- tion was obtained in Ref. [16] for the topology of the N-punctured sphere. Upon comparing our result with the general concept of the geometric quantiza- tion, the holomorphic part ~t[m] of the quantum sector is regarded as a holomorphic section of a line bundle £~ over the moduli space .A4g,N with curva-

c to On the other ture given by the K/ihler form ~ wP. hand, the classical correlation function corresponds to a Hermitian measure factor defining an inner product ( I )c in the Hilbert space which is the space of sec- tions on Lc. Hence, at least for the topology of the N-punctured sphere, the amplitude can be written as .A = ~ Nl'J (WtlW j)~ up to a constant.

Finally it is interesting to give a comment on the central charge. The central charge of the original strong coupling Liouville gravity SL(*) is given by CL = 1 + 12/y 2 as calculated by Takhtajan [ 14]. On the other hand, from the right-hand side of Eq. (19), the total central charge is the sum of the classical one, 12//32 , and (22). We then get the following relation among y, ,8 and Q,

1 1 (~2. y--2 = ~-~ + (25)

Here we assume that y and ,8 are real. There are three phases for the quantum sector according to the values of the coupling constants y and ,8; (i) y2 > ,82, (ii) y2 </32 and the boundary between the two, namely, (iii) y2 = ,82. We set CL = 26- - D, i.e., y - i = V/(25 - D ) / 1 2 in order that the central charge from

338 (1994) 175-180 179

the total physical system, that is, the strong Liouville mode ~ (z , ~) plus string sector XU(z, E) embedded in D-dimensional target space, cancels against that from the ghost system. Before going to the discussions of each phases, it is convenient to rewrite the action of the quantum sector as

1/ K2s~(~) ---, ~ a2z ( o ~ + O~t¢~,~

+ A2e'~(z'~)), (26)

where we have used the normalized field ~ - ~v/~. In the following, we will use the notations c = 12/,82 and D ~ = D -4- c. Let us see these phases separately. (i) In this case, 0 is an imaginary number. This case corresponds to D ~ > 25. Supposing A2 = 0 and de- noting 0 = iat0, the quantum sector is nothing but the minimal CFT with the background charge a0 = v/(D ' - 25)/12 and the central charge being cq = I - 12ate. The screening charges are given by at+ = (x/D' - 25 T V/-D-7 -- 1 ) / v / ~ . The vertex operator

e -j~6 = e -q'~'6, which is the highest weight state in a finite dimensional representation of Uqsl(2, C) ® Uqsl(2,C), is the primary field of the type (2j + 1, 1) ® (2j + 1). The section ~[m, ji] is the confor- mal block of the Virasoro minimal series. (ii) In this case, Q is real. This case corresponds to D t < 25 and the quantum sector may be regarded as the weak coupling gravity. (iii) Since in this case Q = 0, the contribution 1 in the central charge cL comes just from the quantum sector. This reminds us of Takhtajan's result [ 14], where he showed upon a perturbation expansion that 12/y 2 = 25 - D and 1 in cL came from the tree di- agram and quantum corrections, i.e., loop diagrams, respectively. In this case, the field ~(z, f ) can be re- garded as the (D -t- 1 )th component of the string co- ordinates, namely, ~ = X °+l. Remember that, in the ordinary weak coupling gravity when D = 25, the Li- ouville field can be regarded as the time component of the string in the 26-dimensional Minkowski space. In our case, the same situation will happen for a dif- ferent dimension D by choosing the classical central charge c appropriately.

In summary, we gave a description of a possible strong coupling Liouville gravity via the representa- tion ~ . Our investigations were based on the fact (15) and the conjecture (16). That is to say, the strong

180 T. Suzuki/Physics Letters B 338 (1994) 175-180

coupling Liouville gravity associated with ~D will be a total system of the classical Liouville gravity and the quantum sector which is associated with the finite di- mensional representations of Uqsl(2, C). We also dis- cussed some properties of our gravity theory. However we have not discussed the anomalous dimensions of the vertex operators at all. In order to do this within the framework of quantum groups, we have to exam- ine how the braiding-commutation relations in ~) are decomposed into the classical and quantum sectors. Once we know the dimensions, we can find the relation between Q and ~, like (8) in the weak coupling Liou- ville gravity. Further studies will appear elsewhere.

In the light of our results, let us try to give an an- swer to the question: What is quantum space-time? In quantum gravity theory, every point on the space-time manifold fluctuates by a quantum effect. If we can re- gard the fluctuation as a sort of internal space on every point of the manifold, the quantum space-time can be recognized as an object which consists of a classical manifold and the internal spaces on it. As for the case (i) discussed above, the quantum fluctuations, that is, the internal spaces correspond to the minimal matter. In this sense, we can say that the strong coupling Li- ouville gravity for the case (i) is the unified theory of gravity and matter. Recently, another unified pic- ture of gravity and matter has been proposed [ 17] in the framework of the Connes' noncommutative differ- ential geometry [ 18 ]. There, the total system is built on the space M4 ® Z2, where M4 and Z2 are the 4- dimensional classical Minkowski space and the dis- crete "two-point" space, respectively. It must be inter- esting and helpful for our understanding of the quan- tum space-time to search for the relationship between the formulation and our theory.

The author would like to thank Dr. H.W. Braden and Dr. T. Matsuzaki for valuable discussions and col- laborations.

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