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Boundary operators and touching of loops in 2D gravity
ELSEVIER
Nuclear Physics B 501 (1997) 251-268
Boundary operators and touching of loops in 2D gravity* Masahiro Anazawa 1 Department of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560 Japan
Received 26 February 1997; accepted 9 June 1997
Abstract We investigate the correlators in unitary minimal conformal models coupled to two-dimensional gravity from the two-matrix model. We show that simple fusion roles for all of the scaling operators exist. We demonstrate the role played by the boundary operators and discuss its connection to how loops touch each other. @ 1997 Elsevier Science B.V. PACS: 04.60.Nc; 11.25.Pro; Keywords: Matrix model; Two-dimensionalgravity; Fusion rules; Boundary operators
1. Introduction The understanding o f two-dimensional quantum gravity has experienced great progress through the study o f the matrix models. 2 The one-matrix model has an infinite number of critical points which are considered to represent the (2m + 1,2) minimal conformal models coupled to two-dimensional gravity. The two-matrix model has critical points which correspond to the (m + 1, m) unitary minimal conformal models [ 2 - 4 ] . In this paper we investigate the unitary minimal model (m + 1, m) coupled to two-dimensional gravity from the two-matrix model. The emergence o f the infinite number of scaling operators trj is one of the most important properties o f the matrix models. Before being coupled to gravity, the minimal * This work is supported in part by the Grand-in-Aid for Scientific Research Fund (2690) from the Ministry of Education, Science and Culture, Japan. 1JSPS research fellow. E-mail: [email protected] 2 See, for example, Ref. [ 1] for a review. 0550-3213/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII S0550-3213 (97)00369-6
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M. Anazawa/NuclearPhysics B 501 (1997) 251-268
model has a finite number of primary fields. After gravitational dressing, however, an infinite number of scaling operators emerge. This phenomenon can be understood as follows. In the Kac table we can divide the primary fields ~r,s into those which are inside the minimal conformal grid 1 ~< r ~< q - 1, 1 ~< s ~< p - 1 and those outside, which correspond to the null states. Before being dressed by gravity, the fields outside the minimal conformal grid decouple [5] from physical correlators. After gravitational dressing, they cease to decouple [6,7] and become an infinite number of scaling operators. A similar phenomenon has been shown to exist in the continuum framework. Through the examination of the BRST cohomology of the minimal model coupled to Liouville theory, an infinite number of physical states were shown to exist [8]. These states have their counterparts in the matrix models as the scaling operators. Some of the scaling operators do not have their counterparts in the BRST cohomology, which we will discuss later. In ordinary (p, q) minimal conformal model the primary fields satisfy certain fusion rules [5] ; the three-point function (~rl,sl ~r2,s2~r3,s3) is non-vanishing only when 1+
Ir~
- r2] ~< r3 ~< mim{rl + r2 - 1,p},
l+lsl-s2l<<,s3<~mim{sl+s2-1,q},
rl + rz + r3 = odd, sl+s2+s3=odd.
(1.1)
It is interesting to examine how the fusion rules change when matter couples to gravity. The three-point functions involving lower-dimensional scaling operators were examined from the point of view of the generalized KdV flow in [9] in the case of (m + 1, m) unitary matter. It was shown that the gravitational primaries o-j (j = 1 . . . . . m - 1 ) satisfy fusion rules of the BPZ type; for jl + j2 + j3 ~< 2m - 1, (O'jlO'j20"j3) is non-vanishing only when 1 + IJ1 -
J2l ~< j3
~< jl + j2 -- 1 .
(1.2)
The fusion rules were also examined in the continuum framework [7]. As for the gravitational descendants, however, we think clear results have not been obtained. In this paper we would like to clarify the fusion rules for all of the scaling operators including the gravitational descendants in the case of the unitary minimal model. Macroscopic loop correlators, which are the amplitudes of the surfaces with boundaries (loops) of fixed lengths, are the fundamental amplitudes of the matrix models. It was shown [ 10,11 ] that these correlators have more information than those of local operators and that the latter correlators can be extracted from the former explicitly in the case of c = 0, 1/2, 1. The authors of [ 10,11 ] argued that macroscopic loops could be replaced by a sum of local operators whose wave functions satisfy the Wheeler-DeWitt equations. We generalize this idea to the loop correlators [4,12-14] in the cases of the general unitary minimal models, and derive the fusion rules for all of the scaling operators. First we derive the explicit form of the expansion of loops in terms of the scaling operators, and then deduce the three-point correlators from the loop correlators which were calculated in [ 13,14] from the two-matrix model. We show that the three-point correlators of all of the scaling operators satisfy certain simple fusion rules and these
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
253
fusion rules are summarized in a compact form as the fusion rules for three-loop correlators [ 13 ]. In matrix models, there is an infinite subset of the scaling operators ~-j ( j = 0 mod m + 1) which does not have its counterpart in the BRST cohomology of the minimal model coupled to Liouville theory. In the case of the one-matrix model, Martinec, Moore and Seiberg [ 15] argued that these operators are boundary operators which couple to the boundaries of a two-dimensional surface. They proved that one of them is in fact a boundary operator which measures the total loop length. But little has been said about the role played by the rest of these operators. We examine the geometrical meaning of these operators and their connection to the touching of loops in the case of general unitary models. We show that the boundary operators play a role of connecting several parts of the loops together. We also discuss the relation of the boundary operators to the Schwinger-Dyson equations proposed in [ 17].
2. Expansion of loops in local operators We consider the (m + 1,m) unitary minimal model coupled to two-dimensional gravity from the two-matrix model with a symmetric potential. The partition function Z is defined by e z = f dA d/~ e - ~ tr(u(~)+u(~)-~) ,
(2.1)
where A and/~ are hermitian matrices, and U is a certain polynomial. In this article, we limit our discussion to the two-matrix model with a symmetric potential and to the critical points which correspond to the unitary minimal models. In the case of an asymmetric potential, some of the boundaries (loops) of the two-dimensional surface would have fractal dimensions different from the usual dimension of length. In Ref. [ 10], in the case of the one-matrix model, it was shown that the loop operator can be expanded in terms of local operators, that is, the loop can be replaced with an infinite combination of local operators, except for some special cases. It was argued that this is the case for the general (m + 1, m) unitary models. Whether this replacement can be done safely or not is connected with whether the corresponding classical solution exits or not in the limit of small length of the corresponding loop. This claim is quite natural because all of the scaling operators are expressed in term of one matrix A as o~i = Tr(1 - ~)j+1/2 = ~ n an(J )n-l Tr'4n in the one-matrix model. In the two-matrix model, it appears that this idea is not the case since the direct connection of the scaling operators to the operators Tr ~n or Tr/~ n' is not clear. But this expansion is considered to be possible for the following reason. When one of the loops on the two-dimensional surface shrinks to a microscopic loop, the loop represents a local deformation of the surface. The microscopic loop can be replaced by the insertions of local operators. The loop correlators except for the one-loop case are continuous when
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
254
the length of one of the loops approaches zero, so that we expect that a macroscopic loop can also be replaced by a sum of local operators. In this section we derive explicitly the expansion of loops in local operators in the case of the unitary minimal models. Using this relation, we can deduce the amplitudes of local operators and those involving both loops and local operators from the loop correlators. Let us recall that the two-loop correlators in the (m + 1, m) unitary minimal model coupled to two-dimensional gravity are [4,13] m--1
1M ~le2 Z(q_)k_l~.~,(Mel)~.lk(Me2) (W+(el)Wd:(e2)) = m 2 el + (4)me2 k=-I
(2.2)
for e I ://= e2. Here w+(g) and w - ( g ) represent loop operators which create loops made by the matrices ,4 and/~ respectively, and we introduced the notation
K,p(Me) = sinqrlp[g 7r/2 p(Mg) .
(2.3)
Here the parameter M is defined by ( M ) 2 - m~l'/'t
a - A. = - a 2 / x ,
(2.4)
where A. represents the critical value of the bare cosmological constant, a n d / z is the renormalized cosmological constant. The definition of a set of the scaling operators has arbitrariness which comes from the contact terms [ 10]. As a set of local operators, we take the scaling operators ~-j whose wave functions gtj(£) satisfy the (minisuperspace) Wheeler-DeWitt equations -
e~
+ 4~e 2 +
g,j(e) = o.
(2.5)
It was shown [ 10] that these scaling operators correspond to the dressed primary fields of the conformal field theory in the case of the one-matrix model. In terms of these scaling operators, we can obtain simple fusion rules for three-point correlators, which we will show in the next section. We normalize the wave function of ~-j as -m
Kz(Me), m
j / > 1 v~ 0 ( m o d m ) .
g:j(Me)
(2.6)
Note that the normalization factor sin(j/m)cr in in Eq. (2.6) is consistent because there are no scaling operators ~'j for j = 0 (mod m) in the matrix model. We can express the right-hand side of Eq. (2.2) as an infinite sum in terms of ffij(Mg2) for el < g2 (see Appendix A):
255
M. Anazawa/Nuclear Physics B 501 (1997) 251-268 m--I
(w+ ( gl )w+ ( g2) ) = m1 Z
k
oo
~
(±)k-, _m + 2n Ii~,+2nl(Mg,)K~,+2,(Mg2).
k=l n=--oo
(2.7) Comparing Eq. (2.6) with Eq. (2.7), we expect the following expansions of the loop operators in terms of the local operators: 1 Z
" w+(g)
Z
m k=l n=-oo
(.+.)k-1
II~,+2nl(Mg) °'lk+2mnl "
(2.8)
These expansions are the generalizations of those in the case of the one-matrix model [ 10] to the cases of general unitary minimal models coupled to two-dimensional gravity. Since the loop correlators are symmetric under the interchange of two kinds of loops, that is, (w + (gl) w+ (g2)) = (w-(gl) w-(g2)), the wave functions of the scaling operators with respect to loop w-(g) read (O'j W - - ( g ) ) =
( - 1 ) .i-' (~'j w+(g)) .
(2.9)
3. Fusion rules for scaling operators Using the expansion of loops Eq. (2.8), we can obtain the correlators of the scaling operators from loop correlators. 3 In this section we show that there are rather simple fusion rules for all of the scaling operators. The fusion rules involving the gravitational descendants (o-j, j ~> m + 2) have not been clear from the point of view of generalized KdV flow.
3.1. One- and two-point functions Let us examine one- and two-point functions first. Since the one-loop amplitude diverges when the loop length approaches zero, this amplitude includes the contribution which is not represented by the local operators. Extracting the parts proportional to 1~ (v > 0), which can be considered as the contributions from local operators, from the one-loop amplitude
=
@,(M£)-I
2 ,(Mg)-I±(Mg)+l --
--
m
m
±(Mg) --
(3.1)
m
we can obtain the one-point functions of the scaling operators 3 The multi-loop correlators were examined in [ 14] from the two-matrix model. These correlators were also examined in 116] from the viewpoint of random surfaces immersed in Dynkin diagrams.
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
256
(3-1) = - m
,
(3-1+2,,)= m
(3-j) = 0,
j~: 1,1+2m.
,
(3.2) (3.3)
Let us turn to the two-point functions. Substituting Eq. (2.8) into Eq. (2.6), we obtain the two-point functions
(3-i~i) = ~Sijj
(7)
,
i,j ~ 0 (mod m).
(3.4)
Note that we obtain diagonal two-point functions.
3.2. Three-point functions The three-loop correlator from the two-matrix model is obtained in [ 13] as 1--±
m(m + 1) ×
Z
glg2e3
K 1 - ~ (Mgl) K I - ~ (Mg2) K I - ~ (Mg3) ,
(3.5)
(k I --l,k2 --l,k 3 --1) E'D~ m)
Here we have denoted by D~m) 3
79~m) = { ( al , a2, a3 )
~-~ ai -- aj >/ O for j = l , , ~ 3 , i(÷j)
Z 3a i
=
even ~< 2(m -- 2) ,ai = 0, 1,2 . . . . . }
(3.6)
i=1
It was shown [ 13] that the selection rules in Eqs. (3.5) and (3.6) correspond to fusion rules [9,7] for the dressed primaries (~ii, i = 1 . . . . . m - 1) by studying small length behavior of the three-loop correlator. Using the loop expansion (2.8), can show that the selection rules in Eq. (3.5) represent the fusion rules for all of scaling operators in a compact form. Let us show this in the following. Using the formula o~
zKI-Ipl (z) = ~" Z
[p + 2n[ sinTr[p + 2n[ IIP+2nl(Z)
(3.7)
n=--oo
we first expand the three-loop correlator Eq. (3.5) as -2--±
(w+(el)w+(e2)w+(e3))
-
r e ( m + 1)
the the we the
oo
~
E
oo
Z
oo
"D~m) ttl=--OO n2 = - O O ;'/3--OO
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
× I[~+2n~l (Mgl)I1~+2n21 (MgE)II~+2ml (Mg3) •
257
(3.8)
Comparing Eq. (3.8) with Eq. (2.8), we can extract the three-point functions 3
I ( ki + 2mni) (~rlk~+2mn'[O'lk2+2mnzlO'lk3+2mnal) =Cklk2k3 m ( m Sr 1) Ii=1
x ( M ) -2-"~+~]°' ~,lk,+2m,,I ,
(3.9)
where Ck~kzk3 =
1, 0,
(kl -- 1 , k 2 - 1 , k 3 - 1) 6 D~ "~ otherwise
(3.10)
For ni = 0, Eq. (3.9) is nothing but the correlator of the gravitational primaries. For the gravitational primaries, Eq. (3.4) and Eq. (3.9) agree with the correlators obtained in [9] from the generalized KdV flow up to a factor - 2 . Note that we here obtain the correlators of the gravitational descendants as well. In [7], the fusion rules for the gravitational primaries were examined in the continuum framework. We have found here the fusion rules for the gravitational descendants as well as those for the gravitational primaries. These fusion rules are similar to those for the gravitational primaries due to the factor Ck, k2t3 in Eq. (3.9). Introducing the equivalence classes [o'k] by the equivalence relation O'k "~ O'lk+2mnl ,
n E Z,
(3.11 )
we can consider the fusion rules in Eq. (3.9) as the fusion rules among [ok] (k = 1 . . . . . m - 1 ). Note here that the class [~'k] does not correspond to the set which consists of the gravitational primary Ok and its gravitational descendants orI ( O k ) , l = 1,2 . . . . in [9] introduced from the viewpoint of KdV flow. 3.3. Further on the fusion rules
In this subsection, let us examine the fusion rules in Eq. (3.9) further and summarize the relation of the scaling operators to the primary fields in the corresponding conformal field theory. In the (p, q) minimal conformal model, the primary field q0rs has the conformal dimension (pr -- qs) 2 -- (p -- q)2 dr, s = , (3.12) 4pq where r and s are positive integers. Since we have Ar,s = Ar+q,s+p = A q - r , p - s , the corresponding primary fields can be regarded to be the same. The integers r and s can thus be restricted to the range
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M. Anazawa/Nuclear Physics B 501 (1997) 251-268
r
0
p
2p
3p
s
Fig. 1. The range of (r, s). 1 ~r~q--1,
1 ~s
~ 0(modp),
pr
(3.13)
(see Fig. 1). In Fig. 1, the primary fields in the region ((2)) or ( (2))' are the secondary fields of those in the region ((1)). In general, the fields in the region ((n + 1)) or ( ( n + 1) ) t are the secondaries of the fields in ((n)) or ((n)) t. Since the secondary fields correspond to null vectors, those fields decouple. One can thus construct a consistent conformal field theory which includes only the primary fields in the region ((1) ) (i.e. inside the minimal table), that is, the (p, q) minimal model [5]. Coupled to Liouville theory, however, the fields outside the minimal table fail to decouple [6] and an infinite number of physical states emerge accordingly. These states are considered to correspond to the primaries outside the minimal table. This correspondence is implied by the BRST cohomology [ 8] of the coupled system. In the minimal model coupled to Liouville theory, the gravitational dimension zlI,S ~ of the dressed operator for ~r,s is given by the relations d ras = 1 -
Otr,___Ls y '
Otr, s = p + q - [p r - q s [ y 2q '
(3.14)
where r and s take values in the range Eq. (3.13). On the other hand, in the matrix model, the corresponding relation for the scaling operator ~'j is given by O(j y
-
-
p + q- j 2q
(3.15)
From Eq. (3.14) and Eq. (3.15), we should take j=[pr-qs[,
j=l,2 ....
4=0(modq),
(3.16)
for ~'j. Consider now the relation of ~'lk+2mnl tO the primary field ~r,s of the unitary ( m + 1, m) minimal model. Let us first compare the two sets Sk = { I k + 2 n m l l n E Z } , and
(3.17)
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
r
03 02 O1
•
: (711+2toni
0
: OI2+2mnp
259
® : ~13+~,~,,I
p.
0
5
10
15
s
Fig. 2. The scaling operators ~lk+2,,~l in the (5,4) minimal model coupled to 2d gravity.
{[pr-qs[}={[(m+l)r-ms[}={r'+(s-r-1)m},
(3.18)
where r and s are positive integers in the range l~
l <<.s,
r + l <<.s
(3.19)
and r ~ - m - r. Note that we include s = 0 (mod m ÷ 1) here. Decomposing the set Sk into two sets as & =
•
(3.20)
where
S+ = { k + 2nm l n =O, 1,2 . . . . } , {(m-
k) + (2n' + 1)m I n ' = 0 , 1 , 2 . . . . } ,
(3.21)
and comparing Eq. (3.21) and Eq. (3.18), we can express the sets S~- and S~- in terms of [ ( m + 1 ) r - m s [ as S~-={[(m+l)r-msl]r'=k,
s-r=2n+l,n=O,
1. . . . } ,
S k = { l ( m + l ) r - ms[ l r = k, s - r = 2n' + 2, n' = O, 1 . . . . } .
(3.22)
From Eq. (3.22), the following correspondence is obtained:
~rlk+2mn[ (n >/O) ~ ~m-k, r+2n+l (n /> 0) O'[k+2m(-1-n')[ ( n! ~ O) ~-+(1)k, r+2n'+2 ( n! ~ 0) ,
(3.23)
where s ~ 0 (mod m + 1). As for the scaling operators corresponding to s = 0 (mod m + 1 ), we will discuss these in the next section. As an example, we depicted the scaling operators on the r - s plane for the case of m = 4 in Fig. 2. In this figure we showed the equivalence classes [o'k] explicitly.
260
M. Anazawa/NuclearPhysics B 501 (1997) 251-268
(a)
(b)
Fig. 3. (a) A surface with two loops touching each other in a point. (b) When one of the loops shrinks to a microscopic loop, the microscopic loop is equivalent to the insertion of the operator denoted by the dot on the loop.
(a) (b) Fig. 4. The case of a surface with two loops touchingeach other in two points. 4. B o u n d a r y operators 4.1. Boundary operators and touching of loops
The scaling operators ~-j ( j = 0 mod m + 1 ), which correspond to s = 0 (mod m + 1 ) on the r-s plane, do not have their counterparts in the BRST cohomology of the minimal model coupled to Liouville theory. In [ 15] it was proposed that the scaling operators which do not occur in the BRST cohomology of Liouville theory are boundary operators and one of them, o'3 = o-I (O1) in the case of pure gravity, was in fact proven to be a boundary operator for the one-matrix model and the Ising model case. We would like to examine the role played by the operators O'n(m+l)' n = 1,2 . . . . ~ 0 (mod m) as well as ~'m+l for general unitary minimal models. Let us denote these operators by /~n =o'n(m+l),
n = 1,2 . . . . :~ 0 ( m o d m ) .
(4.1)
In the matrix models the loop amplitudes contain the contribution from the surfaces with loops touching each other. In two-loop case, let us consider the surfaces in which the two loops touch each other in n points. When we shrink one of the loops to a microscopic loop, the other loop splits into n loops, which are stuck to each other through the microscopic loop (see Figs. 3-5). Since the microscopic loop represents a sum of the scaling operators, the wave functions of some scaling operators contain the contribution from the surfaces with the split loop. We now show that the boundary operators indeed represent these surfaces. From Eqs. (2.6) and (3.1), the wave function of B, and the one-loop amplitude are
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
(a)
261
(b)
Fig. 5. The case of a surface with two loops touching each other in three points.
( M'~ n(l+~)
(s~nw+(g))=n(l+ 1) \-~] (w+(g))=
(1) 1+ g-'
#2n(l@,)(Mg),
(4.2)
~JI+~(Mg).
(4.3)
We denote the Laplace transformation of any function f ( g l , f2 . . . . ) of loop lengths by oo £[f(gl,~2
oo
.... )] =id~l/d~.2...e-(gl'21+h(2+)f(~-l,g2 0
.... ).
(4.4)
0
In the space of the Laplace transformed coordinate sr, we have f[
Lg-l(~..w+(g))j] = -
2coshn(m+l)0, 1
£ [(w+(g))] = -
(4.5)
1
(7) +m2cosh(m + 1)0,
(4.6)
where we have used the relation £ [-g-'lulR'~(Mg)] = 2coshmz,0,
(4.7)
and ( is parametrized as ( = McoshmO. Note here that w+(f) represents a loop with a marked point and g-lw+(g) represents a loop without a marked point. Since cosh n(m + 1)0 can be expressed as a polynomial of cosh(m + 1)0, 2coshn(m +
1)O=2Tn(cosh(m+ 1)O) In~2] = Z C'r-(n)[2cosh(m +
1)O] n-2r
r--O
@n)--(--l)rn( )------n--r n-r r ' where
7". is the Chebyshev polynomial,
£[-g-l(s~,w+(g))]=
~_~ r---O
(4.8)
we obtain the following relation:
{~--~[--]} n-2r
c~") \ 2]
(4.9)
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
262
In the space of the loop length, the above relation means that the wave function of/3n is equivalent to a sum of the convolutions of disk amplitudes:
[ (n--l)/2]
(~n w+ (~) ) = --e
Z
( M ) 2r(,+.~ )
C'r-(n)
;-l)n-2r[(~)J4~] n-2r (£).
(4.10)
r=0 Here we introduced the notation A + = (w+(e)), and [®.A+]S (g) denotes the convolution of s .A~-(g)'s, for example, [QA+] 2 ( g ) = f f dgldg2 6(gl + g2- g)A+(gl)A~(£2).
(4.11)
0 0 From Eq. (4.10) we can conclude that the operator Bn couples to the point to which s (~< n) parts of the loop are stuck to each other in the case of one-loop amplitudes. When there is more than one loop, we infer that the operator couples to the point to which s parts of several loops are stuck to each other; the operator will not recognize that it is touching different loops this time. Using the following relation:
[2coshx
[(n--I)/2] =
z (n) 2 c o s h ( n - 2r)x,
(up to a constant),
(4.12)
r=O we also obtain
, [Q).A~-]n ( ' ) = (--1)n+'
[(n--I)/2] Z (;) r=0
( M ) 2r(l+~5 )
(~n_2rW+(')).
(4.13)
Here we drop the constant term in Eq. (4.12) when carrying out the inverse Laplace transformation. From Eq. (4.13) we see that the boundary operator coupled to the point on which n parts of loops are touching each other is given by
[(n-l)/2] /3n = ( - 1 ) n + '
Z
[M,~ 2r(l+~,) ^
Bn-2r.
(~)~-2")
(4.14)
r=0 Now let us consider the boundary operators when there are two loops on a twodimensional surface. As for ~1, we expect that (w+(el)w+(e2)131) should be proportional to (el + g2) (w+(gl)w+(g2)). Let us confirm this in the following. From the three-loop correlator (3.5), the expansion of the loop operator (2.8) and the wave function of °'lk+2mnl (2.6), we obtain the following correlator with two loops and a local operator:
(w+ ( e')w+ ( e2) k3+2m"3 ) =
m+ 1 Z
kl .k2
Ck'kzk3
×'1'2 (~- -+- 2n3") ~"1- hm (M'I)K" 1-- ~m (M'2) " (4.15) \m /
M. Anazawa/NuclearPhysicsB 501 (1997)251-268 Consider the amplitude for B1 =/31 = O'm+l = o'[ ~ _ 2 [ only for the case of kx + k2 = m, we have m-
263
Since Ct,,k2,,,_ 1 is non-vanishing
1
(w+(,°.I)W±(,°.2)/3I)=LS_.t(4-)k-I(M)p~I,P,2K~(Mgl)KI_,~_(Mg2). m k "
(4.16'
Comparing Eqs. (4.16) to (2.2) we obtain the desired relation,
(w+(et)w±(e2)&)= {el + (-1)me2} (w+(g~)w±(e2)).
(4.17)
Note here that we have (Blw+(g)) = ( - 1 ) m (BIW-(g)). Next, let us consider/32. Since we infer that the insertion of/32 should play the role of connecting two parts of loops, we expect the relation
(w+(g,)w+(g2)B2)=2g,f < (w+(e',)w+(e2))(w+(e,-e',)) o g2 +2h/de'2 (w+ o
) (w+(e2 - e'2) )
--k2ele2 (w+(el --I-e2))
(4.18)
to hold. The third term in the right-hand side of Eq. (4.18) represents the contribution from the surfaces with loops w+(gl ) and w+(g2) touching each other in a point. Let us confirm the relation (4.18) in the following. In this case, it is convenient to consider it in the space of Laplace transformed coordinates (i- In this space Eq. (4.15) reads
(17V+(f')f'V±((Z)O'[k3+2mn31)= -1 Osrl ~ 2
(M)-}-2+l~+Zn31(k--3m- - + 2 n 3 sinh toO1
kl ,k2
s--:-mnhm-0-22
'
where ff'+((i) - £[w±(gi)]. Due to the relation ~--~C
kl ,k2
rq._xk2sinh(m -
k, k2k3t )
= 2(coshOl
kl )01 sinh(m - k2)02 sinhm02
s-~nhm~
-lqz cosh 02) _(sinh(m-k3)Olsinh nO1 k3s i n h ( m - k3)02) \ -- ( + ) " s--~nhm~ ,
(4.20)
we have
' 2+1.0.+2n31 (if'+ ((~) ff'±(~'2)~lk3+2,,<) = 2(m + 1)
-f~.-~Ta 0 { xvs~v~z
1
- - + 2n3
(sinh(m--k3)Ol --(4-)k sinh(m-k3)02)}3
coshOlqzcoshO2 \
sinhmO~
sinhm02 (4.21)
264
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
Since we should take k3 = 2 for/32 = -o'2(m+l) (for m ~> 3), we obtain the amplitude for/32,
1
(~/+(~1) lf~/4_((2)/32)= ::~1
" ____~ 0q
m
0(1 0(2
{
1
cosh 01 q: cosh 02
(sinh(m -- 2)01 sinh(m--2)02 ~ } × \ sinhmO1 sinhm02 j .
(4.22)
On the other hand, from the amplitudes [4] 0
0
coshOl - cosh02
(I~t/+((1) l~/+((2)) -- 0(1 0(2 In cosh mO1 - cosh m02 a {
1
sinh O,
= 0(2 coshO1 - cosh02 mMsinhmOl
1 } (1 - (2 '
(4.23)
we obtain the relation _2 0
e(1 { (~/+ ((1)Vi/+ ((2)) ( ~ + ( ( 1 ) ) } "~ (1 +--+2)
--2 0-' } 0---O--q-0(2O { (~+((1)) ( ,(1 - (2 (I~7+((2)) 1
2 (M)~' 0 O { 1 m 0(l 0(2 cosh 01 cosh 02
(sinhO, cosh(m+l)O1 sinh mO1
(1,-+2))}. (4.24)
One can easily show that the right-hand side of Eq. (4.24) agrees with that of Eq. (4.22). Putting Eqs. (4.24) and (4.22) together and performing the inverse Laplace transformation, we obtain the desired relation (4.18). As for (W+(gl)W--(g2)/32), from the amplitude [4] 0 O (I~'+((1) I~'-((2) ) - 0(1 0(2 ln(cosh 01 + cosh02) ,
(4.25)
we obtain the relation
gl (w+(e,)w-(e2)/32)
-2g, / de',(w+ <
)w- (
) (w+ ( e, -
))
+2e2/de;(w+ ( el )w- (
) (w- ( h -
)
o g2
(4.26)
o
in a similar way. In this case, the operator/32 does not connect the different kinds of loops w+(gl) and w-(g2) together. We have shown that the operator/32 connects two parts of the same kind of loops together in the case of two loops. We infer that similar phenomena occur in general; the
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
265
operator/3n connects n parts of the same kind of loops together in the case with any number of loops.
4.2. Connection to the Schwinger-Dyson equations A close relationship between the boundary operators and the Schwinger-Dyson equations proposed in [ 17] can be observed. The continuum limit of the Schwinger-Dyson equations for loops in the two- and multi-matrix models were proposed in [ 17] under some assumptions. It was shown [ 17] that these equations for the two-matrix model contain W3 constraints, which were derived explicitly in [ 18]. The integrability of these equations was shown in [ 19]. These facts justify the proposed Schwinger-Dyson equations. Let us consider the connection of the boundary operators with the Schwinger-Dyson equations. For the (m-t-l, m) minimal models, the following Schwinger-Dyson equations were proposed in [ 17]:
g / d ~ . ' (w(1)(et)w (1) (~
et; {']"/(o-) ] J) w(l)(~l)...w(1)(.Pn))t
0 e i (W <1) (e -~-ei; [-7.-/(0-)]j) W(I)(el)... W(1) (ei--1)W (1>(ei+l)... W(1) (en)~! i -[-(W (1) (e; [~'/(O-)]J+I)W(I)(e,)...w(l)(en)) ! ,~0, f o r j = 0 . . . . . m - 2 , +g ~
(4.27)
(W (1) (e; [']"/(O-)] m-l) W(1)(el ) . . . W(1)(en)) t ,-~ O.
(4.28)
Here (...}' represents loop correlators that are not necessarily connected, and w (1) (g) represents a loop operator corresponding to a loop created by the matrix ~(1) in the multi-matrix model. The operator 7-/(o-) describes an operator which changes the 'spin' on a loop locally from 1 to 2. Also w(l)(g; [7-/(o-)]J) describes a loop with [7-/(0-)] ,i inserted. The symbol ~ means that as a function of g, the quantity has its support at g=0. From Eq. (4.27) for j = 0 and n = 1, we have the relation
g, (w (t) (g,; ~(o-)) w(~)(g2))' + g2 (w(1)(gl)w (') (g2;~(o-)))' gl +gl f dg'l (w(D(g/l)W(')(gl - g'l)W(l)(g2)) ' o g2
+g2 f dg~ (w(1)( gl)W(1)(g~z)W(')(g2 -- g~) )' o
+2gglg2 (w(l)(gl + gz) )' "~ O.
(4.29)
266
M. A n a z a w a / N u c l e a r Physics B 501 (1997) 2 5 1 - 2 6 8
The planar part of the above relation agrees with Eq. (4.18). Note that the loop amplitudes in Eq. (4.18) represent connected correlators. This agreement implies that 7-[ would correspond to B2. Taking into account the fact that/3~ (n = 0 mod m) do not exist and Eq. (4.28), it is legitimate to consider that the amplitude (for j = 1. . . . . m) (w+(g,)
...w+(gn)Bj)
(4.30)
corresponds to the connected part of the amplitude /I W(1) (gi--I) W(I) i=1
(w(a)(gl)...
(gi; [~(o'i)]j-l)
W(I)(~ei+l),,. w(l)(gn))
I "
(4.31)
5. Summary In this paper we have examined the correlators in the (m + 1,m) unitary minimal models coupled to two-dimensional gravity from the point of view of the two-matrix model. From the two-loop correlators and the wave function of the scaling operators, we derived the explicit form of the expansion of the loops in terms of the scaling operators. Using this expansion, we deduced the three-point functions from the threeloop operators, and showed that simple fusion rules exist for all of the scaling operators. The three-loop correlator [ 13] can be understood to express these fusion rules in a compact form. At the (m + 1, m) critical point in two-matrix models, the scaling operators ~j (j = 0 mod m ÷ 1 ) have no counterparts in the BRST cohomology of Liouville theory coupled to the corresponding conformal matter. In [15], these operators were argued to be boundary operators which couple to loops in the case of the one-matrix model. It was also shown explicitly that one of them, corresponding to ~'m+l in the case of the unitary matter, is an operator which measures the total length of the loops. We examined the role played by the rest of these operators. We showed, in some examples, that the operator Bn couples to the points to which n parts of the same kind of loops are stuck to each other. In other words, the operator Bn connects n parts of the same kind of loops together. We think that these operators play an important role concerning the touching of macroscopic loops. The emergence of these operators in matrix models can easily be understood from the viewpoint of macroscopic loops and their expansion in terms of local operators.
Acknowledgements We thank Hiroshi Itoyama for helpful discussions and a careful reading of the manuscript. We also thank Keiji Kikkawa for useful discussions on this subject.
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
267
Appendix A
Let us prove that the two-loop correlator Eq. (2.2) can be written as Eq. (2.7). From Eq. (2.2), we have m-1
(w+(g,)w+(&))=Z(±) k-I
{sinTr~ 2
(wk(gl)wk(&)l,
(A.1)
0 1M cgM (wk(61)Wk(62)) = - - ~ --~glg2Kl_~ ( Mgl ) Kl_~,(Mg2),
(A.2)
where we have introduced loop operators Wk(l) which represent loops with some distinct matter boundary condition. Making use of the formula
K.(z)K.(w)=~
K~,(~) exp " 2
t
S
/
(A.3)
o
and replacing t with
tM2, we have
M6162 KI_~,(M~I)KI_~, (M62) oo
1 fdt K
=2it
1-~,
(glfZ)exp( --
tM2
g~+g~'~
2
j
~
.
(A.4)
o
Carrying out the integral with respect to M, and from Eq. (A.2), we have oo
1 /dt (Wk(61)Wk(62))=-4-Tm
gI&K.
T
l
1--;
(gl~2) --
( tM2 exp - 2
g2+g2) . 2t
(A.5)
0
Due to the formula
zgl_lpl(Z)
f E E sinh ~rE KiE(Z), = J d coshTrE - cosTrp
(A.6)
0
the right-hand side of Eq. (A.5) turns into o1~
o{3
1 f dt [dE 4m
t J 0
EsinhTrE KiE ( ( @ ) e x p ( cosh~'E - cosTrp
tM2 2
62 +g2~ 2t
J .
(A.7)
0
Using Eq. (A.3) again, Eq. (A.7) turns out to be 1 2m
/de 0
E sinh ~rE Kie(Mgl)KiE(M&) cosh 1rE - cos ~._k • m
Putting Eqs. (A.8) and (A.1) together, we have
(A.8)
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
268
m-1
1
( s i n ' / r ~ "~ 2
(W+ (el) W'4-( e 2 ) ) = ~--~ ~ m ( q - ) k - I
x --/dE ,/ 0
]
E sinh 7rE Kie(Mfl)Kie(Mg2). cosh ~rE - cos 7r -k
(A.9)
m
The integral in E can be carried out by d e f o r m i n g the contour. The residues for poles E = + i ( ~ + 2 n ) , n = 0, + l , + 2 . . . . . contribute to the integral and, after all, we obtain the f o l l o w i n g e x p a n s i o n for the two-loop correlators (for el < g2):
m--I oo
(w+ ( gl )W + ( g2) ) =
m1 E
Z
(4-)~-1 k + 2 n
II~' ÷2nl( Mgl ) ff ~'+2n( Mg2)
k=l n=-oo (A.10)
References [1] P. Di Francesco, P. Ginsparg and J. Zinn-Justin, 2D Gravity and Random Matrices, hep-th/9306153. [2] M. Douglas, in Proc. 1990 Carg~se Workshop, loc. tit. [3] T. Tada, Phys. Lett. B 259 (1991) 442; S. Odake, Phys. Lett. B 269 (1991) 300; T. Tada and T. Yamaguchi, Phys. Lett. B 250 (1990) 38. [4] J.M. Daul, V.A. Kazakov and I.K. Kostov, Nucl. Phys. B 409 (1993) 311. [5] A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [6] Y. Kitazawa, Phys. Lett. B 265 (1991) 262. [7] L. Alvarez-Gaume, J. Barhon and C. Gomez, Nucl. Phys. B 368 (1992) 57. [8] B.H. Lian and G.J. Zuckerman, Phys. Lett. B 254 (1991) 417; E Bouwknegt, J. McCarthy and K. Pileh, Commun. Math. Phys. 145 (1992) 541. [9] P. DiFrancesco and D. Kutasov, Nucl. Phys. B 342 (1990) 589. [10] G. Moore, N. Seiberg and M. Staudacher, Nucl. Phys. B 362 (1991) 665. [ 11 ] G. Moore and N. Seiberg, Int. J. Mod. Phys. A 7 (1992) 2601. [12] M. Anazawa, A. Ishikawa and H. Itoyama, Phys. Rev. D 52 (1995) 6016. [ 13] M. Anazawa, A. Ishikawa and H. Itoyama, Phys. Lett. B 362 (1995) 59. [14] M. Anazawa and H. Itoyama, Nuel. Phys. B 471 (1996) 334. [15] E. Martinec, G. Moore and Yu.N. Seiberg, Phys. Lett. B 263 (1991) 190. [16] I.K. Kostov, Phys. Lett. B 266 (1991) 42, 317; Nuel. Phys. B 376 (1992) 539; I.K. Kostov and M. Staudacher, Phys. Lett. B 305 (1993) 43; S. Higuchi and I.K. Kostov, Phys. Lett. B 357 (1995) 62. [ 17] M. Ikehara, N. Ishibashi, H. Kawai, T. Mogami, R. Nakayama and N. Sasakura, Phys. Rev. B 50 (1994) 7467. [18] E. Gava and K.S Narain, Phys. Lett. B 263 (1991) 213. [19] R. Nakayama and T. Suzuki, Phys. Lett. B 354 (1995) 69.
Nuclear Physics B 501 (1997) 251-268
Boundary operators and touching of loops in 2D gravity* Masahiro Anazawa 1 Department of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560 Japan
Received 26 February 1997; accepted 9 June 1997
Abstract We investigate the correlators in unitary minimal conformal models coupled to two-dimensional gravity from the two-matrix model. We show that simple fusion roles for all of the scaling operators exist. We demonstrate the role played by the boundary operators and discuss its connection to how loops touch each other. @ 1997 Elsevier Science B.V. PACS: 04.60.Nc; 11.25.Pro; Keywords: Matrix model; Two-dimensionalgravity; Fusion rules; Boundary operators
1. Introduction The understanding o f two-dimensional quantum gravity has experienced great progress through the study o f the matrix models. 2 The one-matrix model has an infinite number of critical points which are considered to represent the (2m + 1,2) minimal conformal models coupled to two-dimensional gravity. The two-matrix model has critical points which correspond to the (m + 1, m) unitary minimal conformal models [ 2 - 4 ] . In this paper we investigate the unitary minimal model (m + 1, m) coupled to two-dimensional gravity from the two-matrix model. The emergence o f the infinite number of scaling operators trj is one of the most important properties o f the matrix models. Before being coupled to gravity, the minimal * This work is supported in part by the Grand-in-Aid for Scientific Research Fund (2690) from the Ministry of Education, Science and Culture, Japan. 1JSPS research fellow. E-mail: [email protected] 2 See, for example, Ref. [ 1] for a review. 0550-3213/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII S0550-3213 (97)00369-6
252
M. Anazawa/NuclearPhysics B 501 (1997) 251-268
model has a finite number of primary fields. After gravitational dressing, however, an infinite number of scaling operators emerge. This phenomenon can be understood as follows. In the Kac table we can divide the primary fields ~r,s into those which are inside the minimal conformal grid 1 ~< r ~< q - 1, 1 ~< s ~< p - 1 and those outside, which correspond to the null states. Before being dressed by gravity, the fields outside the minimal conformal grid decouple [5] from physical correlators. After gravitational dressing, they cease to decouple [6,7] and become an infinite number of scaling operators. A similar phenomenon has been shown to exist in the continuum framework. Through the examination of the BRST cohomology of the minimal model coupled to Liouville theory, an infinite number of physical states were shown to exist [8]. These states have their counterparts in the matrix models as the scaling operators. Some of the scaling operators do not have their counterparts in the BRST cohomology, which we will discuss later. In ordinary (p, q) minimal conformal model the primary fields satisfy certain fusion rules [5] ; the three-point function (~rl,sl ~r2,s2~r3,s3) is non-vanishing only when 1+
Ir~
- r2] ~< r3 ~< mim{rl + r2 - 1,p},
l+lsl-s2l<<,s3<~mim{sl+s2-1,q},
rl + rz + r3 = odd, sl+s2+s3=odd.
(1.1)
It is interesting to examine how the fusion rules change when matter couples to gravity. The three-point functions involving lower-dimensional scaling operators were examined from the point of view of the generalized KdV flow in [9] in the case of (m + 1, m) unitary matter. It was shown that the gravitational primaries o-j (j = 1 . . . . . m - 1 ) satisfy fusion rules of the BPZ type; for jl + j2 + j3 ~< 2m - 1, (O'jlO'j20"j3) is non-vanishing only when 1 + IJ1 -
J2l ~< j3
~< jl + j2 -- 1 .
(1.2)
The fusion rules were also examined in the continuum framework [7]. As for the gravitational descendants, however, we think clear results have not been obtained. In this paper we would like to clarify the fusion rules for all of the scaling operators including the gravitational descendants in the case of the unitary minimal model. Macroscopic loop correlators, which are the amplitudes of the surfaces with boundaries (loops) of fixed lengths, are the fundamental amplitudes of the matrix models. It was shown [ 10,11 ] that these correlators have more information than those of local operators and that the latter correlators can be extracted from the former explicitly in the case of c = 0, 1/2, 1. The authors of [ 10,11 ] argued that macroscopic loops could be replaced by a sum of local operators whose wave functions satisfy the Wheeler-DeWitt equations. We generalize this idea to the loop correlators [4,12-14] in the cases of the general unitary minimal models, and derive the fusion rules for all of the scaling operators. First we derive the explicit form of the expansion of loops in terms of the scaling operators, and then deduce the three-point correlators from the loop correlators which were calculated in [ 13,14] from the two-matrix model. We show that the three-point correlators of all of the scaling operators satisfy certain simple fusion rules and these
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
253
fusion rules are summarized in a compact form as the fusion rules for three-loop correlators [ 13 ]. In matrix models, there is an infinite subset of the scaling operators ~-j ( j = 0 mod m + 1) which does not have its counterpart in the BRST cohomology of the minimal model coupled to Liouville theory. In the case of the one-matrix model, Martinec, Moore and Seiberg [ 15] argued that these operators are boundary operators which couple to the boundaries of a two-dimensional surface. They proved that one of them is in fact a boundary operator which measures the total loop length. But little has been said about the role played by the rest of these operators. We examine the geometrical meaning of these operators and their connection to the touching of loops in the case of general unitary models. We show that the boundary operators play a role of connecting several parts of the loops together. We also discuss the relation of the boundary operators to the Schwinger-Dyson equations proposed in [ 17].
2. Expansion of loops in local operators We consider the (m + 1,m) unitary minimal model coupled to two-dimensional gravity from the two-matrix model with a symmetric potential. The partition function Z is defined by e z = f dA d/~ e - ~ tr(u(~)+u(~)-~) ,
(2.1)
where A and/~ are hermitian matrices, and U is a certain polynomial. In this article, we limit our discussion to the two-matrix model with a symmetric potential and to the critical points which correspond to the unitary minimal models. In the case of an asymmetric potential, some of the boundaries (loops) of the two-dimensional surface would have fractal dimensions different from the usual dimension of length. In Ref. [ 10], in the case of the one-matrix model, it was shown that the loop operator can be expanded in terms of local operators, that is, the loop can be replaced with an infinite combination of local operators, except for some special cases. It was argued that this is the case for the general (m + 1, m) unitary models. Whether this replacement can be done safely or not is connected with whether the corresponding classical solution exits or not in the limit of small length of the corresponding loop. This claim is quite natural because all of the scaling operators are expressed in term of one matrix A as o~i = Tr(1 - ~)j+1/2 = ~ n an(J )n-l Tr'4n in the one-matrix model. In the two-matrix model, it appears that this idea is not the case since the direct connection of the scaling operators to the operators Tr ~n or Tr/~ n' is not clear. But this expansion is considered to be possible for the following reason. When one of the loops on the two-dimensional surface shrinks to a microscopic loop, the loop represents a local deformation of the surface. The microscopic loop can be replaced by the insertions of local operators. The loop correlators except for the one-loop case are continuous when
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
254
the length of one of the loops approaches zero, so that we expect that a macroscopic loop can also be replaced by a sum of local operators. In this section we derive explicitly the expansion of loops in local operators in the case of the unitary minimal models. Using this relation, we can deduce the amplitudes of local operators and those involving both loops and local operators from the loop correlators. Let us recall that the two-loop correlators in the (m + 1, m) unitary minimal model coupled to two-dimensional gravity are [4,13] m--1
1M ~le2 Z(q_)k_l~.~,(Mel)~.lk(Me2) (W+(el)Wd:(e2)) = m 2 el + (4)me2 k=-I
(2.2)
for e I ://= e2. Here w+(g) and w - ( g ) represent loop operators which create loops made by the matrices ,4 and/~ respectively, and we introduced the notation
K,p(Me) = sinqrlp[g 7r/2 p(Mg) .
(2.3)
Here the parameter M is defined by ( M ) 2 - m~l'/'t
a - A. = - a 2 / x ,
(2.4)
where A. represents the critical value of the bare cosmological constant, a n d / z is the renormalized cosmological constant. The definition of a set of the scaling operators has arbitrariness which comes from the contact terms [ 10]. As a set of local operators, we take the scaling operators ~-j whose wave functions gtj(£) satisfy the (minisuperspace) Wheeler-DeWitt equations -
e~
+ 4~e 2 +
g,j(e) = o.
(2.5)
It was shown [ 10] that these scaling operators correspond to the dressed primary fields of the conformal field theory in the case of the one-matrix model. In terms of these scaling operators, we can obtain simple fusion rules for three-point correlators, which we will show in the next section. We normalize the wave function of ~-j as -m
Kz(Me), m
j / > 1 v~ 0 ( m o d m ) .
g:j(Me)
(2.6)
Note that the normalization factor sin(j/m)cr in in Eq. (2.6) is consistent because there are no scaling operators ~'j for j = 0 (mod m) in the matrix model. We can express the right-hand side of Eq. (2.2) as an infinite sum in terms of ffij(Mg2) for el < g2 (see Appendix A):
255
M. Anazawa/Nuclear Physics B 501 (1997) 251-268 m--I
(w+ ( gl )w+ ( g2) ) = m1 Z
k
oo
~
(±)k-, _m + 2n Ii~,+2nl(Mg,)K~,+2,(Mg2).
k=l n=--oo
(2.7) Comparing Eq. (2.6) with Eq. (2.7), we expect the following expansions of the loop operators in terms of the local operators: 1 Z
" w+(g)
Z
m k=l n=-oo
(.+.)k-1
II~,+2nl(Mg) °'lk+2mnl "
(2.8)
These expansions are the generalizations of those in the case of the one-matrix model [ 10] to the cases of general unitary minimal models coupled to two-dimensional gravity. Since the loop correlators are symmetric under the interchange of two kinds of loops, that is, (w + (gl) w+ (g2)) = (w-(gl) w-(g2)), the wave functions of the scaling operators with respect to loop w-(g) read (O'j W - - ( g ) ) =
( - 1 ) .i-' (~'j w+(g)) .
(2.9)
3. Fusion rules for scaling operators Using the expansion of loops Eq. (2.8), we can obtain the correlators of the scaling operators from loop correlators. 3 In this section we show that there are rather simple fusion rules for all of the scaling operators. The fusion rules involving the gravitational descendants (o-j, j ~> m + 2) have not been clear from the point of view of generalized KdV flow.
3.1. One- and two-point functions Let us examine one- and two-point functions first. Since the one-loop amplitude diverges when the loop length approaches zero, this amplitude includes the contribution which is not represented by the local operators. Extracting the parts proportional to 1~ (v > 0), which can be considered as the contributions from local operators, from the one-loop amplitude
=
@,(M£)-I
2 ,(Mg)-I±(Mg)+l --
--
m
m
±(Mg) --
(3.1)
m
we can obtain the one-point functions of the scaling operators 3 The multi-loop correlators were examined in [ 14] from the two-matrix model. These correlators were also examined in 116] from the viewpoint of random surfaces immersed in Dynkin diagrams.
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
256
(3-1) = - m
,
(3-1+2,,)= m
(3-j) = 0,
j~: 1,1+2m.
,
(3.2) (3.3)
Let us turn to the two-point functions. Substituting Eq. (2.8) into Eq. (2.6), we obtain the two-point functions
(3-i~i) = ~Sijj
(7)
,
i,j ~ 0 (mod m).
(3.4)
Note that we obtain diagonal two-point functions.
3.2. Three-point functions The three-loop correlator from the two-matrix model is obtained in [ 13] as 1--±
m(m + 1) ×
Z
glg2e3
K 1 - ~ (Mgl) K I - ~ (Mg2) K I - ~ (Mg3) ,
(3.5)
(k I --l,k2 --l,k 3 --1) E'D~ m)
Here we have denoted by D~m) 3
79~m) = { ( al , a2, a3 )
~-~ ai -- aj >/ O for j = l , , ~ 3 , i(÷j)
Z 3a i
=
even ~< 2(m -- 2) ,ai = 0, 1,2 . . . . . }
(3.6)
i=1
It was shown [ 13] that the selection rules in Eqs. (3.5) and (3.6) correspond to fusion rules [9,7] for the dressed primaries (~ii, i = 1 . . . . . m - 1) by studying small length behavior of the three-loop correlator. Using the loop expansion (2.8), can show that the selection rules in Eq. (3.5) represent the fusion rules for all of scaling operators in a compact form. Let us show this in the following. Using the formula o~
zKI-Ipl (z) = ~" Z
[p + 2n[ sinTr[p + 2n[ IIP+2nl(Z)
(3.7)
n=--oo
we first expand the three-loop correlator Eq. (3.5) as -2--±
(w+(el)w+(e2)w+(e3))
-
r e ( m + 1)
the the we the
oo
~
E
oo
Z
oo
"D~m) ttl=--OO n2 = - O O ;'/3--OO
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
× I[~+2n~l (Mgl)I1~+2n21 (MgE)II~+2ml (Mg3) •
257
(3.8)
Comparing Eq. (3.8) with Eq. (2.8), we can extract the three-point functions 3
I ( ki + 2mni) (~rlk~+2mn'[O'lk2+2mnzlO'lk3+2mnal) =Cklk2k3 m ( m Sr 1) Ii=1
x ( M ) -2-"~+~]°' ~,lk,+2m,,I ,
(3.9)
where Ck~kzk3 =
1, 0,
(kl -- 1 , k 2 - 1 , k 3 - 1) 6 D~ "~ otherwise
(3.10)
For ni = 0, Eq. (3.9) is nothing but the correlator of the gravitational primaries. For the gravitational primaries, Eq. (3.4) and Eq. (3.9) agree with the correlators obtained in [9] from the generalized KdV flow up to a factor - 2 . Note that we here obtain the correlators of the gravitational descendants as well. In [7], the fusion rules for the gravitational primaries were examined in the continuum framework. We have found here the fusion rules for the gravitational descendants as well as those for the gravitational primaries. These fusion rules are similar to those for the gravitational primaries due to the factor Ck, k2t3 in Eq. (3.9). Introducing the equivalence classes [o'k] by the equivalence relation O'k "~ O'lk+2mnl ,
n E Z,
(3.11 )
we can consider the fusion rules in Eq. (3.9) as the fusion rules among [ok] (k = 1 . . . . . m - 1 ). Note here that the class [~'k] does not correspond to the set which consists of the gravitational primary Ok and its gravitational descendants orI ( O k ) , l = 1,2 . . . . in [9] introduced from the viewpoint of KdV flow. 3.3. Further on the fusion rules
In this subsection, let us examine the fusion rules in Eq. (3.9) further and summarize the relation of the scaling operators to the primary fields in the corresponding conformal field theory. In the (p, q) minimal conformal model, the primary field q0rs has the conformal dimension (pr -- qs) 2 -- (p -- q)2 dr, s = , (3.12) 4pq where r and s are positive integers. Since we have Ar,s = Ar+q,s+p = A q - r , p - s , the corresponding primary fields can be regarded to be the same. The integers r and s can thus be restricted to the range
258
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
r
0
p
2p
3p
s
Fig. 1. The range of (r, s). 1 ~r~q--1,
1 ~s
~ 0(modp),
pr
(3.13)
(see Fig. 1). In Fig. 1, the primary fields in the region ((2)) or ( (2))' are the secondary fields of those in the region ((1)). In general, the fields in the region ((n + 1)) or ( ( n + 1) ) t are the secondaries of the fields in ((n)) or ((n)) t. Since the secondary fields correspond to null vectors, those fields decouple. One can thus construct a consistent conformal field theory which includes only the primary fields in the region ((1) ) (i.e. inside the minimal table), that is, the (p, q) minimal model [5]. Coupled to Liouville theory, however, the fields outside the minimal table fail to decouple [6] and an infinite number of physical states emerge accordingly. These states are considered to correspond to the primaries outside the minimal table. This correspondence is implied by the BRST cohomology [ 8] of the coupled system. In the minimal model coupled to Liouville theory, the gravitational dimension zlI,S ~ of the dressed operator for ~r,s is given by the relations d ras = 1 -
Otr,___Ls y '
Otr, s = p + q - [p r - q s [ y 2q '
(3.14)
where r and s take values in the range Eq. (3.13). On the other hand, in the matrix model, the corresponding relation for the scaling operator ~'j is given by O(j y
-
-
p + q- j 2q
(3.15)
From Eq. (3.14) and Eq. (3.15), we should take j=[pr-qs[,
j=l,2 ....
4=0(modq),
(3.16)
for ~'j. Consider now the relation of ~'lk+2mnl tO the primary field ~r,s of the unitary ( m + 1, m) minimal model. Let us first compare the two sets Sk = { I k + 2 n m l l n E Z } , and
(3.17)
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
r
03 02 O1
•
: (711+2toni
0
: OI2+2mnp
259
® : ~13+~,~,,I
p.
0
5
10
15
s
Fig. 2. The scaling operators ~lk+2,,~l in the (5,4) minimal model coupled to 2d gravity.
{[pr-qs[}={[(m+l)r-ms[}={r'+(s-r-1)m},
(3.18)
where r and s are positive integers in the range l~
l <<.s,
r + l <<.s
(3.19)
and r ~ - m - r. Note that we include s = 0 (mod m ÷ 1) here. Decomposing the set Sk into two sets as & =
•
(3.20)
where
S+ = { k + 2nm l n =O, 1,2 . . . . } , {(m-
k) + (2n' + 1)m I n ' = 0 , 1 , 2 . . . . } ,
(3.21)
and comparing Eq. (3.21) and Eq. (3.18), we can express the sets S~- and S~- in terms of [ ( m + 1 ) r - m s [ as S~-={[(m+l)r-msl]r'=k,
s-r=2n+l,n=O,
1. . . . } ,
S k = { l ( m + l ) r - ms[ l r = k, s - r = 2n' + 2, n' = O, 1 . . . . } .
(3.22)
From Eq. (3.22), the following correspondence is obtained:
~rlk+2mn[ (n >/O) ~ ~m-k, r+2n+l (n /> 0) O'[k+2m(-1-n')[ ( n! ~ O) ~-+(1)k, r+2n'+2 ( n! ~ 0) ,
(3.23)
where s ~ 0 (mod m + 1). As for the scaling operators corresponding to s = 0 (mod m + 1 ), we will discuss these in the next section. As an example, we depicted the scaling operators on the r - s plane for the case of m = 4 in Fig. 2. In this figure we showed the equivalence classes [o'k] explicitly.
260
M. Anazawa/NuclearPhysics B 501 (1997) 251-268
(a)
(b)
Fig. 3. (a) A surface with two loops touching each other in a point. (b) When one of the loops shrinks to a microscopic loop, the microscopic loop is equivalent to the insertion of the operator denoted by the dot on the loop.
(a) (b) Fig. 4. The case of a surface with two loops touchingeach other in two points. 4. B o u n d a r y operators 4.1. Boundary operators and touching of loops
The scaling operators ~-j ( j = 0 mod m + 1 ), which correspond to s = 0 (mod m + 1 ) on the r-s plane, do not have their counterparts in the BRST cohomology of the minimal model coupled to Liouville theory. In [ 15] it was proposed that the scaling operators which do not occur in the BRST cohomology of Liouville theory are boundary operators and one of them, o'3 = o-I (O1) in the case of pure gravity, was in fact proven to be a boundary operator for the one-matrix model and the Ising model case. We would like to examine the role played by the operators O'n(m+l)' n = 1,2 . . . . ~ 0 (mod m) as well as ~'m+l for general unitary minimal models. Let us denote these operators by /~n =o'n(m+l),
n = 1,2 . . . . :~ 0 ( m o d m ) .
(4.1)
In the matrix models the loop amplitudes contain the contribution from the surfaces with loops touching each other. In two-loop case, let us consider the surfaces in which the two loops touch each other in n points. When we shrink one of the loops to a microscopic loop, the other loop splits into n loops, which are stuck to each other through the microscopic loop (see Figs. 3-5). Since the microscopic loop represents a sum of the scaling operators, the wave functions of some scaling operators contain the contribution from the surfaces with the split loop. We now show that the boundary operators indeed represent these surfaces. From Eqs. (2.6) and (3.1), the wave function of B, and the one-loop amplitude are
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
(a)
261
(b)
Fig. 5. The case of a surface with two loops touching each other in three points.
( M'~ n(l+~)
(s~nw+(g))=n(l+ 1) \-~] (w+(g))=
(1) 1+ g-'
#2n(l@,)(Mg),
(4.2)
~JI+~(Mg).
(4.3)
We denote the Laplace transformation of any function f ( g l , f2 . . . . ) of loop lengths by oo £[f(gl,~2
oo
.... )] =id~l/d~.2...e-(gl'21+h(2+)f(~-l,g2 0
.... ).
(4.4)
0
In the space of the Laplace transformed coordinate sr, we have f[
Lg-l(~..w+(g))j] = -
2coshn(m+l)0, 1
£ [(w+(g))] = -
(4.5)
1
(7) +m2cosh(m + 1)0,
(4.6)
where we have used the relation £ [-g-'lulR'~(Mg)] = 2coshmz,0,
(4.7)
and ( is parametrized as ( = McoshmO. Note here that w+(f) represents a loop with a marked point and g-lw+(g) represents a loop without a marked point. Since cosh n(m + 1)0 can be expressed as a polynomial of cosh(m + 1)0, 2coshn(m +
1)O=2Tn(cosh(m+ 1)O) In~2] = Z C'r-(n)[2cosh(m +
1)O] n-2r
r--O
@n)--(--l)rn( )------n--r n-r r ' where
7". is the Chebyshev polynomial,
£[-g-l(s~,w+(g))]=
~_~ r---O
(4.8)
we obtain the following relation:
{~--~[--
c~") \ 2]
(4.9)
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
262
In the space of the loop length, the above relation means that the wave function of/3n is equivalent to a sum of the convolutions of disk amplitudes:
[ (n--l)/2]
(~n w+ (~) ) = --e
Z
( M ) 2r(,+.~ )
C'r-(n)
;-l)n-2r[(~)J4~] n-2r (£).
(4.10)
r=0 Here we introduced the notation A + = (w+(e)), and [®.A+]S (g) denotes the convolution of s .A~-(g)'s, for example, [QA+] 2 ( g ) = f f dgldg2 6(gl + g2- g)A+(gl)A~(£2).
(4.11)
0 0 From Eq. (4.10) we can conclude that the operator Bn couples to the point to which s (~< n) parts of the loop are stuck to each other in the case of one-loop amplitudes. When there is more than one loop, we infer that the operator couples to the point to which s parts of several loops are stuck to each other; the operator will not recognize that it is touching different loops this time. Using the following relation:
[2coshx
[(n--I)/2] =
z (n) 2 c o s h ( n - 2r)x,
(up to a constant),
(4.12)
r=O we also obtain
, [Q).A~-]n ( ' ) = (--1)n+'
[(n--I)/2] Z (;) r=0
( M ) 2r(l+~5 )
(~n_2rW+(')).
(4.13)
Here we drop the constant term in Eq. (4.12) when carrying out the inverse Laplace transformation. From Eq. (4.13) we see that the boundary operator coupled to the point on which n parts of loops are touching each other is given by
[(n-l)/2] /3n = ( - 1 ) n + '
Z
[M,~ 2r(l+~,) ^
Bn-2r.
(~)~-2")
(4.14)
r=0 Now let us consider the boundary operators when there are two loops on a twodimensional surface. As for ~1, we expect that (w+(el)w+(e2)131) should be proportional to (el + g2) (w+(gl)w+(g2)). Let us confirm this in the following. From the three-loop correlator (3.5), the expansion of the loop operator (2.8) and the wave function of °'lk+2mnl (2.6), we obtain the following correlator with two loops and a local operator:
(w+ ( e')w+ ( e2) k3+2m"3 ) =
m+ 1 Z
kl .k2
Ck'kzk3
×'1'2 (~- -+- 2n3") ~"1- hm (M'I)K" 1-- ~m (M'2) " (4.15) \m /
M. Anazawa/NuclearPhysicsB 501 (1997)251-268 Consider the amplitude for B1 =/31 = O'm+l = o'[ ~ _ 2 [ only for the case of kx + k2 = m, we have m-
263
Since Ct,,k2,,,_ 1 is non-vanishing
1
(w+(,°.I)W±(,°.2)/3I)=LS_.t(4-)k-I(M)p~I,P,2K~(Mgl)KI_,~_(Mg2). m k "
(4.16'
Comparing Eqs. (4.16) to (2.2) we obtain the desired relation,
(w+(et)w±(e2)&)= {el + (-1)me2} (w+(g~)w±(e2)).
(4.17)
Note here that we have (Blw+(g)) = ( - 1 ) m (BIW-(g)). Next, let us consider/32. Since we infer that the insertion of/32 should play the role of connecting two parts of loops, we expect the relation
(w+(g,)w+(g2)B2)=2g,f < (w+(e',)w+(e2))(w+(e,-e',)) o g2 +2h/de'2 (w+ o
) (w+(e2 - e'2) )
--k2ele2 (w+(el --I-e2))
(4.18)
to hold. The third term in the right-hand side of Eq. (4.18) represents the contribution from the surfaces with loops w+(gl ) and w+(g2) touching each other in a point. Let us confirm the relation (4.18) in the following. In this case, it is convenient to consider it in the space of Laplace transformed coordinates (i- In this space Eq. (4.15) reads
(17V+(f')f'V±((Z)O'[k3+2mn31)= -1 Osrl ~ 2
(M)-}-2+l~+Zn31(k--3m- - + 2 n 3 sinh toO1
kl ,k2
s--:-mnhm-0-22
'
where ff'+((i) - £[w±(gi)]. Due to the relation ~--~C
kl ,k2
rq._xk2sinh(m -
k, k2k3t )
= 2(coshOl
kl )01 sinh(m - k2)02 sinhm02
s-~nhm~
-lqz cosh 02) _(sinh(m-k3)Olsinh nO1 k3s i n h ( m - k3)02) \ -- ( + ) " s--~nhm~ ,
(4.20)
we have
' 2+1.0.+2n31 (if'+ ((~) ff'±(~'2)~lk3+2,,<) = 2(m + 1)
-f~.-~Ta 0 { xvs~v~z
1
- - + 2n3
(sinh(m--k3)Ol --(4-)k sinh(m-k3)02)}3
coshOlqzcoshO2 \
sinhmO~
sinhm02 (4.21)
264
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
Since we should take k3 = 2 for/32 = -o'2(m+l) (for m ~> 3), we obtain the amplitude for/32,
1
(~/+(~1) lf~/4_((2)/32)= ::~1
" ____~ 0q
m
0(1 0(2
{
1
cosh 01 q: cosh 02
(sinh(m -- 2)01 sinh(m--2)02 ~ } × \ sinhmO1 sinhm02 j .
(4.22)
On the other hand, from the amplitudes [4] 0
0
coshOl - cosh02
(I~t/+((1) l~/+((2)) -- 0(1 0(2 In cosh mO1 - cosh m02 a {
1
sinh O,
= 0(2 coshO1 - cosh02 mMsinhmOl
1 } (1 - (2 '
(4.23)
we obtain the relation _2 0
e(1 { (~/+ ((1)Vi/+ ((2)) ( ~ + ( ( 1 ) ) } "~ (1 +--+2)
--2 0-' } 0---O--q-0(2O { (~+((1)) ( ,(1 - (2 (I~7+((2)) 1
2 (M)~' 0 O { 1 m 0(l 0(2 cosh 01 cosh 02
(sinhO, cosh(m+l)O1 sinh mO1
(1,-+2))}. (4.24)
One can easily show that the right-hand side of Eq. (4.24) agrees with that of Eq. (4.22). Putting Eqs. (4.24) and (4.22) together and performing the inverse Laplace transformation, we obtain the desired relation (4.18). As for (W+(gl)W--(g2)/32), from the amplitude [4] 0 O (I~'+((1) I~'-((2) ) - 0(1 0(2 ln(cosh 01 + cosh02) ,
(4.25)
we obtain the relation
gl (w+(e,)w-(e2)/32)
-2g, / de',(w+ <
)w- (
) (w+ ( e, -
))
+2e2/de;(w+ ( el )w- (
) (w- ( h -
)
o g2
(4.26)
o
in a similar way. In this case, the operator/32 does not connect the different kinds of loops w+(gl) and w-(g2) together. We have shown that the operator/32 connects two parts of the same kind of loops together in the case of two loops. We infer that similar phenomena occur in general; the
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
265
operator/3n connects n parts of the same kind of loops together in the case with any number of loops.
4.2. Connection to the Schwinger-Dyson equations A close relationship between the boundary operators and the Schwinger-Dyson equations proposed in [ 17] can be observed. The continuum limit of the Schwinger-Dyson equations for loops in the two- and multi-matrix models were proposed in [ 17] under some assumptions. It was shown [ 17] that these equations for the two-matrix model contain W3 constraints, which were derived explicitly in [ 18]. The integrability of these equations was shown in [ 19]. These facts justify the proposed Schwinger-Dyson equations. Let us consider the connection of the boundary operators with the Schwinger-Dyson equations. For the (m-t-l, m) minimal models, the following Schwinger-Dyson equations were proposed in [ 17]:
g / d ~ . ' (w(1)(et)w (1) (~
et; {']"/(o-) ] J) w(l)(~l)...w(1)(.Pn))t
0 e i (W <1) (e -~-ei; [-7.-/(0-)]j) W(I)(el)... W(1) (ei--1)W (1>(ei+l)... W(1) (en)~! i -[-(W (1) (e; [~'/(O-)]J+I)W(I)(e,)...w(l)(en)) ! ,~0, f o r j = 0 . . . . . m - 2 , +g ~
(4.27)
(W (1) (e; [']"/(O-)] m-l) W(1)(el ) . . . W(1)(en)) t ,-~ O.
(4.28)
Here (...}' represents loop correlators that are not necessarily connected, and w (1) (g) represents a loop operator corresponding to a loop created by the matrix ~(1) in the multi-matrix model. The operator 7-/(o-) describes an operator which changes the 'spin' on a loop locally from 1 to 2. Also w(l)(g; [7-/(o-)]J) describes a loop with [7-/(0-)] ,i inserted. The symbol ~ means that as a function of g, the quantity has its support at g=0. From Eq. (4.27) for j = 0 and n = 1, we have the relation
g, (w (t) (g,; ~(o-)) w(~)(g2))' + g2 (w(1)(gl)w (') (g2;~(o-)))' gl +gl f dg'l (w(D(g/l)W(')(gl - g'l)W(l)(g2)) ' o g2
+g2 f dg~ (w(1)( gl)W(1)(g~z)W(')(g2 -- g~) )' o
+2gglg2 (w(l)(gl + gz) )' "~ O.
(4.29)
266
M. A n a z a w a / N u c l e a r Physics B 501 (1997) 2 5 1 - 2 6 8
The planar part of the above relation agrees with Eq. (4.18). Note that the loop amplitudes in Eq. (4.18) represent connected correlators. This agreement implies that 7-[ would correspond to B2. Taking into account the fact that/3~ (n = 0 mod m) do not exist and Eq. (4.28), it is legitimate to consider that the amplitude (for j = 1. . . . . m) (w+(g,)
...w+(gn)Bj)
(4.30)
corresponds to the connected part of the amplitude /I W(1) (gi--I) W(I) i=1
(w(a)(gl)...
(gi; [~(o'i)]j-l)
W(I)(~ei+l),,. w(l)(gn))
I "
(4.31)
5. Summary In this paper we have examined the correlators in the (m + 1,m) unitary minimal models coupled to two-dimensional gravity from the point of view of the two-matrix model. From the two-loop correlators and the wave function of the scaling operators, we derived the explicit form of the expansion of the loops in terms of the scaling operators. Using this expansion, we deduced the three-point functions from the threeloop operators, and showed that simple fusion rules exist for all of the scaling operators. The three-loop correlator [ 13] can be understood to express these fusion rules in a compact form. At the (m + 1, m) critical point in two-matrix models, the scaling operators ~j (j = 0 mod m ÷ 1 ) have no counterparts in the BRST cohomology of Liouville theory coupled to the corresponding conformal matter. In [15], these operators were argued to be boundary operators which couple to loops in the case of the one-matrix model. It was also shown explicitly that one of them, corresponding to ~'m+l in the case of the unitary matter, is an operator which measures the total length of the loops. We examined the role played by the rest of these operators. We showed, in some examples, that the operator Bn couples to the points to which n parts of the same kind of loops are stuck to each other. In other words, the operator Bn connects n parts of the same kind of loops together. We think that these operators play an important role concerning the touching of macroscopic loops. The emergence of these operators in matrix models can easily be understood from the viewpoint of macroscopic loops and their expansion in terms of local operators.
Acknowledgements We thank Hiroshi Itoyama for helpful discussions and a careful reading of the manuscript. We also thank Keiji Kikkawa for useful discussions on this subject.
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
267
Appendix A
Let us prove that the two-loop correlator Eq. (2.2) can be written as Eq. (2.7). From Eq. (2.2), we have m-1
(w+(g,)w+(&))=Z(±) k-I
{sinTr~ 2
(wk(gl)wk(&)l,
(A.1)
0 1M cgM (wk(61)Wk(62)) = - - ~ --~glg2Kl_~ ( Mgl ) Kl_~,(Mg2),
(A.2)
where we have introduced loop operators Wk(l) which represent loops with some distinct matter boundary condition. Making use of the formula
K.(z)K.(w)=~
K~,(~) exp " 2
t
S
/
(A.3)
o
and replacing t with
tM2, we have
M6162 KI_~,(M~I)KI_~, (M62) oo
1 fdt K
=2it
1-~,
(glfZ)exp( --
tM2
g~+g~'~
2
j
~
.
(A.4)
o
Carrying out the integral with respect to M, and from Eq. (A.2), we have oo
1 /dt (Wk(61)Wk(62))=-4-Tm
gI&K.
T
l
1--;
(gl~2) --
( tM2 exp - 2
g2+g2) . 2t
(A.5)
0
Due to the formula
zgl_lpl(Z)
f E E sinh ~rE KiE(Z), = J d coshTrE - cosTrp
(A.6)
0
the right-hand side of Eq. (A.5) turns into o1~
o{3
1 f dt [dE 4m
t J 0
EsinhTrE KiE ( ( @ ) e x p ( cosh~'E - cosTrp
tM2 2
62 +g2~ 2t
J .
(A.7)
0
Using Eq. (A.3) again, Eq. (A.7) turns out to be 1 2m
/de 0
E sinh ~rE Kie(Mgl)KiE(M&) cosh 1rE - cos ~._k • m
Putting Eqs. (A.8) and (A.1) together, we have
(A.8)
M. Anazawa/Nuclear Physics B 501 (1997) 251-268
268
m-1
1
( s i n ' / r ~ "~ 2
(W+ (el) W'4-( e 2 ) ) = ~--~ ~ m ( q - ) k - I
x --/dE ,/ 0
]
E sinh 7rE Kie(Mfl)Kie(Mg2). cosh ~rE - cos 7r -k
(A.9)
m
The integral in E can be carried out by d e f o r m i n g the contour. The residues for poles E = + i ( ~ + 2 n ) , n = 0, + l , + 2 . . . . . contribute to the integral and, after all, we obtain the f o l l o w i n g e x p a n s i o n for the two-loop correlators (for el < g2):
m--I oo
(w+ ( gl )W + ( g2) ) =
m1 E
Z
(4-)~-1 k + 2 n
II~' ÷2nl( Mgl ) ff ~'+2n( Mg2)
k=l n=-oo (A.10)
References [1] P. Di Francesco, P. Ginsparg and J. Zinn-Justin, 2D Gravity and Random Matrices, hep-th/9306153. [2] M. Douglas, in Proc. 1990 Carg~se Workshop, loc. tit. [3] T. Tada, Phys. Lett. B 259 (1991) 442; S. Odake, Phys. Lett. B 269 (1991) 300; T. Tada and T. Yamaguchi, Phys. Lett. B 250 (1990) 38. [4] J.M. Daul, V.A. Kazakov and I.K. Kostov, Nucl. Phys. B 409 (1993) 311. [5] A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [6] Y. Kitazawa, Phys. Lett. B 265 (1991) 262. [7] L. Alvarez-Gaume, J. Barhon and C. Gomez, Nucl. Phys. B 368 (1992) 57. [8] B.H. Lian and G.J. Zuckerman, Phys. Lett. B 254 (1991) 417; E Bouwknegt, J. McCarthy and K. Pileh, Commun. Math. Phys. 145 (1992) 541. [9] P. DiFrancesco and D. Kutasov, Nucl. Phys. B 342 (1990) 589. [10] G. Moore, N. Seiberg and M. Staudacher, Nucl. Phys. B 362 (1991) 665. [ 11 ] G. Moore and N. Seiberg, Int. J. Mod. Phys. A 7 (1992) 2601. [12] M. Anazawa, A. Ishikawa and H. Itoyama, Phys. Rev. D 52 (1995) 6016. [ 13] M. Anazawa, A. Ishikawa and H. Itoyama, Phys. Lett. B 362 (1995) 59. [14] M. Anazawa and H. Itoyama, Nuel. Phys. B 471 (1996) 334. [15] E. Martinec, G. Moore and Yu.N. Seiberg, Phys. Lett. B 263 (1991) 190. [16] I.K. Kostov, Phys. Lett. B 266 (1991) 42, 317; Nuel. Phys. B 376 (1992) 539; I.K. Kostov and M. Staudacher, Phys. Lett. B 305 (1993) 43; S. Higuchi and I.K. Kostov, Phys. Lett. B 357 (1995) 62. [ 17] M. Ikehara, N. Ishibashi, H. Kawai, T. Mogami, R. Nakayama and N. Sasakura, Phys. Rev. B 50 (1994) 7467. [18] E. Gava and K.S Narain, Phys. Lett. B 263 (1991) 213. [19] R. Nakayama and T. Suzuki, Phys. Lett. B 354 (1995) 69.