KdV recursion relations for the two-matrix models

Physics Letters B 265 ( 1991 ) 321-325 North-Holland

PHYSICS LETTERS B

KdV recursion relations for the two-matrix models ¢r Tim Yen CaliforniaInstituteof Technology,Pasadena,CA 91125, USA Received 9 March 1991

We find a KdV recursion relation for the two-matrix models in addition to that by Dijkgraaf and Witten [Nucl. Phys. B 342 (1990) 486]. We show that together the relations can be used as algebraic recursion relations for correlation functions which express all correlation functions with an insertion of P in each genus in terms of (PP), (PQ), and (QQ) in a unique way. P and Q are the puncture operator and the dilation operator, and also we assume the scaling ansatz. The KdV recursion relations found do not involve explicitly the infinite number of coupling constants as in the case of the recursion relations given by the Virasoro constraints and the W-constraints.

1. Introduction Recently there has been much interest in understanding the role o f the K d V recursion relations and the string equation within the context o f the multi-matrix models a p p r o a c h to conformal m a t t e r coupled to 2D gravity [ 1-4 ]. By recasting the K d V recursion relation for the one-matrix models and the string equation, D i j k g r a a f et al. [ 5 ] d e r i v e d the loop equations which, r e m a r k a b l y have the algebraic structure o f Virasoro constraints. Fuk u m a et al. [ 6 ] also derived the loop equations in a different way. We are interested in treating K d V recursion relations as algebraic recursion relations for the correlation functions. F o r the one-matrix models, we will show that all correlation functions with an insertion o f P at each genus can be d e t e r m i n e d in terms o f (PP) by using the K d V recursion relation for the one-matrix models a n d assuming that correlation functions at each genus scales. F o r the two-matrix models, we will find a K d V recursion relation different from that in ref. [ 1 ]. We will show that together the two recursion relations can be used to express all correlation functions with an insertion o f P in each genus in terms o f (PP), ( P Q ) , and ( Q Q ) , if we assume the scaling ansatz.

2. Two-matrix model KdV reeursion relations The KdV equations defining the two-matrix models in the language o f pseudo-differential operators are [ 1,3,4 ] 0L

Ot,

-[L~+/3, L] , n e {3j+l,3j+2lje Z, j>~O},

(2.1)

where L - - . d 3 + 3{ul, d} + 3u2, ul = (PP) and u2= (PQ), d=O/Ox, t, are the coupling constants o f the scaling operators tr,, tl is the coupling constant o f the puncture o p e r a t o r P, a n d t2 is the coupling constant o f the dilaton o p e r a t o r Q. The + subscript denotes taking the differential o p e r a t o r part o f the pseudo-differential o p e r a t o r and the - s u b s c r i p t will denote d r o p p i n g the differential o p e r a t o r part o f the pseudo-differential operator. W h e r e v e r the index n appears, we will implicitly assume that n is a positive integer that is not divisible by three. We define R , j , for i = 1 a n d 2, by L"_/3 = {R,, 1, d - 1} + R,,2d- 2 + O ( d - 3 ). Then eq. ( 2.1 ) translates to ~r Work supported in part by the US Department of Energy under Contract No. DE-AC0381-ER40050. 0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

321

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(

d+3

PHYSICS LETTERS B

vtn ' ~ O u 2 - r d 3 , {Rn, l , d - l } + R n 2 d -.2 + . . . ] = ( 6 R n , . ) d + 3 R ~ , l

15 August 1991

.

(2.2)

Eq. (2.2) simplifies in compact notation as 0ui 0R.,i 0 t n - 0x '

i=land2.

(2.3)

We can compute to find that Z ~ / 3 = { U l , d - l } + u z d - 2 + O ( d -s) and that LZ_/3={u2, d - J } = O ( d - 2 ) . So Rt,l=ul and Ri,2=uz implies q=x. So R 2 A = U 2 implies u2=. Note that R 1,2= u2 = R2,1. We now find an identity [ 7 ] from which we can derive the KdV recursion relations. On the one hand we have [L~_/3+1 ,

L] = [L, L"_/3+ 1] = ( 6 R . + 3 , 1 ) d + 3R~+3,1 + 3 R ' + 3 . 2 .

(2.4)

On the other hand we have t~./3+1

= ~-t~-~ +1s!'n/3,

L}+½{L"_/3,L}+.

[L•/3+1 , L] =½[{L~_/3, L}+, LI +½[{L~-/3, L}+, L] =½{L, [L, LL/3]}-½[{LL/3, L}+, L ] .

(2.5) (2.6)

Thus we have the identity 0 = [L, L~_/3+1 ] -½ {L, [L, LL/3]}+½[{L"_/3, L}+, L ] .

(2.7)

The identity above will have coefficients of d i and d o only, and we set them to zero to obtain the two KdV recursion relations. Note that [L, L~_/3] = [L, L"_13] + can be used to simplify the computations. The d I coefficient gives 6R'+3,1 = 18u2R", + 12u'2R., + 12u1R~,.2 +6u'lRn.2 +2R~',2.

(2.8)

The d o coefficient gives

3R'n+ 3,2 =9u2 R ~,2 + 3u'2R,,.2 + 3u't R',,,2 -48uZ R'n,I -48ul u'l R~.l + 6u'2R'~.l - 3u'z R'n,2 - 15u'l R".l . . -3u2R..z . . -. 10uIR~31 . . ), - 2 u t R.,I - 3_lo(s) +3u'(Rn.2-9u'(R'~.I +6uzR~,I .... 1 -

(2.9)

We translate eq. (2.8) and eq. (2.9) by using eq. (2.3), and we obtain unique assignments of the genus for the correlation functions by using scaling arguments g =

~

(3 s, g2+2g, g2+ g2 +2g, < a . e > ~ )

gl +g2 = g

+](a.QPPP>g_I . (a.+3PQ>g=

~

(2.10)

( - g, g, g~)

gl +g2 = g

+

~

(3g,(a,,PQ>g2+g,(a,,PQ)g2+2g,(a,,PP)g2

gl + g 2 = g - - 1

+ (PPPP>g, (a.Q>g2 + 2 ( QPPP>g, (~r~P>g2) +

~

(-16(PP>g,(PP>g2(tY,,PP>g3-16g,(PPP>g2(tTnP>g3)

gl +g2 +g3 = g

+

y~ g~ +g2 = g -

( - ~ < P e > g , ~-5~, ~ 1

- 3 g, ( a~e>g: - ~ g, g-2 • 322

(2.11)

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3. One-matrix model correlation functions For the one-matrix models at the kth multicritical points, k~> 2, we will determine all correlation functions with an insertion of P in every genus in terms of ( P P ) of that genus and lower genus. We assume that every correlation function scales as a power of x, the coupling constant of the puncture operator. The procedure will demonstrate the method to be followed for the more complicated two-matrix models. The one-matrix models KdV equations give Ou/Ot.,= OR,,,/Ox, t~ =x, u = ( P P ) , R,,,= (Pam), m>~ 1. The onematrix models KdV recursion relation is [ 1 ]

(a,,,PP)g=

~.,

((am-lP)g~(PPP)g2+2(am-lPP)g~(PP)g~)+(am-IPPPP)g-t,

m>~l.

(3.1)

gl +g2 = g

We will argue inductively genus by genus and build up correlation functions with more and more operator insertions. Let us first examine the genus zero case. The one-matrix model KdV recursion relation for genus zero is

( a,,,PP)o = ( a,,,_lP)o ( PPP)o + 2 ( tTm-lPP)o ( PP)o .

(3.2)

At the kth multicritical point, the scaling ansatz gives ( P P ) o = x J/k and (am) o= am X b''. Plugging them into the KdV recursion relation eq. (3.2), we get that bm= 1 + m / k and a,, = 1/( 1 + m / k ) m . Differentiate the genus zero KdV recursion relation eq. (3.2)

( atemPP)o = (ale,._ i P)o ( PPP)o + (am- l P)o ( atPPP)o + 2 ( crto'm_lPP)o ( PP)o + 2 ( am_tPP)o ( alPP)o .

(3.3)

The RHS is determined once (atam-tPP)o is determined. Also (atam_~P)o is determined by integrating (atam_lPP)o with respect to x. Since (atPPP)o is determined by differentiating (O~)o with respect to x, so (a~mPP) o is determined via induction. Differentiate twice the genus zero KdV recursion relation, eq. (3.2), we get that (a~aramPP)o is determined once (a~aa,~_ tPP)o is determined. So (aia~rmPP)o is determined, and SO on.

For genus one, the scaling ansatz gives (am)~=CmX am. Using the genus one KdV recursion relation gives dm= - 1 + ( m - 1 ) / k and Cm'S determined by Co. Co is undetermined, and this is the reason that the initial data of the recursions are the values of ( P P ) for every genus. For higher genus, there are no new features, and the same arguments follow. There is, however, one crucial point in our arguments which needs more careful attention. In order to use the KdV recursion relation to build up the correlation functions in terms of lower ones, we had to integrate the LHS of both eqs. (3.2) and (3.3) once to eliminate one insertion of P from the insertion of PP, and plug the results back to the RHS of eqs. (3.2) and (3.3), respectively. Usually the integrated correlation function scales with x to a non-zero power, so that the integration constant can be set to zero using the scaling ansatz. In the special case that the integrated correlation function scales as x to the zero, let us examine the problem involved. Rm is a homogeneous polynomial in u, u', u", ... of degree m, where degree(u)= 1, d e g r e e ( u ' ) = { , degree ( u" ) = 2 .... [ 8 ]. The homogeneity of the polynomial is related to scaling behavior. For example, in genus zero Rm = u m, so that (amP) o scales as x "/k. We can use the homogeneity property to determine the integration constants:

R'm=f[u, u', u", ...],

R m = F [ u , u', u", ...]+const.

(3.4)

For each rn>~ 1, Rm is a homogeneous polynomial in u, u', u", ... of degree m, and this determines const. Namely i f F [ u , u', u", ...] is a homogeneous polynomial in u, u', u", ... of degree m, then const, is set to zero, else Rm is not homogeneous since a constant has degree zero. Similarly we can integrate ( (17l~S O/Ot~)R,,)' to 323

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get (1-Ills O/Ott)Rm. Requiring that each (YIz~s O/Ott)Rm be a homogeneous polynomial in u, u', u", ... of degree m, determines the integration constants. The result is that we first express every correlation in terms of u = ( P P ) and its derivatives u', u", .... Then we substitute in u = ( P P ) expressed in terms of x, the scaling variable, to determine every correlation in terms ofx. In passing we will mention that a non-linear recursion relation exists for Rm [ 8 ]: m--I

m--I

R" +5_. Rm = ~ R i--m--l--i 2 L i=0

m--I

m--I

R~R'm-i_i +½u ~. R i R m - t - i +

i=O

i=0

~ RiRm-l,

rn>~2

(3.5)

i= 1

Also Ro = 1 and R t = u. But the non-linear recursion relation above gives the same results as the recursion relation eq. (3.1), which, using the fact that Rm = ( Pam ), can be expressed as R'm-,,,,,_t-~'' +2URm_ l + U ' R m _ ~ .

(3.6)

Note that eq. (3.6) can be solved to give R,,, = R','_ ~+ 2uRm_ ~- ( d / d x ) - ~ u ' R , , , _ l, where ( d / d x ) - l does not have any integration constant [ 8 ]. The key point is that eq. (3.6) can be used to determine Rm recursively.

4. Two-matrix model correlation functions We proceed by induction genus by genus and build up correlation functions in terms of lower ones as in the case of the one-matrix models. Consider genus zero, the KdV recursion relations are ( a,+ 3PP)o = 3 ( o, P e ) o ( e Q ) o + 2 ( P P ) o ( a, e Q ) o + ( P P P ) o ( a~Q)o + 2 ( Q P P ) o ( o n e ) o .

(4.1)

o o - 16 ( P P ) o ( P P ) o ( a ~ P P ) o - 16 ( P P ) o ( P P P ) o ( a , P ) o .

(4.2)

We assume that we know ( P P ) o , ( P Q ) o , and ( Q Q ) o , and we assume the scaling ansatz. Setting n = 1 in eq. (4.1), we can determine ( a 4 P ) o after we integrate out one insertion of P and using the scaling ansatz. Setting n = 1 in eq. (4.2), we can determine ( a 4 Q ) o after integrating out one P. Setting n = 2 in eq. (4.1), we can determine ( a s P ) o. Setting n = 2 in eq. (4.2), we can determine ( a s Q ) o- We can proceed with n = 4, n = 5, n = 7, n = 8 .... in eq. (4.1) and eq. (4.2) to find all ( a , P ) o . Note that it is clear that eq. (4.1) by itself would be incomplete as a recursion relation. Other correlations and higher genus proceed as in the one-matrix models. For the three-matrix models, we can find KdV recursion relations by using an identity similar to eq. (2.7): O = [ L , Lm_/4+l] --~{ l L , [L,L'L'_/4]}+~[{L_ l m/4 , L } + , L ] ,

m ~ { 4 j + l, 4j+ 2, 4j+ 31j~Z,j>~O} .

(4.3)

The identity above will have coefficients of d 2, d ~, and d o only, and we set them to zero to obtain three KdV recursion relations for the three-matrix models. It is clear that we can proceed systematically to find KdV recursion relations for p-matrix models, and that we will find that there are p KdV recursion relations given by identities similar to eqs. (2.7) and (4.3). Lastly, we comment that the KdV recursion relations found for the two-matrix models, eqs. (2.10) and (2.11 ), do not involve explicitly the infinite number of coupling constants as in the case of the recursion relations given by the Virasoro constraints and the W-constraints [ 5,6 ]. The same comment also holds in the cases of the threematrix models and, in general, p-matrix models.

5. Conclusion We conclude by discussing some questions and speculations. 324

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( 1 ) As mentioned, we can proceed systematically to find that there are p KdV recursion relations for p-matrix models. In the one-matrix models, the KdV recursion relation can be used to perturb away from the string equation to obtain the Virasoro constraints [ 5]. In the two-matrix models, one can speculate that the KdV recursion relation in ref. [ 1 ] can be used to perturb away from the string equation to obtain the Virasoro constraints, whereas the KdV recursion relation found in this paper can be used to perturb away from the string equation to obtain the W-constraints. In general, for p-matrix models, one can speculate that one of the KdV recursions is related to the Virasoro constraints and ( p - 1 ) remaining ones are related to the ( p - 1 ) sets o f Wconstraints. Recently, ref. [ 9 ] has derived the W-constraints from the KdV equations and the string equation. (2) There is a question about whether one can recover the KdV equation from the Virasoro constraints [ 5 ]. As we have seen, using the KdV equations and assuming a scaling ansatz, one can determine all correlation functions with an insertion o f P in terms of (PP). At the kth multicritical point, the Virasoro constraints can determine correlation functions with non-primary fields in terms o f primary fields, but it seems that the correlation functions of primary fields are undetermined. Therefore, it seems unlikely that the KdV equations can be recovered from the Virasoro constraints in the kth multicritical point.

Acknowledgement I would like to thank K. Li and M. Douglas for discussions. And I would especially like to thank my research advisor J.H. Schwarz for discussions, and patient guidance.

References [ 1] R. Dijkgraaf and E. Witten, Nucl. Phys. B 342 (1990) 486. [2] M. Douglas and S. Shenker, Nucl. Phys. B 335 (1990) 635; D. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127; E. Br~zin and V. Kazakov, Phys. Lett. B 236 (1990) 144. [3] M. Douglas, Phys. Lett. B 238 (1990) 176. [4] E. Winen, preprint IASSNS-HEP-90/45 (1990), [5] R. Dijkgraaf, H. Verlinde and E. Verlinde, Nucl. Phys. B 348 ( 1991 ) 435. [ 6 ] M. Fukuma, H. Kawai and R. Nakayama, Intern. J. Mod. Phys. A 6 ( 1991 ) 1385. [7] P. Ginsparg, M. Goulian, M. Plesser and J. Zinn-Justin, Nucl. Phys. B 342 (1990) 539. [ 8 ] I. Gel'fand and L. Dikii, Russ. Math. Surv. 30 (5) ( 1975) 77. [9] J. Goeree, Utrecht preprint THU-19 (1990).

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