An Operator Formalism for Unitary Matrix Models

We analyze the double scaling limit of unitary matrix models in terms of trigonometric orthogonal polynomials on the circle. In particular we (cid:12)nd a compact formulation of the string equation at the k th multicritical point in terms of pseudo-di(cid:11)erential operators and a corresponding action principle. We also relate this approach to the mKdV hierarchy which appears in the analysis in terms of conventional orthogonal polynomials on the circle.


Introduction
Random matrix models provide an elegant and powerful way to study the dynamics of random surfaces 1{4]. Random surfaces themselves appear in a wide variety of physical problems 5]. They correspond to statistical mechanical models in which the background geometry is allowed to uctuate. The uctuations of the geometry itself is characteristic of theories of gravity and thus one is really studying matter coupled to 2-dimensional gravity. The simplest models (one-matrix models) are de ned by a partition function which is a nite dimensional ordinary integral over an N N-matrix M: Di erent models correspond to di erent classes of matrices M and di erent universality classes of potentials V (M). The best understood case is when M is Hermitian. The integral (1) may then be expanded in a double power series in 1 N 2 and the coupling constant , and generates a set of Feynman diagrams which are dual to a discrete triangulation of a random manifold. Given powers of N correspond to surfaces of xed genus. The partition function Z may be evaluated in the large-N (planar) limit 6] , corresponding to spherical topology. In string theory, and perhaps in 2D-gravity, one is interested in summing the complete topological expansion. This may be done via the \double scaling" continuum limit in which is tuned to a critical value c and N tends to in nity with the scaling variable z = ( c ? )N 2k 2k+1 xed 7{9]. The order of multicriticality is then k 10]. The case k = 2 corresponds to pure 2D-gravity. N is then related to Newton's constant G 0 (N = e 1 4G 0 ) and to the cosmological constant ( = e ? ). To reach the k th multicritical point requires a potential of order at least 2k for even potentials. In the double scaling limit the speci c heat f k (second derivative of Z with respect to z) is determined by a nonlinear di erential equation of order 2k ? 2. At criticality, however, the potential V k for even-order multicritical points is unbounded from below and Z k is not well-de ned. This problem does not exist for odd-order multicritical points 11,12].
Another case of great interest is that of unitary matrices U Z = Z DU expf? N Tr V (U)g : (2) This may be considered as a model of pure two dimensional QCD 13,14]. It has the virtue of being well de ned at all multicritical points since the integration domain is compact.
Ultimately one would like to formulate four dimensional QCD as a matrix model corresponding to sums over world sheets of string-like chromo-electric ux tubes. It is hoped that the double scaling limit of (2) will yield some new insights into this problem 15]. In this paper we give a di erential operator formulation of the continuum limit of (2) and the associated string equation mimicking as closely as possible the analysis of the hermitian model. The organization of the paper is the following. Section 2 introduces some technical machinery, particularly orthogonal polynomials, appropriate for analyzing unitary matrix models. In section 3 the continuum limit is carefully de ned. In section 4 we analyze the string equation for the k th multicritical point. In section 5 an action principle is given for the string equation of section 4 and the relation to the mKdV hierarchy is derived. Finally we conclude and list open problems.

Unitary Matrix Models
We will consider symmetric unitary matrix models of the form 15{17] where V (U) = X k 0 g k U k ; (4) and DU is the Haar measure for the unitary group. It is easy to show that Z U N reduces to Z U N = Z Y j dz j 2 iz j j (z)j 2 expf? N X i V (z i + z i )g ; (5) where (z) is the Vandermonde determinant Y k<j (z k ? z j ) (6) and z i , the eigenvalues of U, live on the unit circle. The inner product is de ned as a contour integral over the unit circle hA(z); B(z)i = Introducing orthogonal polynomials with respect to this inner product hP n (z); P m (z)i = h n n;m (8) one can show that the P n (z) obey a recursion relation z P n (z) = P n+1 (z) ? S n z n P n (1=z) (9) with S 2 n = 1 ? h n+1 h n : (10) The partition function is as usual given by (1 ? S 2 i ) N?i (11) and is thus determined by the recursion coe cients S n of the multiplication operator z. The dependence of S n on the coe cients g k of the potential V is easily shown to be described by the integral ows of the modi ed Volterra hierarchy 18], the simplest ow being _ S n = @S n @g 1 = ?(1 ? S 2 n )(S n+1 ? S n?1 ) : (12) In the continuum limit the modi ed Volterra hierarchy becomes the modi ed K-dV (mKdV) hierarchy. We will return to this later.
By taking appropriate linear combinations of the orthogonal polynomials fP n (z); P n (z)g which preserve the measure factor j (z)j 2 , it is possible to nd an alternative trigonometric basis of orthogonal polynomials 19] of the form c n = z n + n;n?1 z n?1 + : : : n;n?1 z ?n+1 z ?n = c n (z ?1 ) where n is an integer for U(2N + 1) and a half integer for U(2N). The attractive feature of these polynomials is that they satisfy a three term recursion relation analogous to that of the Hermitian matrix model z + c n (z) = c n+1 (z) ? r n c n (z) + R n c n?1 (z) (14) z ? c n (z) = c n+1 (z) ? q n c n (z) ? Q n c n?1 (z) (15) where z = z 1 z . Let us denote the norms of c n (z) by e n hc n ; c m i = e n ; n;m : (16) The integrable ows analogous to the modi ed Volterra hierarchy are now those of the Toda chain on the half line 20] @ 2 n @g 2 1 = e n+1 ? n ? e n ? n?1 : The norms e n are related to the norms h n of the P n (z) polynomials by e n = 2(1 S 2n?1 ) h 2n?1 (18) and e 0 = h 0 : (19) Then one nds that R n = e ( n ? n?1 ) = (1 S 2n?1 )(1 ? S 2 2n?2 )(1 S 2n?3 ) ; (20) r n = @ n @g 1 = S 2n (1 S 2n?1 ) S 2n?2 (1 S 2n?1 ) and Q n = e ( n ? n?1 ) = (1 S 2n?1 )(1 ? S 2 2n?2 )(1 S 2n?3 ) : (22) Using the relation z + ; z ? ] = 0 one can show that q n = (Q n+1 ? Q n ) + (R n+1 ? R n ) r n ? r n = (1 S 2n?1 )(S 2n + S 2n?2 ) : (23) Next, we compute the action of the operator z@ z z @ @z on the c n basis. One nds that z@ z c n = n c n + N k X r=1 ( z ) n;n?r c n?r ; (24) where ( z ) n;n?r = e ? n?r Z d (c n?r ) (z@ z V (z + )) c n (25) and k is the highest power of z + in the potential. For k = 1, for example, the above relation becomes z@ z c n = n c n ? N Q n c n?1 : (26) The operator z@ z acting on c n is not hermitian and is not appropriate for taking the continuum limit. We need to compute instead the action of z@ z on a basis of functions n orthonormal with respect to the \ at" measure dz 2 iz . Therefore we de ne n (z) = e ? n =2 e ? N 2 V (z + ) c n (z) (27) and nd that h n (z); m (z)i = I dz 2 iz ( n (z)) ( m (z)) = ; n;m : (28) The recursion relations (14) and (15) become z + n (z) = q R n+1 n+1 (z) ? r n n (z) + p R n n?1 (z) ; The action of the operator z@ z on the n (z) basis is found to be (v z ) n;n+r n+r (z) + n s Q n R n ? N 2 (v z ) n;n n (z) (v z ) n;n?r n?r (z) ; where (v z ) n;n?r = I dz 2 iz ( n?r (z)) (z@ z V (z + )) n (z) : (31) The k = 1 case now becomes z@ z n (z) = ? N 2 q Q n+1 n+1 (z) + (n + N 2 q n ) s Q n R n n (z) It is easy to check that the above operator is hermitian. The string equation is now derived from the relation y z@ z ; z ] = ?z 19,21]. We are now ready to calculate the continuum limit of the operators z@ z and z near the critical region.

The Continuum Limit
In this section we wish to study the continuum limit of the operators z and z@ z as de ned in (29) and (30) . At the discrete level, the above-mentioned operators act on an in nite dimensional inner product space of complex functions on the unit circle, spanned by the functions n de ned in (27) . Taking the continuum limit means letting N ! 1. But N appears only as the limit of the product (11) . In the continuum limit, therefore, only the indices n in a small neighbourhood of N will contribute to the singular part of Z U N . For the k th multicritical point the relevant index space is described by the scaling variable 16{17,22{23] t = (1 ? n N )N 2k 2k+1 : and n?m (z) ! n (z) + mN ? 1 2k+1 ( n (z)) 0 + : : : + m r r! N ? r+1 2k+1 ( n (z)) (r) + : : : ; and equations (18){(23), we nd that Q n (t; z) = 1 2N ? 1 2k+1 f(t; z) 2N ? 2 2k+1 f 0 (t; z) + O(N ? 3 2k+1 ) R n (t; z) = 1 + N ? 2 2k+1 ( f 0 (t; z) ? 2f 2 (t; z)) + O(N ? 3 2k+1 ) r n (t; z) = N ? 2 2k+1 ( f 0 (t; z) + 2f 2  Q ? = 2 0 @ t + v @ t ? v 0 : In the above formula v = ?2f, @ t @ @t and z act on the column vector ? + n ? n . In the continuum limit the operator z@ z becomes z@ z ! 1 a k N 1 2k+1 P k ; The matrix operator P k has the form P k = 0 P k P y k 0 ; with P k = @ 2k t + p k;2k?1 @ 2k?1 t + : : : ? a k (t + z) : The coe cient a k may be calculated from the action of z@ z given in Eq.(30) and the k-multicritical potentials found in 17]. The result is a ?1 k = 2(2k + 1) k X l=1 (?1) l l 2k B(k + 1; k + 1) ?(k ? l + 1)?(k + l + 1) : The computation of P k is straightforward, but becomes quite tedious for high values of k. For k = 1, for example, a 1 = ?2 and the explicit form of z@ z is z@ z ! ? By letting n ! N and S 2n ! S (spherical limit), we obtain 2 S 2 = 2S 2 (1 ? S 2 ) or = 1 ? S 2 . As the critical solutions for the k th multicritical point are given by = c (1 ? S 2k ), we deduce that c = 1. In the above computation we have used the minimal k = 1 potential V (z + ) = z + .
The string equation is computed from z@ z ; z ] = ?z . As expected, we nd that v obeys 1 2 @ 2 t v(t; z) ? v(t; z) 3 = ?4v(t; z) (t + z) : Therefore v is a function of x = t + z and is a solution of 1 2 v 00 (x) ? v(x) 3 = ?4v(x) x ; (50) which is the Painlev e II equation. As already noted, the computation of z@ z and of the string equation following the steps described above is quite tedious for general k. In the next section we describe a more elegant way of computing them that will give the operator formalism for the unitary matrix models and its relation to the mKdV hierarchy.

The Operator Formalism and the String Equation
In this section we present the form of the operator P k of Eqs.(45) and (46) and of the string equation (50) for general k. We nd that P k is given as the positive part of a pseudo-di erential operator as in the case of the hermitian one-matrix model 24] and that the string equation is closely related to the mKdV hierarchy as in 17].
The string equation z@ z ; z ] = ?z in terms of the operators P k , Q is given by z@ z ; z + ] = ?z ? ) P k ; Q + ] = a k Q ? ) P k (D ? v)(D + v) ? ( EliminatingP y k (P k ) yields Eq.(55) and its hermitian conjugate respectively. The LHS of Eqs.(56) are di erential operators of order 2k. We get, therefore, a total of 4k+2 equations, which is an overdetermined system of di erential equations for the 2k+1 functions p k;i and v. By checking the rst few values of k we nd that, remarkably, only 2k + 1 of them are independent. We conjecture that this is true for all k, although we have no general proof. If this is the case, Eq.(56) uniquely determines the operatorP k and the string equation.
It is instructive to examine the k = 1 case in this formalism. First note that in this case Eqs.

The Relation to the mKdV Hierarchy and the Action Principle
In this section we discuss the relation between Eq.(60) and the mKdV hierarchy and we nd an action principle from which Eq.(60) is derived 25]. A simple way to see that Eq.(60) is related to the mKdV hierarchy is the following. First observe that where u is related to v by the Miura transformation 26,19]  In order to see the relation of Eq. (71) with the one given in 17], one must use Eq.(68) and R mKdV

Conclusions
We have seen that the basis of trigonometric orthogonal polynomials on the circle allows an analysis of unitary matrix models which closely parallels that of the hermitian models. There is a nite-term recursion relation for the multiplication operators z and the derivative operator z@ z which leads in the continuum limit to an explicit representation in terms of pseudo-di erential operators. The string equation has a simple formulation in terms of these operators and follows from an elegant action principle. The most pressing open problem is to nd a world-sheet interpretation of unitary matrix models analogous to that of 2D-gravity coupled to (p,q) conformal matter in the case of the hermitian models. In this respect some of the recent results of Minahan 28] are interesting.