P and Q operator analysis for the two-matrix model
Tsukasa Tada
Institute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan
and
Masahiro Yamaguchi 1
Department of Physics, Tohoku University, Sendai 980, Japan Received 9 July 1990
The two-matrix model with the sixth order potential is investigated. We construct Douglas' Q and P operators from the matrix model by directly taking the continuum limit of the corresponding matrix operators, and show that the (4, 5 ) and (3, 8 ) minimal models are realized at the critical points.
Recent development of the theory of random surfaces [ 1-3 ] has clarified the remarkable relation between two-dimensional quantum gravity and the theory of the KdV hierarchy [4,5] and the KP hierarchy [6]. One- matrix models were solved directly and it was shown that the string equations are related to the KdV equations.
However in the case of the multi-matrix chain models, the systems become too involved to solve them and find the critical points directly. Actually only the two- and three-matrix models with quartic potentials were closely investigated to obtain the string equations [ 7-12 ]. A more general framework for the multi-matrix chain models has already been presented by Douglas [ 13 ]. He proposed that a critical point of a matrix chain model describes a (q, p) minimal conformal field theory [whose central charge is 1 - 6 ( p - q ) 2 / p q ] , coupled to gravity and that its string equation is simply given by
[Q~q, Q] = 1. (1)
However the direct correspondence between the matrix models and the approach started by Douglas is still obscure.
In this letter, we shall consider the two-matrix model with the sixth order potential. We shall closely investi- gate the Q and P operators by explicitly constructing them and illustrate how eq. ( 1 ) is realized in the continuum limit. It will be shown that this model has critical points which describe the (4, 5 ) and (3, 8 ) minimal models and the commutation relation [P, Q] = 1 coincides with the one given by Douglas.
Let us first explain our model. We consider the following partition function:
Z N = ~ d~V~ dN~2 e x p ( - - f l T r [ V(01 ) + V(~2) - c ~ t ~ 2 ] } , V(0) = ½~2+ ~g~4.q..~h06 ' (2) where ~,z are N × N hermitian matrices and the integration is done with the SU (N) Haar measure. As is well known this integral over matrices can be diagonalized and it reduces to an integral w.r.t, the eigenvalues of the matrices [ 14 ].
JSPS fellow.
Following the standard method, we introduce a set oforthogonal polynomials
Pi(x)
[ 14], such thatf dx dy exp [ -
w(x, y) ]ei(x)Pj(y) = h,~,,j, (3)
where
w(x, y)
= f l [V(x) + V(y) -cxy]. P,(x)
is a polynomial of degree i with the coefficient o f x ~ equal to 1 and admits the following expansion:xPi(x) =Pi+l (x) + RiPi_, (x) .-t.-siei_3(x )
"l-Tiei_5(x).
(4)The number of terms that emerge in the above expansion is determined by the order of the potential and in the present case it is four. This contrasts with the case of the one-matrix models, where the similar recursion relation always contains only two terms. As a result the analysis of two-matrix models is much more difficult than that of the one-matrix models.
In terms ofh~, we can write down the partition function
ZN
asN - - I N
ZN=N]. I-I h,=N~.h~ l--[ f ~-',
(5)i = 0 i = 1
where f - h ~ / h i _
1.
Then the problem is how to o b t a i n f . A standard method is to studyw ~ , ) = - c f, + ~
hi_~ a dxdyexp[-w(x'Y)]P~-~(Y)(x+gx3+hxS)P~(x)'
(6) from which we can d e t e r m i n e r implicitly as follows:i - W ( f ) . ( 7 )
To make the connection with Douglas' proposal, we shall develop a different method.
First we define two matrices/sand (~ using a normalized set of polynomials
P~=Pi/x/~
as follows:t "
Qo - J dx dy exp[ -
w(x, y) lPi(y)x~(x),
Then the commutation relation [ 15 ][P, QI = 1
(8,9)
(10) leads to the lattice version of the string equation. This ought to reduce to ( 1 ) in the continuum limit. The importance of the above commutation relation was emphasized by Douglas [ 13 ].
Note that one can write/5 as
/sij = - fl ~ dx dy exp [ -
w(x, y) lP,(y) ( c y - x - g x 3- hx 5)pAx)
= - Pc(~ji + polynomial of Q. ( 11 )
Thus we find that the essential part of the/5 operator is - f l c Q
t,
because we are not interested in the polynomial of the Q part, which is irrelevant to the string equation [P, Q] = 1. Hence we can easily obtain the/5 operator from the Q operator with the conjugation operator t and adding its polynomial if necessary.One might think that the m o m e n t u m operator P which is constructed as above has no information on the form of the potential (or the coupling constants) and thus the above construction is meaningless. However this is not the case. Notice that/5o equals zero for
i>j
because ( d / d x ) / ~ ( x ) is a polynomial of degree j - 1 and can be expanded in terms of/~_ ~ and lower order polynomials. Calculating/5o for i > j explicitly from (9) and setting them to zero, we get the following nontrivial relations among the ingredients of (~ and the coupling constants:¢Ti=hfifi_lfi_2fi_3fi_4,
( 1 2 )cSi = [ g + h(Ri_3 + Ri-z + R,_ i + R~ + Rz+ 1 ) ] f f - l f - z , (13)
c r y = { 1 +g(R~_t + Ri + Ri+,) +h[Si_3 + Si_2 + Si_l '[-Si~-Si+l
+Rg_ 1 (R~_: +R,_ 1 +R~ +R~+ 1) +R,(R~_ 1 +R~ +R,+ 1 ) +R~+ l(Ri +Ri+I ) +R~+=R~+ 1 ] } f . (14) ( 12 ) - ( 14 ) constitute the usual recursion equations [ 16 ].
Let us consider the continuum limit of the model. Let x = i/fl, e = 1 / N and d e f i n e f ( x ) , A (x), etc. a s f ( x ) = f , A (x) =A, and so on. Writing O explicitly using (4),
O_.=AA,.j+, +~j6,.j_,
+ CA,,j_~ +DA,,j_~,Rj Sj , Dj= ~
Aj=N/ff-j+l, Wj= %//~, Cj-
4 £ £ _ 1 £ _ 2 4 £ £ _ 1 £ _ 2 £ _ 3 £ _ 4 , one can expand Q as e-,0 into(15)
satisfy W' (f~) = W" (f¢) = W " (f¢) = 0 :
c=+2~, g=-T-2/llfc, h = l / l l O f ~ , (20)
c = + 9 , g = ~ 6 / 3 7 f ¢ , h = 3 / 3 7 0 f 2, (21)
c = + l , g = h = 0 . (22)
In other words we have obtained six candidates for the critical points that have multi-criticality higher than three. For the last critical point (22) W ( f ) is always zero,hence this point is trivial. In the cases of negative c for (20) and (21 ), it is seen that f¢ is negative, hence the partition function is ill-defined. Thus we can disregard such cases. On the other hand, the cases with positive c are physically meaningful. One can show that for the case (20) W ~4) (f¢) S 0 and for (21) W ~4) (f¢) =0, W tS) (f~) S0. This means the multi-criticality is four and five, respectively. In both cases X and Y are zero, hence the order of Q is higher than three. With a detailed
Q = A (x) exp(e0x ) + B ( x ) exp( - COx ) + C ( x ) exp( - 3e0x ) + D ( x ) exp( - 5E0:, )
= A + B + C+ D+ ( A - B - 3 C - 5 D ) e O x + (A+ B + 9C+ 25D)eZO2x/2! + .... (16) We shall first find the critical points where the specific heat becomes nonanalytic, f ( x ) -f¢ ~ #l/m. Here # is the cosmological constant and m indicates the degree of multi-criticality. This requirement is equivalent to the following [ 1-3 ]:
W ( f ( x ) ) - 1 ~ [f(x)
--fc] m,
( 17 )where W ( f ) is now evaluated in a planar limit.
A convenient parameterization to study the critical points is Re Sc Tc Rc S¢ + 2 5 ~ = 6 X - Y , Tc
1 - ~-c - 3 f-{2 - 5~c3 --- - Y, 1 + ~-~-¢ +9f-{2 ~-~s ---Z. (18)
The new parameters are chosen so that Q starts from the third order of e when we set X = Y= 0. Now we can calculate the derivatives of W with respect to f f r o m ( 7 ) in association with ( 12 ) - ( 14 ). For example, the first derivative is
W' ( f ) = c ( 1 5 Z X - 5 Z Y - 3 X ) ( 2 0 Z - 6X+ Y + 4 ) Y
50Z 2 - 15ZX+ 5 Z Y + 3 X - 2 ( 19 )
We can survey all the branches of zeros of W'. Then we find that there are only six sets of parameters which
analysis on Q we shall show later that in the former case the (4, 5 ) model is realized, while the latter corresponds to the (3, 8 ) model. Notice that for both cases the potential is bounded below as well as the one-matrix model with the potential of degree six [ 5 ].
Next we are going to investigate the double scaling limit. It is convenient to introduce a new scaling variable and replace x by z as
~2m+l=E, 1--~2mz=x. (23,24)
Then we have
0 0
EUx =-~gz --ad" (25)
In this limit we can writefn, Rn and so on, as
f~-~-d~o f(Z), fn+,-~-d~o f(Z-~)~f(x+~).
(26)We can eliminate the variables Sn and T, directly from (12) and ( 13 ). Now the 0 operator takes the following form:
0,,J--~o Q(z) = A ( z ) e x p ( - ~ d )
+B(z)
exp(~d) +C(z)
exp(3~d)+D(z)
exp(5~d), (27)where
A(z)
= ~ and so on.Then we e x p a n d f ( z ) and R (z) into
f(z)
=fo( 1 + a2f2(z) +a3f3(z) +...),R(z) =Ro(
1 +a2r2(z) +a3r3(z) +...). (28) After these expansions we obtain two sets of string equations from (7). They coincide with the equations ob- tained from three-matrix models as the (4, 5 ) and (3, 8) models respectively [ 10,17] *~In the rest of this letter we concentrate on the study of the P and Q operators. First consider the case (20).
Using the constraints ( 12 ) - (14), we find that the operator 0 starts from the fourth order of ~ as expected:
(~4 term of Q)
=2d4+4f2d2+4f'2d+2f'~
+2f22 +6r4 - 2 f 4 , (29)(~5 term of Q)
=~dS+SfEd3+2(Sf'z
+ 2f3)d2+ 2 ( 2 f ~ + 4 f g -3f22 +6r4 - 2f4)d+ 2 f ~ + 2 f ~ 3) + 4 f ~f2 + 4f3 f2 + (3r4 - f 4 ) ' + 6r5 - 2 f s . (30) Identifying
u=-f2, v=3r4-f4,
(31)and introducing K = d E - U, we find
(~4 term of Q) = 2 ( d 4 - {u,
d2}+uE+v) =2(KE+v),
(32)(~5 term o f t 2 ) = ~ ( d S - ~ { u ,
da}+~{3uE+u ", d}
+~{v,d}-~v'-~6{u',
d E}+~{f3, d 2}
+~uu'
+-~ (3r5 - f s ) +-~uf3 ). (33)In the ~ 0 limit the ~4 term of Q dominates and this is nothing but the operator Q. Deriving P is more subtle.
Recall that P i s almost (2'. The operation t takes the form d*= - d for any differential operator in this limit. Let us call the change of the sign of a differential operator under the t conjugation the Z2 parity. The fourth order term of (2 has a definite Z2 parity, i.e., even. This implies that P can be a Z2 odd operator. In fact, as Q * - Q
#~ The equation which corresponds to the (3, 8) model was first derived by Eguchi and Kato in the study of the two-matrix model with sixth order potential [ 18 ].
starts from the fifth order o f ~ this is what we want for P. We thus find that the P and Q operators o f the case (20) should be
Q=K2+v=d4-{u, d2} +u2+v,
(34)a'} +ff6{3u2+ u", a} a} a}, (35)
which are consistent with the argument given by Douglas [ 13 ].
We can construct the operators P and Q for (21 ) similarly. This time, Q has odd Z2 parity and Q* + Q starts from the sixth order of~/. But we can eliminate this term by adding the square o f Q. It turns out that
Q t + 0 - 9 0 2 (36)
is o f eighth order o f 8. Thus we obtain the following P and Q:
P = d8 + {4f2,
d6} + ~{6f 22-J,~''2, a4} + A{ 1 20f ~-
330f2f~ + 179f2 (4) - 5 8 5 ( f ~) 2 , d 2 }- ~ ( 180f 4 + 9 0 0 / 2 f ~ - 288f2f2 (4) + 1980f2 ( f ~ ) + 223/(26) - 1 8 6 0 / ~ ' f ~ - 1695 ( f ~)2), (37)
Q = d3 + 3{f2, d}. (38)
Thus we f o u n d the (3, 8) model and the (4, 5 ) model which has been considered to be closely connected to three-matrix models. O u r technique is very powerful for studying matrix models, with which we can obtain more information than with the usual methods. We expect that we can investigate more complicated models similarly.
In closing this letter we present two conjectures for the two-matrix models. The first conjecture is that the two-matrix model with 2nth order potential include a critical point that corresponds to the (n + 1, n + 2 ) unitary model. In the 2nth order potential model, we have n + 1 elements in the Q operator. Then we can tune the parameters so that Q starts from the (n + 1 ) th order with respect to e as explained for the (4, 5 ) case and identify P as 0 + ( - 1 ) ~ 0 ' . The second conjecture is related with this construction o f P. It is easy to see in the above construction that P and Q have opposite Z2 parity. This parity is connected to the order o f the differential operator, hence we have only the (even, o d d ) or (odd, even) model. Note that minus the string susceptibility or the inverse o f the multi-criticality equals 2 / (p + q - 1 ). Thus we can construct only integer valued multicrit- ical models f r o m the two-matrix models. But this feature should be different for models that contain more than two matrices.
After the completion o f this work we have received ref. [ 19 ] which concerns a similar construction o f P and Q operators.
We would like to thank Professor T. Yoneya for discussions and for carefully reading the manuscript. We are also grateful to Professor A. Jevicki for discussions. This work is supported in part by the Grant-in-Aid for Scientific Research from the Ministry o f Education, Science and Culture o f Japan No. 02952019.
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