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The Ising model, the Yang-Lee edge singularity, and 2D quantum gravity
Volume 237, number 2
PHYSICS LETTERS B
15 March 1990
T H E I S I N G M O D E L , T H E Y A N G - L E E E D G E S I N G U L A R I T Y , A N D 2D Q U A N T U M GRAVITY Q c d o m i r CRNKOVI~" ~'~, Paul G I N S P A R G t,.2 and Gregory M O O R E ~'3 Department o f Phystcs. Yale L'nt versity. New llaven. ( T 06511-816 7, US,I b l~vmanLaboratorv()fPhvsics, ltarvardUntversity. Cambridge. M..I 02138. {'S.4
Received 21 December 1989
We consider one-and two-matrix models recently used to define nonperturbative 2D quantum gravity. We prove lhe equivalence of the m = 3 multicritical one-matrix model and the Yang-Lec edge singularity coupled to 2D quantum gravity. We also derive formulae tor the correlation functions for the Yang-Lec edge singularity and for the Ising model in their summed over topologies formulations.
Recently there has been some very. interesting progress in the theory o f r a n d o m surfaces [ I - 4 ] . While it has been known for some time that the theory o f r a n d o m surfaces gives an elegant definition o f thc path integral o f 2D conformal field theory coupled to q u a n t u m gravity, previous studies have always bccn limitcd to the case o f fixed topology o f the two-dimensional spacetime. In refs. [ I - 3 ], a continuum limit that includes the sum over topologies o f two-surfaces was defined for pure q u a n t u m gravity. There, the diserctized 2D q u a n t u m gravity path integral was identified with the large N expansion of the frcc energy o f a hermitian matrix model with potential g~ Tr(0~) I ' ( O ) = T r ( 0 2 ) + ~ ,V~/2-1 It was found that for generic couplings g~. the free energy can bc described by a solution to the Painlev6 type I equation. F o r special valucs of the gk, corresponding to ruth order multicritical points ( m >~3) of a related potential l , t ' ( r ) , the charactcr o f the theory changes drastically and the free energy is describcd by a solution to a nonlincar diffcrcntial equation o f o r d c r 2 m - 2. Bitnet address: c r n k o v i c ~ y a l p h y . h c p n c t , or ,~:,yalehep.bitnet 2 Bitnet address: ginsparg.'~huhepl.hcpnet, .~,huhepl.bitnet, or :~ h u h e p l . h z r v a r d . c d u ' Bitnel address: moore:-_ayalphy.hcpnet, or ".ayalchep.bitnet
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Unfortunately. the correct physical interpretation o f the thcories defined in rcfs. [ 1-3] is not entirely clear. This is especially true o f the m>~ 3 theories, which were originally conjectured [5] to be equivalent to the ruth m e m b c r o f the unitary discrete series coupled to 2D q u a n t u m gravity. It was, however, noted in rcf. [ 4 ], for example, that the free cnergy o f the lsing model, c o m p u t e d using a two-matrix model formulation, satisfies a different differential equation from that o f the m = 3 multicritical point of the one-matrix theory, thereby disproving the above conjecturc. Thc (gcnus zcro) results o f ref. [6] m o r e o v e r suggested that the correct interpretation o f the m = 3 multicritical point is given by the YangLee edge singularity coupled to 2D q u a n t u m gravity. In this letter we complete the identification o f these two models, in the sum over topologies formulation, by calculating thcir free energies and correlation functions. A simple modification o f our calculations also gives results for all correlation functions of lhc Ising modcl. Let us first compute the correlation functions of the m = 3 one-matrix model. Wc follow closely thc derivation and notation o f rcf. [2 ]. Beginning with the potential potential .~ , I , ' ( 2 ) = 2 7 + N k 4 + g2}, -2 A~ wc define orthogonal polynomials o f norm hk (see rcf. [7] and rcferenccs therein for n o t a t i o n ) . The parti-
0370-2693/90/$ 03.50 c'¢)Elscvier Science Publishers B.V. ( Norlh-Holland )
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tion function is easily expressed in terms of the quantities fk=hk/hk_,, which satisfy recursion relations derived via integration by parts. In the large N limit, we write x = k/N, fk-,Nf(x). In terms o f t ( x ) - g f ( x ) , the tree level ( N ~ o o ) equation is [7] g x = 14"(t)- 2 ( t + 6 t 2 + 30yt 3) .
(1)
We now consider the scaling limit defined in refs. [ 13 ]. Thus we define
g)c=g,.( 1 --a22), where g¢= 14"(t~) is the value of the potential at the tricritical point t=t,., defined by 14" I,c= 14'" I,c=0. The correlation functions of refs. [3,4] correspond here to the g-,g,, scaling functions of the derivatives of the free energy with respect to 7 at the tricritical point. They can be calculated by first expanding t(x) around the solution of W'=O, i.e. t(x)=to(y) +a2/3f(z), where W' It0~, = 0 , and then allowing ), to scale to its value at the tricritical point. The equation I4,"=0 gives t o ( 7 ) = ( - l + x / l - ~ y ) / 1 5 y , and 14"" = 0 gives 7=-~, so that to= --~. The recursion relations satisfied here by t(x),
½gx=t(x)+ 2 t ( x ) [ t ( x - ~ ) + t ( x ) + t ( x + ~ ) ] + 3yt(x){t(x)2 + t ( x - ~ ) 2 +t(x+~) 2 +2t(x)[t(x-c)
+t(x+~)] +t(x+2~)t(x+c)
+t(x-2~)t(x-~)+t(x+~)t(x-~)},
(2)
where ~= 1/N follow from the results of ref. [7]. To obtain a well-defined continuum limit, we take ~ ~ a 7/3 and let ;~ scale to the tricritical points as y = ~(I-a4/~T2), while t o = - l / 6 ( l + a 2 / 3 T ) (thus preserving W ' = 0 ) and t(x)=to(7)+a2/3f(z). Matching powers o f a 2 in (2) and rescaling, we find
z=f~-ff"-~.f'2+~f''+T(fZ-~f
'') .
(3)
Setting T=O, we recover the result of rcfs. [ 1-3] for the tricritical one-matrix model. [Actually, for small T, the result (3) is obvious, since, to lowest nontrivial order in a rescaled T we have t(x)=t,. + a 2 / 3 ( f - t - T). Thus, we could substitute f--,f+ 7" in the equation for the free energy and keep lowest order terms in T t o recover (3).] As discussed in refs. [ 1-3 ], the scaling function for the second derivative of the free energy with respect to the cosmological constant is given b y f ( z ; T = 0 ) , where z = x -6/7 and
15 March 1990
x 2 is the string coupling constant. E x p a n d i n g f ( z ; 7") in powers of 7, the coefficient of 7 "k gives the second derivative with respect to the cosmological constant of the correlation function o f k insertions of a certain basic operator of the theory (expressible as a linear combination of traces of powers of 0). The above derivation is easily extended to the mth multicritical model using the formalism of ref. [ 3 ], as we shall discuss below. We now perform the analogous calculation for the Yang-Lee edge singularity coupled to 2D quantum gravity, formulated in terms of an [sing model in an (imaginary) magnetic field. As in ref. [8], we consider the large N expansion of a two-matrix model with free energy F(g, c, H; N)
I ,&2 /v2 -- log
+ ~-
d
Ud
U4+ ~
Vexp - T r
I;4
)]}
( U 2.q_ 2 V - 2cUV
.
(4)
the two types of quartic vertices in the graphical expansion of the RHS above are identified with the two possible spin states of the ]sing model on a random lattice with coordination number 4, and the propagator generates the link interactions. II in (4) denotes the magnetic field, and the parameter c (with physical range 0 < c~< I ) is related to the ]sing nearest neighbor coupling fl via c = e- 2B. The logarithm picks out the connected graphs so that F in (4) corresponds to the partition function of the ]sing model summed over random surfaces. In ref. [8], the free energy (4) was expressed in terms of norms hk of polynomials orthogonal with respect to a bilocal potential. These norms can in turn be expressed in terms ofJ~ =hk/hk_,--,Nf(x). Defining the quantity t(x) =6g[(x)/c, the recursion relations of ref. [ 8 ] take the form (e = I / N )
0 = - 6r(x) +t(x)(l+
~ e-lt[t(x+~)+t(x)+t(x-~)]
+ ]{t(x+~)[r(x+2~) +r(x+¢)+r(x)]
+ t ( x ) [ r ( x + ~ ) +r(x) + r ( x - ~ ) ] +t(x-~)[r(x-2~)+r(x-c)+r(x)]}),
(5a) 197
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t(x) =to(c, H) + a 2 / 3 f ( z ) ,
~gx= - ~ c 2 t ( x ) +r(x){l +2 eH[r(x+~)+r(x)+r(x-~)]
6)(¢, H) = t . ( c ) +a2/3T.
+~c:[ t(x)t(x-c
From (7) and (8), it follows that ( t o - t . ) 2 ~ t t II(¢). Using the relation cd/cb¢= - a ~ / 3 2 d / d z and the expansion
) t ( x - 2~ )
+ t(x+ ~)t(x)t(x-
c)
+ t ( x ) t ( x + ~ ) t ( x + 2~) ] .
(5b)
The potential defined by the genus zero recurston relation is in this case
gx = W( t, c, H) (6)
As explained in ref. [ 9 ], the leading critical behavior is determined by the smallest absolute value solution, t=to(C, tl), of (O/Ot) W = 0 . Taking the first derivative of (6), we find that to(C, H) is given implicitly by the relation 2 cosh I I =
r~(x) = ro +a2/3rl (z) +a4/3r2 (z) +a2r3(z) in (Sa), (5b), we find
r2 =
2_3 ~c~t+ t ( l + t e - " ) ( l + t e " ) -v~ct 3(1_t2)2
C2(1 __/(])4__ (to4+
6t(] + 1 )
2to( 1 +t(])
(7)
and substituting this relation into the second derivative we find
(9 c o n t ' d )
t,)(3 +t(]) + e - " (1 + 3t(])
6(I _I(])3
(['2_t_ ~22to.f,,) ,
I (.f3A+22f,2B +)2ff,,C+)4f,,,,D) ' r3 -- 216( 1 - t ( ] ) 4 " "" (lO) where A, B, C, D are known polynomials in to and e - ' . After some algebra (done by hand and checked by computer) and appropriate rescaling o f f z, and T, we find the result
z=.f3+ff"+½(f')2+~f(4'+T(fZ+~f"),
(11)
The tricritical "t point, which defines the Ising phase transition and Yang-Lee edge singularity, is given by setting the RHS of (8) to zero. Together with (7), this condition determines t ! = H ( c ) and t. ( c ) - t o ( C , H (c)). For ]
identically reproducing eq. (3). This completes the identification of the sum over topologies versions of the Yang-Lee edge singularity and the rn = 3 multicritical one-matrix model. In ref. [6], a mapping was defined between the two-matrix model and a dimer model (described by a one-matrix model with a sixth order potential) at a special value of (c, I l L namely H ( ¢ ) = ½in. Such a mapping, suitably extended to all c > ~ and all genera, would provide a more conceptual explanation of our result. A slight modification of the above calculations allows us to write a generating function for correlation functions in the lsing model in zero magnetic field at the critical temperature, after summing over topologies. The equations are different since, as c approaches the critical value, c--, ¼, and II--,0, the coefficients r2, r3..... see (10), in the expansion of r ( x ) develop singular terms. Therefore we define a new scaling limit,
Na T/3=)t ,
t= - 1 - t - a 2 / 3 f ( z ) ,
02 --0t2 W[ '°("'")= c2 (St~+3to(10t(]+l + t o2 ) I ) - 1
(8)
(9)
to(C, t i ) = - 1 +a2/3S,
c=~-a2/3T, ~'~ We do not mean tricritical here in the conventional sense of a tricritical phase transition, which Ising is certainly not, but rather in the sense ofa multicritical matrix model (and regret the conflict with standard terminology ). 198
and define the expansion
(12)
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r(x) = - ~ + a 2 Q f ( z )
PHYSICS LETTERS B
+aoJ(z) + a a / 3 r 2 ( z )
+ aS/3rs/2 ( z ) + a2r3 ( z ) .
(13)
Substituting (12) into (7) we find that the physical (real) magnetic field is given by H = a s/3 × x/~S~4T - a 5/3~. Then substituting these expansions into the recursion relations (5a), (5b), we find that rs/2 and r3 drop out to order a 2 and that r2 may be eliminated from the remaining equations, thus giving the pair of coupled equations (again after appropriate rescaling)
z= T(JO--4f '' ) - 0.)2+ f 3 - J f ' ' - -~(f' )2 + 2 f ( 4 )
,
,~¢=.fog- ~¢o" .
(14)
The properties of these differential equations ,2 can be analyzed in a way analogous to the treatments of refs. [ 1-4]. In particular, using the physical normalization of ref. [4] the residues of the second order poles remain integers. Since T in (12) parametrizes the scaling deviation away from criticality, eqs. (14) with ~o=0 can also be regarded to define a massive continuum limit of the Ising model in the sum over topologies formulation. (Relevant perturbations of the nonunitary one-matrix models were considered in refs. [2,3].) Genus zero correlation functions of the energy operator e and spin field a are generated by taking derivatives with respect to 7"and .~ of the logarithm of the solutionf(z; T, .~¢ ) to the tree level equation z =f3 + T/~_ f~-"
( 15 )
15 March 1990
Note that all correlators with an odd number of a's vanish, and that the correlation functions involving only e are identical to the genus zero Yang-Lee correlators. [The quantities in (16) are actually second derivatives with respect to the cosmological constant, proportional to z, of the Ising partition function with the indicated operator insertions, and not normalized by dividing by the partition function. The inferred susceptibilities agree with those given by the formula -~;, fl, l a - ½ z Q / a = - X, fl, l a + ~ of ref. [l 1], where the gravitationally dressed scaling weights are fl/a=~, ~ respectively for the c = ~ operators a and e. These weights are also evident from inspection of (15). For the Yang-Lee edge case, the single weight ~ operator corresponds to the identity operator with weight taken with respect to the negative dimension operator, in accord with the interpretation ofref. [4].] We now discuss and speculate on the physical meaning of the above results. Since a general perturbation away from a multicritical point should define the generic m = 2 pure gravity model, the occurrence of the Painlev6 equation multiplying T in (3) and ( 11 ) could have been easily foreseen. Referring back to the potential IV(r) that arises in the genus zero recursion relation and using the formalism of ref. [2,3 ], we can make this argument more precise and extend it to the general multicritical one-matrix model. By rescaling the hermitian matrix 0, we may take the action V ( O ) = ( N g / g c ) ( ~ T r 0 2 + . . . ) . The perturbation of an ruth order multicritical potential m--I
W,~(r)--l-(l-r)"~14",,(r)+
~ (z(1-r) / /=2
We thereby find the following rather curious correlation functions for the Ising model: (e)
<~) (eee)
= - ~ , =~z -'~ =
~Z -2/3
(eeee) = O,
,
=-~z -~/~ = -- 8z -3
(aaee) = 0
(16)
corresponds to some perturbation of the original potential V(~0) --. V(q~) + 3"t (/V/(q~). Using the recursion relation in the form of eq. (12) of ref. [3], we see that scaling (/= (Ng/gc)-2t"-1~/t2'"+')Ti [equivalent to the scaling used above in the derivations of (3), (11), and (14)] leads to the differential equation m- I
~
T,r-/t[f],
(up to rescalings of the energy and spin operators).
z=;-/,,[.f]+
(17)
.2 These equations have been found independently by Douglas [10]. who argues that the linear term in .~ in the second equation may lead to a nonperturbative breaking of the Ising z~2 symmetry.
where r%[f] is the lth member of the hierarchy of differential equations defined in refs. [1-3] (i.e. 5"-/2Lf ] =.f2 _ ~f,,, .c¢3i f ] =f3 _if,, _ ~f,2 + ~of'', etc. ).
/=2
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One interesting corollary of this argument is that the operator e, that perturbs :'1,,,I f ] ~ ~,,, i f ] + "1"~,~_ ~I f ] has a one-point function given exactly bv (e,) = - 1/ m. This follows sincc, by a simple generalization of thc argument following (3), we must have
~"If+
l T'l = ~ ' " [ f ] +
T~'~-j[j']+O(T~)
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tions to the Yang-Lee correlation functions. The onepoint function receives no nonperturbative corrections, as insured by our argument following ( 18 ), and the leading nonperturbative correction to the twopoint function, for example, is 6(ee,) ~ -3.~ ~t,, ( 1 - i x / 5 ) x - 3 / " e x p ( - ~ & / x )
"
-22-~b2( 1 + i x / 5 ) x -3/ ~ e x p ( - ~ b : / x ) + .... (20) The solution to ( 17 ) is analytic in T = T,,,_ ~, but from (18) we see that f + ( I / m ) T solves a differential equation whose coefficients involve powers of Tlarger than or equal to two. Thus f(c; T ) = - ( l / r e ) T + .~'(z; T) w h e r e f i s analytic in Twith no term linear in 7: In particular, we see that the Yang-Lee one-point function (r,) = - ~ is nonperturbatively exact. In the case of the perturbed Ising equations (14), we do not obtain the correctly normalized Painlev6 equation. Intuitively, this indicates the existence o f other operators in the theory.. Since the coefficients are precisely correct in the one-matrix case we have evidence that the operators V~, which correspond to Kazakov's scaling operators [ 5 ], form a complete set of relevant operators and hence the mth multicritical model is the minimal model (2, 2 m - 1 ) coupled to 2D quantum gravity, as conjectured in ref. [ 4 ]. One can also investigate the effects of the "nonperturbative parameters" discovered in refs. [ 1-3 ]. For eq. ( 3 ) and eq. ( 14 ) with o9= 0, the analysis of the nonperturbative parameters proceeds as in the case T = 0 : we let.f= ~ (z, T) +e, whercJ~ is the asymptotic perturbative solution for z ~ at fixed 7, and e<< 1 in this regime. The nonperturbative corrections then satisfy' the homogenous equation 0 = (3f~ + 2 7 J ~ .f '~')g,- f '~e' - ( ~ +fiT)e" + ae"', where (0~,/3) = (~, ). (~, 4) respectively' for Yang-Lee and Ising. For the Yang-Lee case (3), the WKB approximation gives the asymptotic expansion o f a physically acceptable solution as
.f(z; 7, 2~, 2,) ~.~(z: T) + 2 1 z " e x p ( - 6 t h z T / 6 ) (1 +...) +22zVexp( --~,'2" ok, _7/6~, (1 +...) ,
(19)
whcre b~.2= ( 5 + i / v / 5 ) ~/2 (with the root choscn to give a positive real part), v,.2 = - :~, and 2 t,2 arc all indcpendent o f T. Computing the subdominant contributions to (19) gives the nonperturbativc correc200
Note, in particular, that the partition function and correlation functions pick up imaginary parts, suggestive of an instability, unless o f course 2, =,,13. In the lsing case ( 14 ), we find a nonperturbative correction to the one-point function. Combining the above discussions of the perturbations of differential equations with the analysis of the nonperturbative parameters, we are led to spcculatc about a physical principle by which these parameters might be fixed. The perturbations (17) of the frcc energy differential equations should bc regarded as defining a version of rcnormalization group flow between minimal models, as suggested in refs. [2,3]. Now note that if./'(z: 7; 2,, 2,) is a solution o f (3), then 7'(3'; T. 2 ~, 22) - T 2/5.I"( "1"l/S.l'; "/', 2t, 22) satisfies y=.72-
.{I'" + I "
• --7/5
' (7 - . ~ 3
,,
--'~?"-'--T~., -~ ' ?'.... ).
(21)
Therefore, J'f(y)=]imr .~,7()'. 7"; 2,, ).2) exists it must be a solution of the Pain]ev6 equation. A twoparameter space of functions.f (z: T, ). ]. 22) would then be mapped to the two-dimensional space of solutions of the Painlev6 equation. It is not obvious that the resulting solutions to the latter will automatically lie within the one-dimensional physically acceptable subspace (those with no exponentially growing term in the asymptotic expansion), so to insure this the two original parameters (,;q, ,-2) might have to be related, leaving only one independent. An even more dramatic possibility is that the resultingT(y) is completely independent of the choices of 2~, )-2, i.e. that the "flow" from a solution f ( z ; T, 2t, 22) of ( 11 ) determines a unique (physically acceptable ) solution]'(y) to the l'ainlcv• equation. Naively, this property is suggested by the limiting behavior liml .,,, T-~"5( T2/SY)"exp[ - h(1"2/5y) 7/6 ] -~-0 of the 2 dependent terms in (19). If the argument could be made rigorous, the "'nonperturbative parameter" ofrefs. [ I-3 ] would be completely fixed by
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the requirement that it can be obtained by "renormalization group flow" from a htgher multicritical point. In the interesting event that the conclusion is correct, we would expect as well that a straightforward generalization will show that the m - 1 physically acceptable parameters of the ruth order multicritical point are completely determined by a similar flow from a higher model. We close with some c o m m e n t s on the correlation functions. It was pointed out in ref. [4] that a nonvanishing one-point function of the energy operator in the lsing case violates the physical intuition based on the picture of q u a n t u m gravity as a (coupled) product of matter and gravity path integrals ~3. Since the energy operator, defined by differentiation with respect to 7", only makes sense up to shift by a constant times the identity, it seems natural to us to define a shifted operator g - e,+ ~z - '/3~. From ( 16 ) we find that ~ satisfies ( ~ ) = 0 , (~2)=~z-,/3, ( ~ : 3 ) = 0 , (?,,4) = - 7 ~ z - ' . Correlation functions for the Ising model coupled to gravity are also expected to have certain positivity properties - positive definite correlation functions of the ordinary lsing modcl should remain so if the integral over 2D metrics defines a positive measure. The objects in ( 16 ) first of all need to be integrated twice with respect to the cosmological constant to obtain correlation functions. Although various changes in conventions ( f o r f z, T a n d ,~2) can then be used to shift the signs around, it is not obvious that all of the undesired negative signs can be eliminated. On the other hand, the integrations with respect to thc cosmological constant generate nonuniversal terms in the correlation functions that may act to restore the desired positivity. The proper notion of unitarity in q u a n t u m gravity, as applied to the leading singular behavior of the scaling limit of the theories we consider here, may in any event bc very subtle. In conclusion, the matrix model formalism has thus far produced some tantalizing new results on 2D q u a n t u m gravity. It does not yet provide a clear intuitive picture of all of the p h e n o m e n a it describes, however, and answers to certain fundamental physical questions remain obscure. Wc hope that further investigation will ultimately lead to an entirely novel
~3 In ref. [ 12 ], on the other hand, it is argued that the one-point function vanishes.
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framework for investigating the nonperturbative effects of s u m m i n g over topologies. C.C and G.M. are supported by DE-AC0276ER03075; P.G. by NSF contract PHY-82-15249, DOE OJI grant FG-84ER40171, and by the A.P. Sloan foundation. We thank M. Douglas, A. Morozov, R. Shankar, S. Shatashvili, S. Shenker, C. Vafa, and J. Zinn-Justin for useful discussions, and M. Douglas, N. Seiberg, and S. Shenker for comments on a preliminary version of the manuscript. We also thank A. Migdal and S. Shenker for providing us with copies of ref. [ 6 ].
Note added. After completion of this paper we received a paper [ 13 ] with related results on the YangLee edge singularity. While this sentence was being typed, we were thrilled to receive more papers [ 10,12 ] with further results.
References [ I ] E. Brezin and V. Kazakov, Phys. Lett. B 236 (1990) 144. [2] M. Douglas and S. Shenker, Strings in less than one dimension, Rutgers preprint RU-89-34. [ 3 ] D. Gross and A. Migdal, Nonperturbative two dimensional quantum gravity, Princeton preprint PUPT-1148. [4] E. Br6zin, M. Douglas, V. Kazakov and S. Shenker, Phys. Left. B 237 (1990). [ 5 ] V. Kazakov.The appearance of matter fields from quantum fluctuationsof 2D gravity,NielsBohr Institute preprint NBIHE-89-25. [6]M. Staudacher, The Yang-Lee edge singularity on a dynamical planar random surface, Urbana preprint ILL(TH)-89-54. [ 7 ] D. Bessis, C. Itzykson and J.-B. Zuber, Adv. Appl. Math. I (1980) 109. [8] D. Boulatovand V. Kazakov, Phys. Lett. B 186 (1987) 379. [9] V. Kazakov, Phys. Left. A 199 (1986) 140. [10] M. Douglas, Strings in less than one dimension and the generalized KdV hierarchies, Rutgers preprint RU-89-51. [ 11 ] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Len. A 3 (1988) 819; F. David, MOd. Phys. Lett. A 3 ( 1988 ) 1651; J. Distler and H. Kawai, Nucl. Phys. B 321 ( 1989 ) 509. [121T. Banks, M. Douglas, N. Seiberg and S. Shenker. Microscopic and macroscopic loops in non-perturbativetwo dimensional gravity, Rutgers preprint RU-89-50. [13] D. Gross and A. Migdal, Nonperlurbative solution of the [sing model on a random surface, Princetonpreprint PUPT1156. 201
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T H E I S I N G M O D E L , T H E Y A N G - L E E E D G E S I N G U L A R I T Y , A N D 2D Q U A N T U M GRAVITY Q c d o m i r CRNKOVI~" ~'~, Paul G I N S P A R G t,.2 and Gregory M O O R E ~'3 Department o f Phystcs. Yale L'nt versity. New llaven. ( T 06511-816 7, US,I b l~vmanLaboratorv()fPhvsics, ltarvardUntversity. Cambridge. M..I 02138. {'S.4
Received 21 December 1989
We consider one-and two-matrix models recently used to define nonperturbative 2D quantum gravity. We prove lhe equivalence of the m = 3 multicritical one-matrix model and the Yang-Lec edge singularity coupled to 2D quantum gravity. We also derive formulae tor the correlation functions for the Yang-Lec edge singularity and for the Ising model in their summed over topologies formulations.
Recently there has been some very. interesting progress in the theory o f r a n d o m surfaces [ I - 4 ] . While it has been known for some time that the theory o f r a n d o m surfaces gives an elegant definition o f thc path integral o f 2D conformal field theory coupled to q u a n t u m gravity, previous studies have always bccn limitcd to the case o f fixed topology o f the two-dimensional spacetime. In refs. [ I - 3 ], a continuum limit that includes the sum over topologies o f two-surfaces was defined for pure q u a n t u m gravity. There, the diserctized 2D q u a n t u m gravity path integral was identified with the large N expansion of the frcc energy o f a hermitian matrix model with potential g~ Tr(0~) I ' ( O ) = T r ( 0 2 ) + ~ ,V~/2-1 It was found that for generic couplings g~. the free energy can bc described by a solution to the Painlev6 type I equation. F o r special valucs of the gk, corresponding to ruth order multicritical points ( m >~3) of a related potential l , t ' ( r ) , the charactcr o f the theory changes drastically and the free energy is describcd by a solution to a nonlincar diffcrcntial equation o f o r d c r 2 m - 2. Bitnet address: c r n k o v i c ~ y a l p h y . h c p n c t , or ,~:,yalehep.bitnet 2 Bitnet address: ginsparg.'~huhepl.hcpnet, .~,huhepl.bitnet, or :~ h u h e p l . h z r v a r d . c d u ' Bitnel address: moore:-_ayalphy.hcpnet, or ".ayalchep.bitnet
196
Unfortunately. the correct physical interpretation o f the thcories defined in rcfs. [ 1-3] is not entirely clear. This is especially true o f the m>~ 3 theories, which were originally conjectured [5] to be equivalent to the ruth m e m b c r o f the unitary discrete series coupled to 2D q u a n t u m gravity. It was, however, noted in rcf. [ 4 ], for example, that the free cnergy o f the lsing model, c o m p u t e d using a two-matrix model formulation, satisfies a different differential equation from that o f the m = 3 multicritical point of the one-matrix theory, thereby disproving the above conjecturc. Thc (gcnus zcro) results o f ref. [6] m o r e o v e r suggested that the correct interpretation o f the m = 3 multicritical point is given by the YangLee edge singularity coupled to 2D q u a n t u m gravity. In this letter we complete the identification o f these two models, in the sum over topologies formulation, by calculating thcir free energies and correlation functions. A simple modification o f our calculations also gives results for all correlation functions of lhc Ising modcl. Let us first compute the correlation functions of the m = 3 one-matrix model. Wc follow closely thc derivation and notation o f rcf. [2 ]. Beginning with the potential potential .~ , I , ' ( 2 ) = 2 7 + N k 4 + g2}, -2 A~ wc define orthogonal polynomials o f norm hk (see rcf. [7] and rcferenccs therein for n o t a t i o n ) . The parti-
0370-2693/90/$ 03.50 c'¢)Elscvier Science Publishers B.V. ( Norlh-Holland )
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tion function is easily expressed in terms of the quantities fk=hk/hk_,, which satisfy recursion relations derived via integration by parts. In the large N limit, we write x = k/N, fk-,Nf(x). In terms o f t ( x ) - g f ( x ) , the tree level ( N ~ o o ) equation is [7] g x = 14"(t)- 2 ( t + 6 t 2 + 30yt 3) .
(1)
We now consider the scaling limit defined in refs. [ 13 ]. Thus we define
g)c=g,.( 1 --a22), where g¢= 14"(t~) is the value of the potential at the tricritical point t=t,., defined by 14" I,c= 14'" I,c=0. The correlation functions of refs. [3,4] correspond here to the g-,g,, scaling functions of the derivatives of the free energy with respect to 7 at the tricritical point. They can be calculated by first expanding t(x) around the solution of W'=O, i.e. t(x)=to(y) +a2/3f(z), where W' It0~, = 0 , and then allowing ), to scale to its value at the tricritical point. The equation I4,"=0 gives t o ( 7 ) = ( - l + x / l - ~ y ) / 1 5 y , and 14"" = 0 gives 7=-~, so that to= --~. The recursion relations satisfied here by t(x),
½gx=t(x)+ 2 t ( x ) [ t ( x - ~ ) + t ( x ) + t ( x + ~ ) ] + 3yt(x){t(x)2 + t ( x - ~ ) 2 +t(x+~) 2 +2t(x)[t(x-c)
+t(x+~)] +t(x+2~)t(x+c)
+t(x-2~)t(x-~)+t(x+~)t(x-~)},
(2)
where ~= 1/N follow from the results of ref. [7]. To obtain a well-defined continuum limit, we take ~ ~ a 7/3 and let ;~ scale to the tricritical points as y = ~(I-a4/~T2), while t o = - l / 6 ( l + a 2 / 3 T ) (thus preserving W ' = 0 ) and t(x)=to(7)+a2/3f(z). Matching powers o f a 2 in (2) and rescaling, we find
z=f~-ff"-~.f'2+~f''+T(fZ-~f
'') .
(3)
Setting T=O, we recover the result of rcfs. [ 1-3] for the tricritical one-matrix model. [Actually, for small T, the result (3) is obvious, since, to lowest nontrivial order in a rescaled T we have t(x)=t,. + a 2 / 3 ( f - t - T). Thus, we could substitute f--,f+ 7" in the equation for the free energy and keep lowest order terms in T t o recover (3).] As discussed in refs. [ 1-3 ], the scaling function for the second derivative of the free energy with respect to the cosmological constant is given b y f ( z ; T = 0 ) , where z = x -6/7 and
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x 2 is the string coupling constant. E x p a n d i n g f ( z ; 7") in powers of 7, the coefficient of 7 "k gives the second derivative with respect to the cosmological constant of the correlation function o f k insertions of a certain basic operator of the theory (expressible as a linear combination of traces of powers of 0). The above derivation is easily extended to the mth multicritical model using the formalism of ref. [ 3 ], as we shall discuss below. We now perform the analogous calculation for the Yang-Lee edge singularity coupled to 2D quantum gravity, formulated in terms of an [sing model in an (imaginary) magnetic field. As in ref. [8], we consider the large N expansion of a two-matrix model with free energy F(g, c, H; N)
I ,&2 /v2 -- log
+ ~-
d
Ud
U4+ ~
Vexp - T r
I;4
)]}
( U 2.q_ 2 V - 2cUV
.
(4)
the two types of quartic vertices in the graphical expansion of the RHS above are identified with the two possible spin states of the ]sing model on a random lattice with coordination number 4, and the propagator generates the link interactions. II in (4) denotes the magnetic field, and the parameter c (with physical range 0 < c~< I ) is related to the ]sing nearest neighbor coupling fl via c = e- 2B. The logarithm picks out the connected graphs so that F in (4) corresponds to the partition function of the ]sing model summed over random surfaces. In ref. [8], the free energy (4) was expressed in terms of norms hk of polynomials orthogonal with respect to a bilocal potential. These norms can in turn be expressed in terms ofJ~ =hk/hk_,--,Nf(x). Defining the quantity t(x) =6g[(x)/c, the recursion relations of ref. [ 8 ] take the form (e = I / N )
0 = - 6r(x) +t(x)(l+
~ e-lt[t(x+~)+t(x)+t(x-~)]
+ ]{t(x+~)[r(x+2~) +r(x+¢)+r(x)]
+ t ( x ) [ r ( x + ~ ) +r(x) + r ( x - ~ ) ] +t(x-~)[r(x-2~)+r(x-c)+r(x)]}),
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t(x) =to(c, H) + a 2 / 3 f ( z ) ,
~gx= - ~ c 2 t ( x ) +r(x){l +2 eH[r(x+~)+r(x)+r(x-~)]
6)(¢, H) = t . ( c ) +a2/3T.
+~c:[ t(x)t(x-c
From (7) and (8), it follows that ( t o - t . ) 2 ~ t t II(¢). Using the relation cd/cb¢= - a ~ / 3 2 d / d z and the expansion
) t ( x - 2~ )
+ t(x+ ~)t(x)t(x-
c)
+ t ( x ) t ( x + ~ ) t ( x + 2~) ] .
(5b)
The potential defined by the genus zero recurston relation is in this case
gx = W( t, c, H) (6)
As explained in ref. [ 9 ], the leading critical behavior is determined by the smallest absolute value solution, t=to(C, tl), of (O/Ot) W = 0 . Taking the first derivative of (6), we find that to(C, H) is given implicitly by the relation 2 cosh I I =
r~(x) = ro +a2/3rl (z) +a4/3r2 (z) +a2r3(z) in (Sa), (5b), we find
r2 =
2_3 ~c~t+ t ( l + t e - " ) ( l + t e " ) -v~ct 3(1_t2)2
C2(1 __/(])4__ (to4+
6t(] + 1 )
2to( 1 +t(])
(7)
and substituting this relation into the second derivative we find
(9 c o n t ' d )
t,)(3 +t(]) + e - " (1 + 3t(])
6(I _I(])3
(['2_t_ ~22to.f,,) ,
I (.f3A+22f,2B +)2ff,,C+)4f,,,,D) ' r3 -- 216( 1 - t ( ] ) 4 " "" (lO) where A, B, C, D are known polynomials in to and e - ' . After some algebra (done by hand and checked by computer) and appropriate rescaling o f f z, and T, we find the result
z=.f3+ff"+½(f')2+~f(4'+T(fZ+~f"),
(11)
The tricritical "t point, which defines the Ising phase transition and Yang-Lee edge singularity, is given by setting the RHS of (8) to zero. Together with (7), this condition determines t ! = H ( c ) and t. ( c ) - t o ( C , H (c)). For ]
identically reproducing eq. (3). This completes the identification of the sum over topologies versions of the Yang-Lee edge singularity and the rn = 3 multicritical one-matrix model. In ref. [6], a mapping was defined between the two-matrix model and a dimer model (described by a one-matrix model with a sixth order potential) at a special value of (c, I l L namely H ( ¢ ) = ½in. Such a mapping, suitably extended to all c > ~ and all genera, would provide a more conceptual explanation of our result. A slight modification of the above calculations allows us to write a generating function for correlation functions in the lsing model in zero magnetic field at the critical temperature, after summing over topologies. The equations are different since, as c approaches the critical value, c--, ¼, and II--,0, the coefficients r2, r3..... see (10), in the expansion of r ( x ) develop singular terms. Therefore we define a new scaling limit,
Na T/3=)t ,
t= - 1 - t - a 2 / 3 f ( z ) ,
02 --0t2 W[ '°("'")= c2 (St~+3to(10t(]+l + t o2 ) I ) - 1
(8)
(9)
to(C, t i ) = - 1 +a2/3S,
c=~-a2/3T, ~'~ We do not mean tricritical here in the conventional sense of a tricritical phase transition, which Ising is certainly not, but rather in the sense ofa multicritical matrix model (and regret the conflict with standard terminology ). 198
and define the expansion
(12)
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r(x) = - ~ + a 2 Q f ( z )
PHYSICS LETTERS B
+aoJ(z) + a a / 3 r 2 ( z )
+ aS/3rs/2 ( z ) + a2r3 ( z ) .
(13)
Substituting (12) into (7) we find that the physical (real) magnetic field is given by H = a s/3 × x/~S~4T - a 5/3~. Then substituting these expansions into the recursion relations (5a), (5b), we find that rs/2 and r3 drop out to order a 2 and that r2 may be eliminated from the remaining equations, thus giving the pair of coupled equations (again after appropriate rescaling)
z= T(JO--4f '' ) - 0.)2+ f 3 - J f ' ' - -~(f' )2 + 2 f ( 4 )
,
,~¢=.fog- ~¢o" .
(14)
The properties of these differential equations ,2 can be analyzed in a way analogous to the treatments of refs. [ 1-4]. In particular, using the physical normalization of ref. [4] the residues of the second order poles remain integers. Since T in (12) parametrizes the scaling deviation away from criticality, eqs. (14) with ~o=0 can also be regarded to define a massive continuum limit of the Ising model in the sum over topologies formulation. (Relevant perturbations of the nonunitary one-matrix models were considered in refs. [2,3].) Genus zero correlation functions of the energy operator e and spin field a are generated by taking derivatives with respect to 7"and .~ of the logarithm of the solutionf(z; T, .~¢ ) to the tree level equation z =f3 + T/~_ f~-"
( 15 )
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Note that all correlators with an odd number of a's vanish, and that the correlation functions involving only e are identical to the genus zero Yang-Lee correlators. [The quantities in (16) are actually second derivatives with respect to the cosmological constant, proportional to z, of the Ising partition function with the indicated operator insertions, and not normalized by dividing by the partition function. The inferred susceptibilities agree with those given by the formula -~;, fl, l a - ½ z Q / a = - X, fl, l a + ~ of ref. [l 1], where the gravitationally dressed scaling weights are fl/a=~, ~ respectively for the c = ~ operators a and e. These weights are also evident from inspection of (15). For the Yang-Lee edge case, the single weight ~ operator corresponds to the identity operator with weight taken with respect to the negative dimension operator, in accord with the interpretation ofref. [4].] We now discuss and speculate on the physical meaning of the above results. Since a general perturbation away from a multicritical point should define the generic m = 2 pure gravity model, the occurrence of the Painlev6 equation multiplying T in (3) and ( 11 ) could have been easily foreseen. Referring back to the potential IV(r) that arises in the genus zero recursion relation and using the formalism of ref. [2,3 ], we can make this argument more precise and extend it to the general multicritical one-matrix model. By rescaling the hermitian matrix 0, we may take the action V ( O ) = ( N g / g c ) ( ~ T r 0 2 + . . . ) . The perturbation of an ruth order multicritical potential m--I
W,~(r)--l-(l-r)"~14",,(r)+
~ (z(1-r) / /=2
We thereby find the following rather curious correlation functions for the Ising model: (e)
<~) (eee)
= - ~ , =~z -'~ =
~Z -2/3
(eeee) = O,
,
=-~z -~/~ = -- 8z -3
(aaee) = 0
(16)
corresponds to some perturbation of the original potential V(~0) --. V(q~) + 3"t (/V/(q~). Using the recursion relation in the form of eq. (12) of ref. [3], we see that scaling (/= (Ng/gc)-2t"-1~/t2'"+')Ti [equivalent to the scaling used above in the derivations of (3), (11), and (14)] leads to the differential equation m- I
~
T,r-/t[f],
(up to rescalings of the energy and spin operators).
z=;-/,,[.f]+
(17)
.2 These equations have been found independently by Douglas [10]. who argues that the linear term in .~ in the second equation may lead to a nonperturbative breaking of the Ising z~2 symmetry.
where r%[f] is the lth member of the hierarchy of differential equations defined in refs. [1-3] (i.e. 5"-/2Lf ] =.f2 _ ~f,,, .c¢3i f ] =f3 _if,, _ ~f,2 + ~of'', etc. ).
/=2
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One interesting corollary of this argument is that the operator e, that perturbs :'1,,,I f ] ~ ~,,, i f ] + "1"~,~_ ~I f ] has a one-point function given exactly bv (e,) = - 1/ m. This follows sincc, by a simple generalization of thc argument following (3), we must have
~"If+
l T'l = ~ ' " [ f ] +
T~'~-j[j']+O(T~)
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tions to the Yang-Lee correlation functions. The onepoint function receives no nonperturbative corrections, as insured by our argument following ( 18 ), and the leading nonperturbative correction to the twopoint function, for example, is 6(ee,) ~ -3.~ ~t,, ( 1 - i x / 5 ) x - 3 / " e x p ( - ~ & / x )
"
-22-~b2( 1 + i x / 5 ) x -3/ ~ e x p ( - ~ b : / x ) + .... (20) The solution to ( 17 ) is analytic in T = T,,,_ ~, but from (18) we see that f + ( I / m ) T solves a differential equation whose coefficients involve powers of Tlarger than or equal to two. Thus f(c; T ) = - ( l / r e ) T + .~'(z; T) w h e r e f i s analytic in Twith no term linear in 7: In particular, we see that the Yang-Lee one-point function (r,) = - ~ is nonperturbatively exact. In the case of the perturbed Ising equations (14), we do not obtain the correctly normalized Painlev6 equation. Intuitively, this indicates the existence o f other operators in the theory.. Since the coefficients are precisely correct in the one-matrix case we have evidence that the operators V~, which correspond to Kazakov's scaling operators [ 5 ], form a complete set of relevant operators and hence the mth multicritical model is the minimal model (2, 2 m - 1 ) coupled to 2D quantum gravity, as conjectured in ref. [ 4 ]. One can also investigate the effects of the "nonperturbative parameters" discovered in refs. [ 1-3 ]. For eq. ( 3 ) and eq. ( 14 ) with o9= 0, the analysis of the nonperturbative parameters proceeds as in the case T = 0 : we let.f= ~ (z, T) +e, whercJ~ is the asymptotic perturbative solution for z ~ at fixed 7, and e<< 1 in this regime. The nonperturbative corrections then satisfy' the homogenous equation 0 = (3f~ + 2 7 J ~ .f '~')g,- f '~e' - ( ~ +fiT)e" + ae"', where (0~,/3) = (~, ). (~, 4) respectively' for Yang-Lee and Ising. For the Yang-Lee case (3), the WKB approximation gives the asymptotic expansion o f a physically acceptable solution as
.f(z; 7, 2~, 2,) ~.~(z: T) + 2 1 z " e x p ( - 6 t h z T / 6 ) (1 +...) +22zVexp( --~,'2" ok, _7/6~, (1 +...) ,
(19)
whcre b~.2= ( 5 + i / v / 5 ) ~/2 (with the root choscn to give a positive real part), v,.2 = - :~, and 2 t,2 arc all indcpendent o f T. Computing the subdominant contributions to (19) gives the nonperturbativc correc200
Note, in particular, that the partition function and correlation functions pick up imaginary parts, suggestive of an instability, unless o f course 2, =,,13. In the lsing case ( 14 ), we find a nonperturbative correction to the one-point function. Combining the above discussions of the perturbations of differential equations with the analysis of the nonperturbative parameters, we are led to spcculatc about a physical principle by which these parameters might be fixed. The perturbations (17) of the frcc energy differential equations should bc regarded as defining a version of rcnormalization group flow between minimal models, as suggested in refs. [2,3]. Now note that if./'(z: 7; 2,, 2,) is a solution o f (3), then 7'(3'; T. 2 ~, 22) - T 2/5.I"( "1"l/S.l'; "/', 2t, 22) satisfies y=.72-
.{I'" + I "
• --7/5
' (7 - . ~ 3
,,
--'~?"-'--T~., -~ ' ?'.... ).
(21)
Therefore, J'f(y)=]imr .~,7()'. 7"; 2,, ).2) exists it must be a solution of the Pain]ev6 equation. A twoparameter space of functions.f (z: T, ). ]. 22) would then be mapped to the two-dimensional space of solutions of the Painlev6 equation. It is not obvious that the resulting solutions to the latter will automatically lie within the one-dimensional physically acceptable subspace (those with no exponentially growing term in the asymptotic expansion), so to insure this the two original parameters (,;q, ,-2) might have to be related, leaving only one independent. An even more dramatic possibility is that the resultingT(y) is completely independent of the choices of 2~, )-2, i.e. that the "flow" from a solution f ( z ; T, 2t, 22) of ( 11 ) determines a unique (physically acceptable ) solution]'(y) to the l'ainlcv• equation. Naively, this property is suggested by the limiting behavior liml .,,, T-~"5( T2/SY)"exp[ - h(1"2/5y) 7/6 ] -~-0 of the 2 dependent terms in (19). If the argument could be made rigorous, the "'nonperturbative parameter" ofrefs. [ I-3 ] would be completely fixed by
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the requirement that it can be obtained by "renormalization group flow" from a htgher multicritical point. In the interesting event that the conclusion is correct, we would expect as well that a straightforward generalization will show that the m - 1 physically acceptable parameters of the ruth order multicritical point are completely determined by a similar flow from a higher model. We close with some c o m m e n t s on the correlation functions. It was pointed out in ref. [4] that a nonvanishing one-point function of the energy operator in the lsing case violates the physical intuition based on the picture of q u a n t u m gravity as a (coupled) product of matter and gravity path integrals ~3. Since the energy operator, defined by differentiation with respect to 7", only makes sense up to shift by a constant times the identity, it seems natural to us to define a shifted operator g - e,+ ~z - '/3~. From ( 16 ) we find that ~ satisfies ( ~ ) = 0 , (~2)=~z-,/3, ( ~ : 3 ) = 0 , (?,,4) = - 7 ~ z - ' . Correlation functions for the Ising model coupled to gravity are also expected to have certain positivity properties - positive definite correlation functions of the ordinary lsing modcl should remain so if the integral over 2D metrics defines a positive measure. The objects in ( 16 ) first of all need to be integrated twice with respect to the cosmological constant to obtain correlation functions. Although various changes in conventions ( f o r f z, T a n d ,~2) can then be used to shift the signs around, it is not obvious that all of the undesired negative signs can be eliminated. On the other hand, the integrations with respect to thc cosmological constant generate nonuniversal terms in the correlation functions that may act to restore the desired positivity. The proper notion of unitarity in q u a n t u m gravity, as applied to the leading singular behavior of the scaling limit of the theories we consider here, may in any event bc very subtle. In conclusion, the matrix model formalism has thus far produced some tantalizing new results on 2D q u a n t u m gravity. It does not yet provide a clear intuitive picture of all of the p h e n o m e n a it describes, however, and answers to certain fundamental physical questions remain obscure. Wc hope that further investigation will ultimately lead to an entirely novel
~3 In ref. [ 12 ], on the other hand, it is argued that the one-point function vanishes.
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framework for investigating the nonperturbative effects of s u m m i n g over topologies. C.C and G.M. are supported by DE-AC0276ER03075; P.G. by NSF contract PHY-82-15249, DOE OJI grant FG-84ER40171, and by the A.P. Sloan foundation. We thank M. Douglas, A. Morozov, R. Shankar, S. Shatashvili, S. Shenker, C. Vafa, and J. Zinn-Justin for useful discussions, and M. Douglas, N. Seiberg, and S. Shenker for comments on a preliminary version of the manuscript. We also thank A. Migdal and S. Shenker for providing us with copies of ref. [ 6 ].
Note added. After completion of this paper we received a paper [ 13 ] with related results on the YangLee edge singularity. While this sentence was being typed, we were thrilled to receive more papers [ 10,12 ] with further results.
References [ I ] E. Brezin and V. Kazakov, Phys. Lett. B 236 (1990) 144. [2] M. Douglas and S. Shenker, Strings in less than one dimension, Rutgers preprint RU-89-34. [ 3 ] D. Gross and A. Migdal, Nonperturbative two dimensional quantum gravity, Princeton preprint PUPT-1148. [4] E. Br6zin, M. Douglas, V. Kazakov and S. Shenker, Phys. Left. B 237 (1990). [ 5 ] V. Kazakov.The appearance of matter fields from quantum fluctuationsof 2D gravity,NielsBohr Institute preprint NBIHE-89-25. [6]M. Staudacher, The Yang-Lee edge singularity on a dynamical planar random surface, Urbana preprint ILL(TH)-89-54. [ 7 ] D. Bessis, C. Itzykson and J.-B. Zuber, Adv. Appl. Math. I (1980) 109. [8] D. Boulatovand V. Kazakov, Phys. Lett. B 186 (1987) 379. [9] V. Kazakov, Phys. Left. A 199 (1986) 140. [10] M. Douglas, Strings in less than one dimension and the generalized KdV hierarchies, Rutgers preprint RU-89-51. [ 11 ] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Len. A 3 (1988) 819; F. David, MOd. Phys. Lett. A 3 ( 1988 ) 1651; J. Distler and H. Kawai, Nucl. Phys. B 321 ( 1989 ) 509. [121T. Banks, M. Douglas, N. Seiberg and S. Shenker. Microscopic and macroscopic loops in non-perturbativetwo dimensional gravity, Rutgers preprint RU-89-50. [13] D. Gross and A. Migdal, Nonperlurbative solution of the [sing model on a random surface, Princetonpreprint PUPT1156. 201