- Email: [email protected]
Nonperturbative analysis of the three matrix model
Volume 247, number
PHYSICS
1
Nonperturbative Hiroshi
Kunitomo
LETTERS
B
6 September
1990
analysis of the three matrix model
‘J,~ and Satoru
Odake
2,4
Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Received
26 March
1990
Recently developed techniques are applied to give exact nonperturbative solutions of the three matrix model. There exist three different critical points and the specific heats of the model corresponding to these points satisfy different differential equations (string equations). The critical point expected to correspond to the tricritical Ising model is connected to the Ising model by a critical line. The string equations on each scaling limit coincide with the type (p, q)= (4, 5), (2, 7) and (3, 8) equations of Douglas’s general argument.
In spite of beautiful developments in string theory, we are still in the dark about its nonperturbative properties. Recently, some remarkable progress has been made in the nonperturbative approach to two dimensional gravity or noncritical string theory [l3 ]. An exact nonperturbative solution has been given for discretized two dimensional gravity by using the large-Nmatrix method with some double scaling limit for large N and critical coupling. The other solutions exhibiting multicritical behaviour and indexed by a positive integer m ( = 2, 3, ...) has also been obtained. These solutions were initially believed to correspond to minimal unitary conformal models coupled to quantum gravity. It was shown, however, that this conjecture is not true and that the m = 3 multicritical theory #’ corresponds to the Yang-Lee edge singularity [ 41 which is described by a nonunitary CFT with c= -y [5]. Furthermore, it has been found that the correct exact specific heat of the Ising model, which is realized by the two matrix model [6-S] actually satisfies a different string equation from that of the m = 3 mul’ E-mail address: [email protected] 2 Fellow of the Japan Society for the Promotion of Science. 3 Address after April 1, 1990: Research Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan. 4 E-mail address: [email protected] #i The parameter m is related to the general formula for the minimal conformal series as c= l-6(p-q)*/pq with p=2, q=2m- 1. 0370-2693/90/$03.50
0 1990 - Elsevier Science Publishers
ticritical model [9-l 11. Therefore, we can easily imagine that the next candidates for the models describing the unitary CFTs are given by generalizing the two matrix model [ 12,13 ] to n matrix models. The solvable n matrix models were first introduced in ref. [ 141 and described by the n NX N hermitian matrices M, (i= 1-n) as #’ Z,=
fi dft4, exp -tr I r=l (
i: V,(M,) i=l
n-1
1 1 +l'ixJ.
X.(x)=$x2+
gj!
ia
The nonperturbative properties of these models were studied by Douglas [ 18 ] from the point of view of the generalized KdV hierarchies (GKH). He claimed that these models are governed by some series of equations
[Q,QP’l=l
>
(2)
with the pth order differential Q=
ap+ ,d a,
a-
.
operator (3)
X2 In the limit n+a, these models become the one dimensional string theory (CFT with c= 1) [ 1S-171. This fact also supports the above expectation.
B.V. (North-Holland)
57
Volume 247, number 1
PHYSICS LETTERS B
6 September 1990
Nonperturbative analysis of the three matrix model Hiroshi K u n i t o m o ~,2,3 a n d Satoru O d a k e 2,4 Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Received 26 March 1990
Recently developed techniques are applied to give exact nonperturbative solutions of the three matrix model. There exist three different critical points and the specific heats of the model corresponding to these points satisfy different differential equations (string equations). The critical point expected to correspond to the tricritical Ising model is connected to the Ising model by a critical line. The string equations on each scaling limit coincide with the type (p, q ) = (4, 5 ), (2, 7) and (3, 8) equations of Douglas's general argument.
In spite of beautiful developments in string theory, we are still in the dark about its nonperturbative properties. Recently, some remarkable progress has been made in the nonperturbative approach to two dimensional gravity or noncritical string theory [ 13 ]. An exact nonperturbative solution has been given for discretized two dimensional gravity by using the large-N matrix method with some double scaling limit for large N and critical coupling. The other solutions exhibiting multicritical behaviour and indexed by a positive integer m ( = 2, 3 .... ) has also been obtained. These solutions were initially believed to correspond to minimal unitary conformal models coupled to quantum gravity. It was shown, however, that this conjecture is not true and that the m = 3 multicritical theory ~l corresponds to the Yang-Lee edge singularity [ 4 ] which is described by a nonunitary CFT with
[51. Furthermore, it has been found that the correct exact specific heat of the Ising model, which is realized by the two matrix model [6-8] actually satisfies a different string equation from that of the m = 3 mulE-mail address: [email protected] 2 Fellow of the Japan Society for the Promotion of Science. 3 Address after April 1, 1990: Research Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan. 4 E-mailaddress: [email protected] ~ The parameter m is related to the general formula for the minimal conformal series as c=l-6(p--q)2/pq with p = 2 ,
q= 2 m - 1.
ticritical model [9-11 ]. Therefore, we can easily imagine that the next candidates for the models describing the unitary CFTs are given by generalizing the two matrix model [ 12,13 ] to n matrix models. The solvable n matrix models were first introduced in ref. [ 14 ] and described by the n N × N hermitian matrices Mi ( i = l - n ) as ~2 Z~ = f f i
dMi exp ( - t r ~
i=1
Vi(Mi) 1
°-'
)
+ tr ~ ciMiMi+ 1
,
i=1
V ( x ) = ½x2 + ~ 1 giJxJ " j>~3 N j / z - t j!
( 1)
The nonperturbative properties of these models were studied by Douglas [ 18 ] from the point of view of the generalized KdV hierarchies ( G K H ) . He claimed that these models are governed by some series of equations [Q, Q~{p] = 1,
(2)
with the pth order differential operator P
Q = 0 P + ~] a,O p-~.
(3)
i=2
~2 In the limit n - * ~ these models become the one dimensional string theory (CFT with c = 1 ) [15-17]. This fact also supports the above expectation.
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
57
V o l u m e 247, n u m b e r 1
PHYSICS LETTERS B
The minimal unitary series is identified with (p, q ) = ( m , m + l ) , m>_-3. However, we think that Douglas's arguments are still obscure and need further confirmation by solving the models concretely. In this paper, we will solve the three matrix model explicitly, and show that there is a critical point on which the system is actually controlled by the expected equation, eq. (2), with (p, q) = (4, 5). Moreover, we will find two more critical points which are governed by the string equation, eq. (2), with (p, q ) = (2, 7) and (3, 8). Let us first summarize the general formulae for the n-matrix model, which are applicable to asymmetric coupling constants. The method for solving the n matrix chain model described by eq. (1) with the restricted coupling constants ci . ~ - Cand g~j= gj for Vi, has been formulated in ref. [ 14] by means of the orthogonal polynomial method. We give here some formulae for more general coupling constants as in eq. ( 1 ). In this model, we can simultaneously diagonalize the matrices M~ as M~= U,tX,•Ui using the unitary matrix U~, and the partition function eq. ( 1 ) is rewritten by diagonal matrices X~ as [ 12-14 ]
Z~=K f f l dX, J(X~)LJ(X,) i=l
Vi( XJ + tr " ~ ciXiX,+ l )
×exp (-tr ~
i= 1
N--1
=KN! Y" h~,
(4)
i=o /
n
\tt~=l
NN(N--1)/2
ci/
6 S e p t e m b e r 1990
those in ref. [ 14 ]. If we introduce the notation
J[+ f(x) = ~ dy exp [ - V + (y) + c +xy] f l y ) , E ( x r , j - ~ ? k ) , , j ) ~ ... ~e~ ~ ? U ( x ) = o J
(0~
(8)
d r , 7-fliTj)Pf +\ ~ /~-~ (x)=O ,
c : ~ 0+)
--'~ v ~ ' ( ,
+~_,~) --C/--+ 10L ~+- - 2 )
(l
(10)
with cff o~(-1) ± =fl± , [Or(o), fl-+ ] = 1, and
Ot~k_l)i,jhj=ot(+_k)j, ihi
(l~
after integrating out the angular matrices U~. Here J ( X ) is the Vandermonde determinant and the h~'s are defined by the following conditions of two orthogonal monic polynomials P+ (x) and P - (y):
hic~,j= ~ f i dxkP)-(X,,)P+(Xl)
(11)
In particular, the last equation eq. ( 11 ) is most important and includes all the informations we need ~3 In the case with Zz-symmetric coupling constants such that E = In+ t_~ and c~= c,_~, simplifications occur, because of the fact that we can choose the orthogonal polynomials as P+ (x) = P ; - (x), and then o~~) = o~~), fl+ = f l - . Only these simpler formulae are needed for our purpose. Let us apply the general formulae summarized above to the three matrix model with Zz-symmetric coupling constants,
=j
i=1
(9)
where cy_i=cF =c~, Vy+~_i= V,+ =V,, the expansion coefficients o ~ ) can be computed from the formulae
Z3= j dAdBdCexp[-S(A,B, C ) ] ,
N
(7)
(12)
d X d Y d Z A ( X ) J ( Z ) e x p [ - S ( X , Y, Z ) ] , (13)
S(A, B, C) = ½t r ( A 2 + B 2 + C 2) I i
+ ~.
tr(gA4+g'B4+gC 4) - c tr(AB+BC). (14)
k=l
×exp 1
Vk(Xk)+ ~ CkXkXk+l .
(6)
k=l
The free energy of the model, therefore, can be calculated from these constants h, In order to obtain these constants h~, we can use similar arguments to 58
It should be noted here that the two coupling constants g and g' are different in general. If we set g' = 0, the B integration becomes gaussian and we get the two matrix model after integrating it out. The rela~3 In these formulae,we assumen >/2. For the one matrix model, only one relation V' (a) = fl is enough to solveit.
Volume 247, number 1
PHYSICS LETTERS B
tion betwwen the coupling constants of the two models is C(two)= c2/( 1 - c 2) and g(two) = g~ ( 1 - c 2) 2 in our notation ( 1 ). The three matrix model, therefore, includes the two matrix model, i.e. the Ising model, as a special case. For other coupling constants, we assume g ¢ 0, c ¢ 0 hereafter. We use a familiar notation for a(o) in the general formulae:
xPi(x) =P i+, (x) + RiPi_l (x) + siei_ 3(x) +
[Note that the absence of the terms Pi, Pi-2 .... in this recursion relation is a consequence o f the fact that Pt(x) has a definite parity P i ( - x ) = ( - )~P~(x) since our action S is quartic (even) ]. The next coefficients ot~) are calculated from this recursion relation as [14] for l i - j [ >_-3, g 6N
-
forj=i+3,
- 6N g f f-l f-2
forj=i-
= 1 + ~ g (Ri+Ri+l +Ri+2)
for j = 1 + 1,
forj=i-
1,
3,
(16)
where f = hJh~_ i. From relation ( 11 ), we obtain
g' a3~,) ) c & = ( a~L) + g-~
(17)
, i,i-- 3
c(f+Ri)
:
-OL(1 ol(1)) + -6N
)
i,i-i
,
Ri 1 + ~ "~- ~g ( S i ~ - S i + 1 @Si+2)-COQl)i,i_ 1 = i .
by considering the naive large-N limit. In the naive large-N limit, we have
i U ~x, --
1 ~f
f(x) g
1
,
~Ri
R(x) - - ,g
(20)
which give the possibility to derive from (18) and (19) the algebraic equations
c 4 ( f + R ) = f ( l +½R)
. . . .
(15)
COL(I ) i,j = 0
6 September 1990
x c2+~-[~J3+~f2(l+½R)+~(l+lR)21
,
(21) (R-f)(1 + ½R) + ~
g' [@zf3+f(l+½R)Z+(l+lR)3] )
C2+g
=gx.
(22)
Here we have eliminated & by using eq. (17) ~4. In principle, we can further eliminate R from the LHS of eq. (22) by solving eq. (21). Then we define a "potential" W ( f ) by the LHS of eq. (22) as the function o f f Since the scaling laws arise from the singular behaviour o f f ( x ) near x = 1, the form of W ( f ) in the neighbourhood of this point controls the criticality. Therefore, in order to tune all the couplings g, g' and c on critical, we must adjust it such that the first three derivatives of W ( f ) vanish,
W' (fo)= W" (fo)= W" (fo)=O .
(23)
The coupling g is related to W as g = W(fo). F r o m the practical standpoint, however, it is not useful to solve eq. (21 ) and to treat W ( f ) as the function o f f only. We solve these criticality conditions under the constraint eq. (21) without solving it in the actual calculation. Although it is a little lengthy, it presents no difficulties to solve the criticality conditions eqs. (23) and (21 ). We find just three different solutions ~5.
(19 )
These three equations ( 17 ) - ( 19 ) are sufficient to get the necessary information about the nonperturbative solution. We now try to find the critical points of the model
~4 It is worthwhile to note that using eq. (17) Si can be eliminated without any approximation. ~s Strictly speaking, the solution (24) gives 0/0 if we put it into eq. (23). This is interpreted as the limit t-~ _ 7 on a line explained later. 59
Volume 247, number 1 (i)
fo= -7
,
g,/g= 11767 9801 , (ii)
PHYSICS LETTERSB
Ro - -
23
--22 ,
245
g~
--~
fo=½(l_+x/7),
g'/g=-81,
Ro - -
3 --2~
(25) c2=_9
g = - ~ 35.
(26)
These solutions would yield three different critical points of the model on the sphere. We investigate the critical point (i) in detail since it leads to a string equation expected to represent the tricritical Ising model (the unitary CFT with c = 7 ) . To keep the higher genus contributions, we must expand eqs. ( 18 ) and ( 19 ) in 1/N. As in refs. [ 1-3 ], it is convenient to introduce a lattice spacing a so that the renormalized "cosmological" constant is given by IZR=(gc--g)/gc a2. Then we have the following expansions: lf+l~f---° ( 1 + ~ g\ k~
aZk/mfk(z--lal/m)) ,
~R~+z~ lvl f__o\fo + ~ a2k/mrk(z--la l/m) ) ,
(27)
(28)
k>~l
with the scaled variable z i --
N
~ 1
--aZz,
(29)
and the critical values (24) for couplings. The parameter m is fixed by the order of criticality [ 2 ] and m = 4 in our case. The coefficient function fi is related to the free energy F a s f l (z) = F " (z). Substituting in eqs. ( 18 ) and ( 19 ) these expansions, we find differential equations f o r f and r~. These differential equations reduce to a single equation forfl in the case of all the nonperturbatively solved models so far [ 13,9,19]. In our case, however, the situation is a little different. The string equation satisfied by f~ does not become a single equation but two coupled differential equations:
f"'+lOff"+5f'2+lOfa+lOr"+2Ofr=O, 60
-1- 15f 4 -
1 4 r " - 7Ofr" - 20f ' r' + 20f" r + 60r 2
- - 770..~.Z ,
Ro=-Y-x/~fo,
g=~8(50+ 35x/7), fo=½,
(24)
g'/g=}(32+_lOx/~),
c2=~2 (5 +_xf7),
(iii)
f(6) q_ 13ff,,,,+23f,,2+60f2f,, + 4 5 f f ,2
C2=4~6
,
6 September 1990
(30)
(
30 cont'd )
where f ~i)= d~C/dzi, f = f i and r = r 2 - f > The constant ~770- in front of z can be taken to 1 by appropriate rescaling. We will see next that these string equations can also be derived from the G K H standpoint. Before considering the concrete case, let us rewrite eq. (2) in a more explicit and convenient form. If we write p
aia p-i,
Q=Op+ ~
(31)
i=2
aq/P=Oq-k - ~ biO q-i ,
(32)
i=2
and use the fact that [Q, Qq+/p]= [Qq_/p, Q], eq. (2) imposes #6 (bi+...)' = 0
for(q+l~i~p+q-2),
-p(bp+o_l +...)'=l ,
(33)
where "..." represents some differential polynomials of aj which disappear if all the integration constants are equal to zero. Since the coefficient functions bi are uniquely determined as the differential polynomial ofai, eq. (33) become the differential equations ofai. In particular, we note that the last expression in eq. (33) is always integrated out to the form of the familiar string equation. Eq. (33) are the most general equations without any restriction. If we wish it to have 712-symmetry Q must be restricted to be a self-adjoint operator. This is equivalent to imposing that Q have a definite parity with respect to 0 ~ - 0 when we write it in the symmetric form p
Q=0p+ ~
(ciOP-%OV-'ci).
(34)
i=2
Now consider the case (p, q) = (4, 5 ). Q is a fourth order differential operator in this case Q = 04 dr (U2 02-~-02U2) "l- (U3 0"1-0U3) -~"U4 ,
(35)
#6 T h e a u t h o r s w o u l d like to t h a n k T. E g u c h i f o r p o i n t i n g o u t t h i s fact.
Volume 247, number 1
PHYSICS LETTERS B
and we should take u3 = 0 for Z2-symmetry. The commutation relation is calculated to be [Q, Q5+/4] = _ 4 b ~ 02_ (4b~ + 6 b ~ ) 0 - ( 4b'8 + 6b~ + 4b~' + 2a2b'6 + a'2b6 ) ,
(36)
with 32b6 = u'~" - 10u2 u~ - 15u~2 - lOu 3 + lOu~ + 20u2 u4,
(37)
128b8 - u ~ 6) + 6 u 2 u z"' + 1 2 u z. .u.z. . . + 9 u 2 2 +50u22u~ ..]_50U2 U ~2 + 3 5 U z 4- { - 2 U 4.
-. - 2. 0 U. 2 U. 4 -. b 2. 0 U. z U 4
+20u~u4 +20u24-60uZ2u4,
(38)
and 64b7= - 3 2 b ~ . Then we get two coupled differential equations b~ = 0 ,
(39)
- 4 b ~ = 1.
(40)
This coincides with eq. (30) up to integration constants with the identification f = u2, r = u 4 - u~. Let us now study a few properties of the string equation eq. (30). If we assume that the solution o f eq. (30) f o r f ( z ) has at most a second order moving pole singularity as in the case of Painlev6 type equations [ 9,19 ], f ( z ) and r ( z ) can be expanded as f ( z ) = ~. c i ( z - z o ) i-2 , i=o
(41)
r(z)= ~
(42)
i=0
di(z--z0)
i-4 ,
and these expansions converge in the vicinity of Zo. The coefficients c~ and d~ are determined such that f ( z ) and r ( z ) are the solution ofeq. (30). It is known that if we can arrange the number o f free parameters in expansions (41) to be equal to the order o f the differential equation, then the solution of such an equation has no moving cut singularity in general [20] ~7. In our case, eq. (30), the expansions (41) must have eight free parameters ~8. The explicit cal~7 We invoke the Painlev6 test to see whether a nonlinear differential equation has enough free parameters. #8 This number is obtained as follows: If we allow a nonpolynomial differential equation it is possible to eliminate r(z) from eq. (30) by solving the third order algebraic equation. Then the resultant differential equation off is of eighth order.
6 September 1990
culation leads, if we choose Co= - 2 , to eight parameters Zo, c2, c3, Ca, c5, c6, c7 and C,o remaining unfixed and to the vanishing of the simple pole term, c, = 0. Therefore our string equation eq. (30) does not admit a solution with any moving branch point ~9 The fact that Co= - 2 and c, = 0 is important. Since f i s the second derivative o f the free energy F, this double pole singularity corresponds exactly to logarithmic singularities of the free energy. Thus the partition function Z = e F is entire in z is and only if the residue Co is equal to a negative integer and if and only if there is no simple pole. Until now we have analyzed the critical point which is expected to correspond to the tricritical Ising model coupled to gravity. We c o m m e n t here on the behaviour o f the other two, (25) and (26). First, consider the case (25). The scaling property of this point is governed by the differential equations indexed by (p, q) = (2, 7) of the G K H classification. Since it is known that this equation is derived from the m = 4 multicritical one matrix model, this critical point may be expected to correspond to some YangLee type edge singularity described by some non-unitary CFT. The case (26) needs a little discussion. Although we find this point to be a third-order critical point (the highest order we can tune with three couplings in general), the fourth-order derivative o f the potential W accidentally vanishes in this point so criticality jumps to fourth order. Because o f this, we must change the lattice spacing expansions to (27) and (28) with r n = 5 rather than m = 4 . With this change one can derive the string equation as follows: 4f(8) + 60ff(6) +
1 8 0 f " f ~5)+ 4 0 2 " f " ' + 306f2f ....
+ 252f"'2+ 1224ff'f" +918ff"2+891f'2f " + 630f3f " + 945f2f'2 + 1 0 8 f s = - 4 2 0 z ,
(43)
wheref=f~. This differential equation eq. (43) was ~9 We also find that this situation is common for all the known string equations. The location of the second order pole Zo is always a free parameter and other ones cj appear at j = 6 , for m = 2 , j = 2 , 5, 8 for m = 3 , j = 2 , 4~~4 for the multicritical one matrix model and at j = 3, 4, 8 for the Ising model with the choice Co= - 2. In all the cases, we have no simple pole singularity, Cl = 0.
61
Volume 247, number 1
PHYSICS LETTERS B
first found in the analysis of the two matrix model with higher order criticality [ 21 ]. Eq. (43) can also be derived from the G K H with (p, q) = (3, 8). The operator Q in this case is
Q : O 3 q- ~(U20+ Obla)q-bl3 ,
(44)
and we set u3 = 0 for the Zz-symmetry. The commutation relation becomes [Q, Q8/3] = _ 3b'9 O - 3 (b',o + b~)
(45)
with b9 = 0 ,
1296blo =4u~ 8) + 6 0 U 2 U~ 6) + 180U~ bt~5) + 420u~u~" + 306u~ u~" +252u~ '2 + 1224u2u'2u'j' +918u2u~ 2 +891U2t 2 U2tt +630U3U~ +945UzUz 2 ,2 + 108U25 . (46) Therefore G K H gives the same equation with eq. (43) up to the integration constant. The fact that the criticality jumps by two steps at the point (26) is in accord with the description by GKH. Since there is no G K H equation for (p, q) = (3, 6) ( [ Q, Q6/3 ] = 0), the system has to jump from (3, 4) to (3, 8) [21 ]. The Painlev6 test for this equation gives a result similar to the other cases. The free parameters are Zo, C3, C4, C5, C6, C7, C8, C12 with Co= - 2 and Cl =0. Finally, we speculate a little about model identification. We may expect from our result that the critical point (24) realizes the tricritical Ising model coupled to gravity. As a fact supporting this expectation, we find a critical line connecting the point (24) and the one of the Ising model. In the above investigation we have considered the case of all the coupling constants being tuned to critical values by imposing the condition (23). If we displace one of these constants off critical, by relaxing the condition (23) to W' (fo)= W" OCo)=O ,
(47)
a line which represents lower criticality can be found: fo=t,
Ro=-3t-2,
g' _ g
729(5t+2) t(212t+ 101) 2'
62
c2=
81t 2 4(212t+ 101) ' (48)
6 September 1990
10t(187t+91) g= 243
(48 cont'd)
One can easily see this line connects the critical point (24) ( t = - 7 ) to that of the Ising model ( t = 2 The scaling behaviour of the model on this line is governed by the differential equation f,,,,+9ff,,+gvc,2+6f3 = 5 ( 1 8 7 t + 9 1 ) z, 3(22t+7)
(49)
except for the point t= - 7 . This equation is a string equation for the Ising model derived in refs. [9,19] if we rescale f a n d z appropriately. In the point (24) the critical behaviour of the system jumps to one controlled by eq. (30). This criticality structure corresponds to the tricritical Ising model, in which the chemical potential of the lattice vacancy parametrizes the critical line. A detailed analysis of the phase structure will be reported in a separate paper [22]. We have also checked the positivity of the partition function [19] by means of a concrete calculation. It is found that each term of the genus expansion is positive up to the 60th order. The correspondence between the three matrix model and the tricritical Ising model should be further tested by alternative constructions such as that of Kostov [23 ]. The authors would like to thank T. Eguchi for valuable discussions and reading the manuscript. The authors also would like to acknowledge useful discussions with M. Fukuma, A. Kato, H. Kawai, T. Kawai and K. Ogawa. This work is supported in part by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan No. 63790157 and No. 01790191.
References [ 1 ] E. Br6zin and V. Kazakov, Phys. Lett. B 236 (1990) 144. [2] M. Douglas and S. Shenker, Nucl. Phys. B 335 (1990) 635. [3] D. Gross and A. Migdal, Princeton preprint PUTP-1148 (1989). [4] M. Staudacher, Illinois preprint ILL-(TH)-89-54 (1989). [ 5 ] J. Cardy, Phys. Rev. Lett. 54 ( 1985 ) 1354. [6] M. Bershadsky and A. Migdal, Phys. Lett. B 174 (1986) 393. [ 7 ] V. Kazakov, Phys. Lett. A 119 (1986) 140. [ 8 ] D. Boulatov and V. Kazakov, Phys. Lett. B 186 ( 1987 ) 369.
PHYSICS
1
Nonperturbative Hiroshi
Kunitomo
LETTERS
B
6 September
1990
analysis of the three matrix model
‘J,~ and Satoru
Odake
2,4
Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Received
26 March
1990
Recently developed techniques are applied to give exact nonperturbative solutions of the three matrix model. There exist three different critical points and the specific heats of the model corresponding to these points satisfy different differential equations (string equations). The critical point expected to correspond to the tricritical Ising model is connected to the Ising model by a critical line. The string equations on each scaling limit coincide with the type (p, q)= (4, 5), (2, 7) and (3, 8) equations of Douglas’s general argument.
In spite of beautiful developments in string theory, we are still in the dark about its nonperturbative properties. Recently, some remarkable progress has been made in the nonperturbative approach to two dimensional gravity or noncritical string theory [l3 ]. An exact nonperturbative solution has been given for discretized two dimensional gravity by using the large-Nmatrix method with some double scaling limit for large N and critical coupling. The other solutions exhibiting multicritical behaviour and indexed by a positive integer m ( = 2, 3, ...) has also been obtained. These solutions were initially believed to correspond to minimal unitary conformal models coupled to quantum gravity. It was shown, however, that this conjecture is not true and that the m = 3 multicritical theory #’ corresponds to the Yang-Lee edge singularity [ 41 which is described by a nonunitary CFT with c= -y [5]. Furthermore, it has been found that the correct exact specific heat of the Ising model, which is realized by the two matrix model [6-S] actually satisfies a different string equation from that of the m = 3 mul’ E-mail address: [email protected] 2 Fellow of the Japan Society for the Promotion of Science. 3 Address after April 1, 1990: Research Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan. 4 E-mail address: [email protected] #i The parameter m is related to the general formula for the minimal conformal series as c= l-6(p-q)*/pq with p=2, q=2m- 1. 0370-2693/90/$03.50
0 1990 - Elsevier Science Publishers
ticritical model [9-l 11. Therefore, we can easily imagine that the next candidates for the models describing the unitary CFTs are given by generalizing the two matrix model [ 12,13 ] to n matrix models. The solvable n matrix models were first introduced in ref. [ 141 and described by the n NX N hermitian matrices M, (i= 1-n) as #’ Z,=
fi dft4, exp -tr I r=l (
i: V,(M,) i=l
n-1
1 1 +l'ixJ.
X.(x)=$x2+
gj!
ia
The nonperturbative properties of these models were studied by Douglas [ 18 ] from the point of view of the generalized KdV hierarchies (GKH). He claimed that these models are governed by some series of equations
[Q,QP’l=l
>
(2)
with the pth order differential Q=
ap+ ,d a,
a-
.
operator (3)
X2 In the limit n+a, these models become the one dimensional string theory (CFT with c= 1) [ 1S-171. This fact also supports the above expectation.
B.V. (North-Holland)
57
Volume 247, number 1
PHYSICS LETTERS B
6 September 1990
Nonperturbative analysis of the three matrix model Hiroshi K u n i t o m o ~,2,3 a n d Satoru O d a k e 2,4 Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Received 26 March 1990
Recently developed techniques are applied to give exact nonperturbative solutions of the three matrix model. There exist three different critical points and the specific heats of the model corresponding to these points satisfy different differential equations (string equations). The critical point expected to correspond to the tricritical Ising model is connected to the Ising model by a critical line. The string equations on each scaling limit coincide with the type (p, q ) = (4, 5 ), (2, 7) and (3, 8) equations of Douglas's general argument.
In spite of beautiful developments in string theory, we are still in the dark about its nonperturbative properties. Recently, some remarkable progress has been made in the nonperturbative approach to two dimensional gravity or noncritical string theory [ 13 ]. An exact nonperturbative solution has been given for discretized two dimensional gravity by using the large-N matrix method with some double scaling limit for large N and critical coupling. The other solutions exhibiting multicritical behaviour and indexed by a positive integer m ( = 2, 3 .... ) has also been obtained. These solutions were initially believed to correspond to minimal unitary conformal models coupled to quantum gravity. It was shown, however, that this conjecture is not true and that the m = 3 multicritical theory ~l corresponds to the Yang-Lee edge singularity [ 4 ] which is described by a nonunitary CFT with
[51. Furthermore, it has been found that the correct exact specific heat of the Ising model, which is realized by the two matrix model [6-8] actually satisfies a different string equation from that of the m = 3 mulE-mail address: [email protected] 2 Fellow of the Japan Society for the Promotion of Science. 3 Address after April 1, 1990: Research Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan. 4 E-mailaddress: [email protected] ~ The parameter m is related to the general formula for the minimal conformal series as c=l-6(p--q)2/pq with p = 2 ,
q= 2 m - 1.
ticritical model [9-11 ]. Therefore, we can easily imagine that the next candidates for the models describing the unitary CFTs are given by generalizing the two matrix model [ 12,13 ] to n matrix models. The solvable n matrix models were first introduced in ref. [ 14 ] and described by the n N × N hermitian matrices Mi ( i = l - n ) as ~2 Z~ = f f i
dMi exp ( - t r ~
i=1
Vi(Mi) 1
°-'
)
+ tr ~ ciMiMi+ 1
,
i=1
V ( x ) = ½x2 + ~ 1 giJxJ " j>~3 N j / z - t j!
( 1)
The nonperturbative properties of these models were studied by Douglas [ 18 ] from the point of view of the generalized KdV hierarchies ( G K H ) . He claimed that these models are governed by some series of equations [Q, Q~{p] = 1,
(2)
with the pth order differential operator P
Q = 0 P + ~] a,O p-~.
(3)
i=2
~2 In the limit n - * ~ these models become the one dimensional string theory (CFT with c = 1 ) [15-17]. This fact also supports the above expectation.
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
57
V o l u m e 247, n u m b e r 1
PHYSICS LETTERS B
The minimal unitary series is identified with (p, q ) = ( m , m + l ) , m>_-3. However, we think that Douglas's arguments are still obscure and need further confirmation by solving the models concretely. In this paper, we will solve the three matrix model explicitly, and show that there is a critical point on which the system is actually controlled by the expected equation, eq. (2), with (p, q) = (4, 5). Moreover, we will find two more critical points which are governed by the string equation, eq. (2), with (p, q ) = (2, 7) and (3, 8). Let us first summarize the general formulae for the n-matrix model, which are applicable to asymmetric coupling constants. The method for solving the n matrix chain model described by eq. (1) with the restricted coupling constants ci . ~ - Cand g~j= gj for Vi, has been formulated in ref. [ 14] by means of the orthogonal polynomial method. We give here some formulae for more general coupling constants as in eq. ( 1 ). In this model, we can simultaneously diagonalize the matrices M~ as M~= U,tX,•Ui using the unitary matrix U~, and the partition function eq. ( 1 ) is rewritten by diagonal matrices X~ as [ 12-14 ]
Z~=K f f l dX, J(X~)LJ(X,) i=l
Vi( XJ + tr " ~ ciXiX,+ l )
×exp (-tr ~
i= 1
N--1
=KN! Y" h~,
(4)
i=o /
n
\tt~=l
NN(N--1)/2
ci/
6 S e p t e m b e r 1990
those in ref. [ 14 ]. If we introduce the notation
J[+ f(x) = ~ dy exp [ - V + (y) + c +xy] f l y ) , E ( x r , j - ~ ? k ) , , j ) ~ ... ~e~ ~ ? U ( x ) = o J
(0~
(8)
d r , 7-fliTj)Pf +\ ~ /~-~ (x)=O ,
c : ~ 0+)
--'~ v ~ ' ( ,
+~_,~) --C/--+ 10L ~+- - 2 )
(l
(10)
with cff o~(-1) ± =fl± , [Or(o), fl-+ ] = 1, and
Ot~k_l)i,jhj=ot(+_k)j, ihi
(l~
after integrating out the angular matrices U~. Here J ( X ) is the Vandermonde determinant and the h~'s are defined by the following conditions of two orthogonal monic polynomials P+ (x) and P - (y):
hic~,j= ~ f i dxkP)-(X,,)P+(Xl)
(11)
In particular, the last equation eq. ( 11 ) is most important and includes all the informations we need ~3 In the case with Zz-symmetric coupling constants such that E = In+ t_~ and c~= c,_~, simplifications occur, because of the fact that we can choose the orthogonal polynomials as P+ (x) = P ; - (x), and then o~~) = o~~), fl+ = f l - . Only these simpler formulae are needed for our purpose. Let us apply the general formulae summarized above to the three matrix model with Zz-symmetric coupling constants,
=j
i=1
(9)
where cy_i=cF =c~, Vy+~_i= V,+ =V,, the expansion coefficients o ~ ) can be computed from the formulae
Z3= j dAdBdCexp[-S(A,B, C ) ] ,
N
(7)
(12)
d X d Y d Z A ( X ) J ( Z ) e x p [ - S ( X , Y, Z ) ] , (13)
S(A, B, C) = ½t r ( A 2 + B 2 + C 2) I i
+ ~.
tr(gA4+g'B4+gC 4) - c tr(AB+BC). (14)
k=l
×exp 1
Vk(Xk)+ ~ CkXkXk+l .
(6)
k=l
The free energy of the model, therefore, can be calculated from these constants h, In order to obtain these constants h~, we can use similar arguments to 58
It should be noted here that the two coupling constants g and g' are different in general. If we set g' = 0, the B integration becomes gaussian and we get the two matrix model after integrating it out. The rela~3 In these formulae,we assumen >/2. For the one matrix model, only one relation V' (a) = fl is enough to solveit.
Volume 247, number 1
PHYSICS LETTERS B
tion betwwen the coupling constants of the two models is C(two)= c2/( 1 - c 2) and g(two) = g~ ( 1 - c 2) 2 in our notation ( 1 ). The three matrix model, therefore, includes the two matrix model, i.e. the Ising model, as a special case. For other coupling constants, we assume g ¢ 0, c ¢ 0 hereafter. We use a familiar notation for a(o) in the general formulae:
xPi(x) =P i+, (x) + RiPi_l (x) + siei_ 3(x) +
[Note that the absence of the terms Pi, Pi-2 .... in this recursion relation is a consequence o f the fact that Pt(x) has a definite parity P i ( - x ) = ( - )~P~(x) since our action S is quartic (even) ]. The next coefficients ot~) are calculated from this recursion relation as [14] for l i - j [ >_-3, g 6N
-
forj=i+3,
- 6N g f f-l f-2
forj=i-
= 1 + ~ g (Ri+Ri+l +Ri+2)
for j = 1 + 1,
forj=i-
1,
3,
(16)
where f = hJh~_ i. From relation ( 11 ), we obtain
g' a3~,) ) c & = ( a~L) + g-~
(17)
, i,i-- 3
c(f+Ri)
:
-OL(1 ol(1)) + -6N
)
i,i-i
,
Ri 1 + ~ "~- ~g ( S i ~ - S i + 1 @Si+2)-COQl)i,i_ 1 = i .
by considering the naive large-N limit. In the naive large-N limit, we have
i U ~x, --
1 ~f
f(x) g
1
,
~Ri
R(x) - - ,g
(20)
which give the possibility to derive from (18) and (19) the algebraic equations
c 4 ( f + R ) = f ( l +½R)
. . . .
(15)
COL(I ) i,j = 0
6 September 1990
x c2+~-[~J3+~f2(l+½R)+~(l+lR)21
,
(21) (R-f)(1 + ½R) + ~
g' [@zf3+f(l+½R)Z+(l+lR)3] )
C2+g
=gx.
(22)
Here we have eliminated & by using eq. (17) ~4. In principle, we can further eliminate R from the LHS of eq. (22) by solving eq. (21). Then we define a "potential" W ( f ) by the LHS of eq. (22) as the function o f f Since the scaling laws arise from the singular behaviour o f f ( x ) near x = 1, the form of W ( f ) in the neighbourhood of this point controls the criticality. Therefore, in order to tune all the couplings g, g' and c on critical, we must adjust it such that the first three derivatives of W ( f ) vanish,
W' (fo)= W" (fo)= W" (fo)=O .
(23)
The coupling g is related to W as g = W(fo). F r o m the practical standpoint, however, it is not useful to solve eq. (21 ) and to treat W ( f ) as the function o f f only. We solve these criticality conditions under the constraint eq. (21) without solving it in the actual calculation. Although it is a little lengthy, it presents no difficulties to solve the criticality conditions eqs. (23) and (21 ). We find just three different solutions ~5.
(19 )
These three equations ( 17 ) - ( 19 ) are sufficient to get the necessary information about the nonperturbative solution. We now try to find the critical points of the model
~4 It is worthwhile to note that using eq. (17) Si can be eliminated without any approximation. ~s Strictly speaking, the solution (24) gives 0/0 if we put it into eq. (23). This is interpreted as the limit t-~ _ 7 on a line explained later. 59
Volume 247, number 1 (i)
fo= -7
,
g,/g= 11767 9801 , (ii)
PHYSICS LETTERSB
Ro - -
23
--22 ,
245
g~
--~
fo=½(l_+x/7),
g'/g=-81,
Ro - -
3 --2~
(25) c2=_9
g = - ~ 35.
(26)
These solutions would yield three different critical points of the model on the sphere. We investigate the critical point (i) in detail since it leads to a string equation expected to represent the tricritical Ising model (the unitary CFT with c = 7 ) . To keep the higher genus contributions, we must expand eqs. ( 18 ) and ( 19 ) in 1/N. As in refs. [ 1-3 ], it is convenient to introduce a lattice spacing a so that the renormalized "cosmological" constant is given by IZR=(gc--g)/gc a2. Then we have the following expansions: lf+l~f---° ( 1 + ~ g\ k~
aZk/mfk(z--lal/m)) ,
~R~+z~ lvl f__o\fo + ~ a2k/mrk(z--la l/m) ) ,
(27)
(28)
k>~l
with the scaled variable z i --
N
~ 1
--aZz,
(29)
and the critical values (24) for couplings. The parameter m is fixed by the order of criticality [ 2 ] and m = 4 in our case. The coefficient function fi is related to the free energy F a s f l (z) = F " (z). Substituting in eqs. ( 18 ) and ( 19 ) these expansions, we find differential equations f o r f and r~. These differential equations reduce to a single equation forfl in the case of all the nonperturbatively solved models so far [ 13,9,19]. In our case, however, the situation is a little different. The string equation satisfied by f~ does not become a single equation but two coupled differential equations:
f"'+lOff"+5f'2+lOfa+lOr"+2Ofr=O, 60
-1- 15f 4 -
1 4 r " - 7Ofr" - 20f ' r' + 20f" r + 60r 2
- - 770..~.Z ,
Ro=-Y-x/~fo,
g=~8(50+ 35x/7), fo=½,
(24)
g'/g=}(32+_lOx/~),
c2=~2 (5 +_xf7),
(iii)
f(6) q_ 13ff,,,,+23f,,2+60f2f,, + 4 5 f f ,2
C2=4~6
,
6 September 1990
(30)
(
30 cont'd )
where f ~i)= d~C/dzi, f = f i and r = r 2 - f > The constant ~770- in front of z can be taken to 1 by appropriate rescaling. We will see next that these string equations can also be derived from the G K H standpoint. Before considering the concrete case, let us rewrite eq. (2) in a more explicit and convenient form. If we write p
aia p-i,
Q=Op+ ~
(31)
i=2
aq/P=Oq-k - ~ biO q-i ,
(32)
i=2
and use the fact that [Q, Qq+/p]= [Qq_/p, Q], eq. (2) imposes #6 (bi+...)' = 0
for(q+l~i~p+q-2),
-p(bp+o_l +...)'=l ,
(33)
where "..." represents some differential polynomials of aj which disappear if all the integration constants are equal to zero. Since the coefficient functions bi are uniquely determined as the differential polynomial ofai, eq. (33) become the differential equations ofai. In particular, we note that the last expression in eq. (33) is always integrated out to the form of the familiar string equation. Eq. (33) are the most general equations without any restriction. If we wish it to have 712-symmetry Q must be restricted to be a self-adjoint operator. This is equivalent to imposing that Q have a definite parity with respect to 0 ~ - 0 when we write it in the symmetric form p
Q=0p+ ~
(ciOP-%OV-'ci).
(34)
i=2
Now consider the case (p, q) = (4, 5 ). Q is a fourth order differential operator in this case Q = 04 dr (U2 02-~-02U2) "l- (U3 0"1-0U3) -~"U4 ,
(35)
#6 T h e a u t h o r s w o u l d like to t h a n k T. E g u c h i f o r p o i n t i n g o u t t h i s fact.
Volume 247, number 1
PHYSICS LETTERS B
and we should take u3 = 0 for Z2-symmetry. The commutation relation is calculated to be [Q, Q5+/4] = _ 4 b ~ 02_ (4b~ + 6 b ~ ) 0 - ( 4b'8 + 6b~ + 4b~' + 2a2b'6 + a'2b6 ) ,
(36)
with 32b6 = u'~" - 10u2 u~ - 15u~2 - lOu 3 + lOu~ + 20u2 u4,
(37)
128b8 - u ~ 6) + 6 u 2 u z"' + 1 2 u z. .u.z. . . + 9 u 2 2 +50u22u~ ..]_50U2 U ~2 + 3 5 U z 4- { - 2 U 4.
-. - 2. 0 U. 2 U. 4 -. b 2. 0 U. z U 4
+20u~u4 +20u24-60uZ2u4,
(38)
and 64b7= - 3 2 b ~ . Then we get two coupled differential equations b~ = 0 ,
(39)
- 4 b ~ = 1.
(40)
This coincides with eq. (30) up to integration constants with the identification f = u2, r = u 4 - u~. Let us now study a few properties of the string equation eq. (30). If we assume that the solution o f eq. (30) f o r f ( z ) has at most a second order moving pole singularity as in the case of Painlev6 type equations [ 9,19 ], f ( z ) and r ( z ) can be expanded as f ( z ) = ~. c i ( z - z o ) i-2 , i=o
(41)
r(z)= ~
(42)
i=0
di(z--z0)
i-4 ,
and these expansions converge in the vicinity of Zo. The coefficients c~ and d~ are determined such that f ( z ) and r ( z ) are the solution ofeq. (30). It is known that if we can arrange the number o f free parameters in expansions (41) to be equal to the order o f the differential equation, then the solution of such an equation has no moving cut singularity in general [20] ~7. In our case, eq. (30), the expansions (41) must have eight free parameters ~8. The explicit cal~7 We invoke the Painlev6 test to see whether a nonlinear differential equation has enough free parameters. #8 This number is obtained as follows: If we allow a nonpolynomial differential equation it is possible to eliminate r(z) from eq. (30) by solving the third order algebraic equation. Then the resultant differential equation off is of eighth order.
6 September 1990
culation leads, if we choose Co= - 2 , to eight parameters Zo, c2, c3, Ca, c5, c6, c7 and C,o remaining unfixed and to the vanishing of the simple pole term, c, = 0. Therefore our string equation eq. (30) does not admit a solution with any moving branch point ~9 The fact that Co= - 2 and c, = 0 is important. Since f i s the second derivative o f the free energy F, this double pole singularity corresponds exactly to logarithmic singularities of the free energy. Thus the partition function Z = e F is entire in z is and only if the residue Co is equal to a negative integer and if and only if there is no simple pole. Until now we have analyzed the critical point which is expected to correspond to the tricritical Ising model coupled to gravity. We c o m m e n t here on the behaviour o f the other two, (25) and (26). First, consider the case (25). The scaling property of this point is governed by the differential equations indexed by (p, q) = (2, 7) of the G K H classification. Since it is known that this equation is derived from the m = 4 multicritical one matrix model, this critical point may be expected to correspond to some YangLee type edge singularity described by some non-unitary CFT. The case (26) needs a little discussion. Although we find this point to be a third-order critical point (the highest order we can tune with three couplings in general), the fourth-order derivative o f the potential W accidentally vanishes in this point so criticality jumps to fourth order. Because o f this, we must change the lattice spacing expansions to (27) and (28) with r n = 5 rather than m = 4 . With this change one can derive the string equation as follows: 4f(8) + 60ff(6) +
1 8 0 f " f ~5)+ 4 0 2 " f " ' + 306f2f ....
+ 252f"'2+ 1224ff'f" +918ff"2+891f'2f " + 630f3f " + 945f2f'2 + 1 0 8 f s = - 4 2 0 z ,
(43)
wheref=f~. This differential equation eq. (43) was ~9 We also find that this situation is common for all the known string equations. The location of the second order pole Zo is always a free parameter and other ones cj appear at j = 6 , for m = 2 , j = 2 , 5, 8 for m = 3 , j = 2 , 4~
61
Volume 247, number 1
PHYSICS LETTERS B
first found in the analysis of the two matrix model with higher order criticality [ 21 ]. Eq. (43) can also be derived from the G K H with (p, q) = (3, 8). The operator Q in this case is
Q : O 3 q- ~(U20+ Obla)q-bl3 ,
(44)
and we set u3 = 0 for the Zz-symmetry. The commutation relation becomes [Q, Q8/3] = _ 3b'9 O - 3 (b',o + b~)
(45)
with b9 = 0 ,
1296blo =4u~ 8) + 6 0 U 2 U~ 6) + 180U~ bt~5) + 420u~u~" + 306u~ u~" +252u~ '2 + 1224u2u'2u'j' +918u2u~ 2 +891U2t 2 U2tt +630U3U~ +945UzUz 2 ,2 + 108U25 . (46) Therefore G K H gives the same equation with eq. (43) up to the integration constant. The fact that the criticality jumps by two steps at the point (26) is in accord with the description by GKH. Since there is no G K H equation for (p, q) = (3, 6) ( [ Q, Q6/3 ] = 0), the system has to jump from (3, 4) to (3, 8) [21 ]. The Painlev6 test for this equation gives a result similar to the other cases. The free parameters are Zo, C3, C4, C5, C6, C7, C8, C12 with Co= - 2 and Cl =0. Finally, we speculate a little about model identification. We may expect from our result that the critical point (24) realizes the tricritical Ising model coupled to gravity. As a fact supporting this expectation, we find a critical line connecting the point (24) and the one of the Ising model. In the above investigation we have considered the case of all the coupling constants being tuned to critical values by imposing the condition (23). If we displace one of these constants off critical, by relaxing the condition (23) to W' (fo)= W" OCo)=O ,
(47)
a line which represents lower criticality can be found: fo=t,
Ro=-3t-2,
g' _ g
729(5t+2) t(212t+ 101) 2'
62
c2=
81t 2 4(212t+ 101) ' (48)
6 September 1990
10t(187t+91) g= 243
(48 cont'd)
One can easily see this line connects the critical point (24) ( t = - 7 ) to that of the Ising model ( t = 2 The scaling behaviour of the model on this line is governed by the differential equation f,,,,+9ff,,+gvc,2+6f3 = 5 ( 1 8 7 t + 9 1 ) z, 3(22t+7)
(49)
except for the point t= - 7 . This equation is a string equation for the Ising model derived in refs. [9,19] if we rescale f a n d z appropriately. In the point (24) the critical behaviour of the system jumps to one controlled by eq. (30). This criticality structure corresponds to the tricritical Ising model, in which the chemical potential of the lattice vacancy parametrizes the critical line. A detailed analysis of the phase structure will be reported in a separate paper [22]. We have also checked the positivity of the partition function [19] by means of a concrete calculation. It is found that each term of the genus expansion is positive up to the 60th order. The correspondence between the three matrix model and the tricritical Ising model should be further tested by alternative constructions such as that of Kostov [23 ]. The authors would like to thank T. Eguchi for valuable discussions and reading the manuscript. The authors also would like to acknowledge useful discussions with M. Fukuma, A. Kato, H. Kawai, T. Kawai and K. Ogawa. This work is supported in part by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan No. 63790157 and No. 01790191.
References [ 1 ] E. Br6zin and V. Kazakov, Phys. Lett. B 236 (1990) 144. [2] M. Douglas and S. Shenker, Nucl. Phys. B 335 (1990) 635. [3] D. Gross and A. Migdal, Princeton preprint PUTP-1148 (1989). [4] M. Staudacher, Illinois preprint ILL-(TH)-89-54 (1989). [ 5 ] J. Cardy, Phys. Rev. Lett. 54 ( 1985 ) 1354. [6] M. Bershadsky and A. Migdal, Phys. Lett. B 174 (1986) 393. [ 7 ] V. Kazakov, Phys. Lett. A 119 (1986) 140. [ 8 ] D. Boulatov and V. Kazakov, Phys. Lett. B 186 ( 1987 ) 369.