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One-Point Functions of Loops and Constraints

arXiv:hep-th/9112057v1 19 Dec 1991CLNS 91/1118One-Point Functions of Loops and ConstraintsEquations of the Multi-Matrix Models at finite NChangrim Ahn⋆F. R. Newman Lab.

of Nuclear Studies,Cornell UniversityIthaca,NY 14853andKazuyasu Shigemoto†Department of Physics,Tezukayama UniversityNara 631, JapanABSTRACTWe derive one-point functions of the loop operators of Hermitian matrix-chainmodels at finite N in terms of differential operators acting on the partition func-tions. The differential operators are completely determined by recursion relationsfrom the Schwinger-Dyson equations.

Interesting observation is that these gen-erating operators of the one-point functions satisfy W1+∞-like algebra. Also, weobtain constraint equations on the partition functions in terms of the differentialoperators.

These constraint equations on the partition functions define the sym-metries of the matrix models at off-critical point before taking the double scalinglimit.⋆E-mail address: ahn@cornella.bitnet† E-mail address: shigemot@jpnrifp.bitnet

1. IntroductionRecently much progress has been made on the matrix model formulation ofthe 2D gravity to study the non-perturbative effects,[1] and an interesting connec-tion with the integrable systems has been made in the double scaling limit, inwhich the size of the matrix N becomes infinite and the matrix potentials havecritical forms.

[1] In this limit, the non-perturbative results can be obtained fromnon-linear integrable differential equations, such as KdV equations. Furthermore,the correlation functions satisfy their hierarchical equations.

[2−3]It has been noticed recently that the integrability of the matrix models ismaintained even at off-critical points (finite N) before taking the double scalinglimit. At finite N, the Lax pair, zero-curvature conditions, and infinite number ofconserved quantities of the matrix model have been derived and related to inte-grable systems in more clear and direct way.

The underlying integrable systemshave been identified with 1D Toda hierarchy for one-matrix model[4] which be-comes the KdV hierarchy in the scaling limit,[5] 2D Toda hierarchy,[4,6] and 2D Todamulti-component hierarchy[7] for the two-matrix model and for the general multi-matrix models, respectively. The partition functions are the ‘τ-functions’ of theseintegrable systems.Next object of interest is the correlation functions of local operators.

For theoperators appearing in the action, the correlation functions are simply given bythe derivatives of the partition functions with respect to the coefficients of theoperators in the action. It requires, however, non-trivial analysis for the operatorswhich do not appear in the action.

In this paper, we derive one-point functions forthe general local operators which are the ‘loops’ in the matrix models in terms ofdifferential operators acting on the partition functions. These operators satisfy therecursion relations coming from the Schwinger-Dyson equations.

We notice thatthe commutation relations of these differential operators are similar to those of theW1+∞algebra and become exact in the continuum limit.One important related problem is the symmetry structure of the matrix models.2

The Virasoro and Wp+1 algebras have been conjectured for the p multi-matrixmodels as constraint equations on the partition functions in the double scalinglimit. [8−9] The derivation of these symmetries, however, has not been made exceptfor the one-matrix model (the Virasoro algebra) and a special two-matrix model[10](the W3 algebra).

This derivation may be possible if one consider the constraintequations of the matrix models at finite N first. Indeed, it is at finite N that theVirasoro algebras have been derived for the one-matrix model[5,11,12] and for themulti-matrix model.

[7] In this paper, we derive most general constraint equationsfor the multi-matrix models in terms of the generators of the W1+∞-like algebra.These constraint equations seem to be consistent with the conjectures made in thedouble scaling limit.2. Two-Matrix ModelThe partition function of the Hermian two-matrix model is given byZ [{tk}; {sk}, c] =ZDUDV e−S,S = V1(U) + V2(V ) −cUV,V1(U) =∞Xk=1tkUk,V2(V ) =qXk=1skV k.(1)Note the difference in the two potentials V1 and V2; V1 is arbitrary polynomialpotential and V2 is with fixed order.

We want to express correlation functionsin terms of tk’s and their derivatives acting on the partition functions.Thesedifferential operators depend explicitly on the another parameters, sk’s.The most interesting loop operators in the two-matrix models are Tr(V nUm).The one-point functions of these loops areDTrV nUmE=ZDUDV e−S hTrV nUmi. (2)From the Schwinger-Dyson (SD) equations,NXi,j=1ZDUDV∂∂XijV nUmije−S= 0,(X = U, V ),(m, n ≥0),(3)3

one can derive two recursion relations as follows:cDTrV n+1UmE=DTrV nUmV′1(U)E−m−1Xr=0DTrV nUm−r−1TrUrE, (4)cDTrV nUm+1E=DTrV nV′2(V )UmE−n−1Xs=0DTrV sTrV n−s−1UmE. (5)Differential operators generating the one-point functions, defined byW (n+1)m−n Z [{tk}; {sk}, c] ≡−cnDTrV nUmE,(6)satisfy the recursion relation from Eq.

(4),W (n+2)m−n−1 =m−1Xr=0∂∂tr◦W (n+1)m−n−r−1 +∞Xr=1rtrW (n+1)m−n+r−1,W (1)m =∂∂tm,∂∂t0≡−N,form, n ≥0,(7)where the symbol ◦is defined by (A◦B)Z = A(BZ). This recursion relation canbe rewritten in a simple formW (n+1)m=m+n−1Xr=−∞××Jr◦W (n)m−r×× (n ≥0, m ≥−n),Jr = ∂/∂tr,if r > 0rtr,if r < 0,(8)where×× · · ·×× denotes the normal ordering.

The operator Jr’s satisfy U(1) currentalgebra [Jm, Jn] = mδm+n,0 (J(z) = Pm Jmz−m+1 = ∂zφ ). Eq.

(8) defines recur-sively the generating differential operators of the one-point functions. If there isno upper limit in the summation range (m + n →∞), it is obvious thatW (n)(z) =XmW (n)m z−m+n =××∂zφn××.

(9)These infinite number of currents W (n)(z) (n = 1, 2, · · ·) generate the W1+∞al-gebra. [13] For the finite values of m + n, however, the commution relations are not4

exactly same as those of the W1+∞algebra. This W1+∞-like algebra generates theone-point functions of the loops.

It is remarkable that in the continuum limit theloop operators are given by the operators like Tr(UM) with ‘lattice spacing’ a →0and ‘lattice size’ M →∞while keeping aM finite. Therefore, the W1+∞algebragenerates the one-point functions in the double scaling limit.For explicit examples and later use, we write explicit expressions for W (2)m , W (3)m ,W (2)m =mXr=0∂r∂m−r +Xr>0rtr∂m+r,W (3)m =m+1Xr=0m−rXs=0∂r∂s∂m−r−s +Xr>0rtr m+1Xs=0∂s∂m+r−s +m+rXs=0∂s∂m+r−s!+∞Xr,s=0rtrsts∂m+r+s + (m + 2)(m + 1)2∂m,(10)where ∂m = ∂/∂tm and W (2)mcan be identified with the Virasoro generator Lm asit satisfies the classical Virasoro algebra [Lm, Ln] = (m −n)Lm+n.Now consider the constraint equations on the partition function.One canexpress Eq.

(5) with the one-point generating operators as follows:cW (n+1)mZ[{tk}; {sk}, c] = 0,forn ≥0, m ≥−n,cW (n+1)m= W (n+1)m−qXk=1kskck W (k+n)m−k−n−1Xk=0W (n−k)m+k◦W (k+1)−k. (11)Not all of these constraints are independent.

In fact, we can prove that only cW (1)m ’sare independent by showing the following relation from Eqs. (4) and (11)cW (n+1)m=m+n−1Xr=−∞××Jr◦cW (n)m−r××(n ≥0, m ≥−n).

(12)If cW (1)m Z = 0, cW (n>1)mZ = 0 are automatically satisfied. Therefore, the constraint5

equations for the two-matrix models becomecW (1)m Z ="∂∂tm−qXk=1kskck W (k)m−k#Z = 0. (13)As one can see in Eq.

(13), the constraints are linear combinations of the generatorsof the W1+∞-like algebra with the coefficients of the second potential V2. In thecontinuum limit, the constraint equations are given by the W1+∞algebra.

Further-more, for a special potential V2, the operators cW (1)m ’s may generate a subalgebraof the W1+∞, say, the Wn algebra.One can realize the W1+∞-like algebra as a symmetry of the matrix model inthe context of quantum field theory. In the ordinary quantum field theory, thesymmetry can be found as infinitesimal changes of the quantum fields which leavethe action invariant.For the matrix model, this can be done by the followinggenerators Am,n, Bm,n defined byAm,nhe−Si=NXi,j=1∂∂UijUmV ni,je−S,Bm,nhe−Si=NXi,j=1∂∂VijUmV ni,je−S.

(14)The generators Am,n and Bm,n satisfy the following closed commutation relations:[Am,0, Bk,l] = kBm+k−1,l,[Am,n, B0,l] = −nAm,n+l−1,[Am,1, B1,l] = Bm,l+1 −Am+1,l. (15)Note thatDAm,n[e−S]E=DBm,n[e−S]E= 0 from the SD equations (3), whichbecome the constraint equations as shown above.

This realization makes it simpleto prove the statement that only the n = 0 constraints are independent. Thiscomes from the fact thatDBm,n[e−S]E= 0 can be obtained fromDBm,0[e−S]E= 0by using the commutation relations (15).

In fact, it is not difficult to see that{A0,0, A2,1, B1,0, B0,2} are enough to generate the constraints.6

3. Multi-Matrix ModelsThe multi-matrix models with p Hermitian matrix variables Ua have the par-tition functionZ[{tk}; {sa,k}, {ca}] =ZpYa=1DUae−S,S = Tr( pXa=1Va(Ua) −p−1Xa=1caUaUa+1).

(16)We will choose the matrix potentialsV1(U1) =∞Xk=1tkUk1 ,Va(Ua) =qaXk=1sa,kUka ,(17)considering tk’s as variables and sa,k’s as fixed parameters. The loop operatorsin the multi-matrix models are given by TrUnpp · · · Un22 Un11.

Again, we want toexpress one-point functions of these operators in terms of the linear differentialoperators.We start with the SD equations:NXi,j=1Z[DU]∂∂(U1)ijhUn11 Unpp · · · Un22ij e−Si= 0,NXi,j=1Z[DU]∂∂(Ua)ijhUna−1a−1 · · ·Un11 Unpp · · · Una+1a+1ij e−Si= 0, (2 ≤a ≤p −1)NXi,j=1Z[DU]∂∂(Up)ijUnp−1p−1 · · · Un11 Unppij e−S= 0.(18)Eq. (18) can be used to derive recursion relations for the one-point functions ofthe loop operators:c1DTrUnpp · · · Un2+12Un11E= −n1−1Xr=0DTrUr1TrUnpp · · · Un22 Un1−r−11E+DTrUnpp · · ·Un22 Un11 V′1(U1)E,(19)7

ca−1DTrUnpp · · · Una+1a+1 Una−1+1a−1· · ·Un11E+ caDTrUnpp · · · Una+1+1a+1Una−1a−1 · · · Un11E=DTrUnpp · · · Una+1a+1 V′a(Ua)Una−1a−1 · · · Un11E,2 ≤a ≤p −1(20)cp−1DTrUnpp Unp−1+1p−1· · · Un11E= −np−1Xr=0DTrUrpTrUnp−r−1pUnp−1p−1 · · ·Un11E+DTrV′p(Up)Unpp Unp−1p−1 · · · Un11E.(21)Eq. (19) can be rewritten in the form of the recursion relations in terms of thedifferential operatorsW(n+1)m(n3, · · ·, np) =m+n−1Xr=−∞××Jr◦W(n)m−r(n3, · · · , np)××,W(n+1)m−n (n3, · · ·, np)Z[{tk}] = −cn1cn32 · · · cnpp−1DTrUnpp · · · Un2 Um1E.

(22)We want to show that any one-point function can be expressed in terms of thevariables tk’s and their derivatives. Since Eq.

(22) for n3 = · · · = np = 0 with theidentification c = c1 is exactly same as Eq. (8), W(n+1)m(0, · · ·, 0) = W (n+1)mof thetwo-matrix model.To derive other one-point functions, we consider other recursion formulae com-ing from Eq.

(20), (2 ≤a ≤p −1)W(n+1)m(n3, · · ·, na−1, 0, na+1 + 1, · · ·, np)= −ca−1ca−2W(n+1)m(n3, · · ·, na−1 + 1, 0, na+1, · · · , np)+qaXk=1ksa,kck−1a−1W(n+1)m(n3, · · · , na−1, k −1, na+1, · · ·, np). (23)Assuming that we can express all operators W(n)m (0, · · · , 0, na, na+1, · · ·, np) interms of tk’s, we can find W(n)m (0, · · ·, 0, na+1, · · · , np) by repeatedly using Eq.

(23).Therefore, we showed inductively that all the one-point functions can be foundas differential operators of tk’s acting on the partition function. Finally, if one8

finds all the operators in terms of tk’s, one can find the constraint equations onthe partition function from Eq.(21). Among others, the case of np = 0 gives thefollowing equations:cW(1)m (n3, · · · , np−1 + 1, 0)Z[{tk}; {sa,k}, {ca}] = 0,cW(1)m (n3, · · · , np−1 + 1, 0) = W(1)m (n3, · · ·, np−1 + 1, 0)−cp−2qpXk=1ksp,kckp−1W(1)m (n3, · · · , np−1, k −1),(p ≥4)(24)and we must treat carefully the index n in Eqs.

(23) and (24) for p ≤3.We apply above general analysis to the three-matrix model. DefiningW(n+1)m−n (l)Z[{tk}] = −cn1cl2DTrW lV nUmE,(25)they satisfy the following recursion relations:W(n+1)m(l) =m+n−1Xr=−∞××Jr◦W(n)m−r(l)××,W(1)m (l + 1) = −c1W(1)m+1(l) +q2Xk=2ks2,kck−11W(k)m (l).

(26)As explained above, from W(n)m (0) = W (n)mone can find W(1)m (1)’s from the secondequation and W(n)m (l)’s from the first one. Continuing this step, one can find allW(n)m (l)’s.

Finally, the constraint equations come from Eq.(24). For an explicitexample, consider the following potentials: V2(V ) = v2V 2 + v3V 3 and V3(W) =9

w2W 2 + w3W 3. From Eq.

(26), one can find the explicit expressions:W(1)m (1) = −c1W (1)m+1 + 2v2c1W (2)m + 3v3c21W (3)m .W(2)m (1) =mXr=0∂∂tr◦W(1)m−r(1) +∞Xr=1rtrW(1)m+r(1),W(3)m (1) =m+1Xr=0∂∂tr◦W(2)m−r(1) +∞Xr=1rtrW(2)m+r(1),W(1)m (2) = −c1W(1)m+1(1) + 2v2c1W(2)m (1) + 3v3c21W(3)m (1),(27)and the constraint equations arecW(2)m−1(0)Z =W (2)m−1 −2c1w2c22W(1)m (1) −3c1w3c32W(1)m (2)Z = 0,(28)where W (2)m , W (3)mare given in Eq.(10).4. DiscussionsIn this paper, we computed one-point functions of the multi-matrix models interms of the differential operators acting on the partition functions.

The operatorsare completely determined by the recursion formulae, derived from the SD equa-tions and generate the W1+∞-like algebra. Furthermore, we derived the constraintequations on the partition functions using these operators.Since the partitionfunctions are the ‘τ functions’ of the 2D Toda hierarchies,[4,6,7] this means the one-point functions as well as the partition functions are completely determined by theintegrable systems and symmetry structures.

Our method can be generalized tothe multi-point functions. Again, the generating operators are determined by therecursion relations which comes from the SD equations.It is also possible to consider the sk’s as variables such that one can introduceanother differential operators for the two-matrix model.

For the potentials V1(U) =10

Pk tkUk, V2(V ) = Pk skV k, one can defineW (n+1)m−n Z [{tk}, {sk}] ≡−cnDTrV nUmE,W(n+1)m−n Z [{tk}, {sk}] ≡−cnDTrUnV mE,(29)where both W (n)mand W(n)msatisfy the recursion relation Eq.(8). The constraintequations are justhcmW (n+1)m−n −cnW(m+1)n−miZ [{tk}, {sk}] = 0.

(30)Another interesting point we want to mention is that the constraint equationsfor the multi-matrix models like Eq. (28) have very similar form as those of the two-matrix models Eq.(24).

The coefficients of the second potential of the two-matrixmodel can be decided by those of the multi-matrix potentials. The correspondence,however, is not quite exact.

There appear some terms in the constraint equationsof the multi-matrix models which do not exist in those of the two-matrix model.This observation reminds us of the recent conjectures that all the multi-criticalpoints can be achieved by the two-matrix model. [14] If the extra terms at finiteN are suppressed in the double scaling limit, our observation can be a proof ofthis claim.

Related to this and other motivations, it would be very interestingto consider the double scaling limit of our formalism.Our discovery that thecorrelation functions are generated by the W1+∞algebra acting on the ‘τ-functions’of 2D Toda hierarchy seems to be consistent with the results in the double scalinglimit in that the correlation functions of the one-matrix model are given by KdV-hierarchy equations[2] and that the τ-function of the p reduced KP-hierarchy satisfythe W1+p constraint equations. [9]Recently, there have been several papers which mention the W1+∞algebra.Our W1+∞algebra is different from that of ref.

[15] in that the latter comes from thehigher order terms under the change of M to M + δM. Therefore, this constraintsexist even for the one-matrix model.

Our W1+∞constraints exist only for the11

multi-matrix models and will have direct connection with the Wn algebra structuresconjectured in [8] in the double scaling limit. Also, the W1+∞algebra appears fromthe KP hierarchy in the double scaling limit.

[16] This is a direct p →∞limit of theW1+p constraint equations considered in [9]. It would be interesting to considerthe p →∞limit of our result to understand these results from the matrix modelpoint of view.Note: While typing this paper, we received a paper [17] from Y.-X.

Cheng whereconstraint equations for the two-matrix model like Eq. (30) have been obtained.Acknowledgements:C.A.

thanks Tezukayama Univ. and Yukawa Institute of Kyoto Univ.

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