MATRIX MODELS OF TWO-DIMENSIONAL GRAVITY ... - Science Direct

Nuclear Physics B357 (1991) 565-618 North-Holland

MATRIX MODELS OF TWO-DIMENSIONAL GRAVITY AND TODA THEORY A. G E R A S I M O V u, A. M A R S H A K O V 2, A. M I R O N O V 2, A. M O R O Z O V t AND A. O R L O V 3

tinstitute of Theoretical and Experbnental Physics, 117259 Moscow~ USSR ZDepartment of Theoretical Physics, P.N. Lebedev Physical Institute, LeninskT Prospect 53, Moscow 117924, USSR 3Oceanology b~stitute, Moscow USSR Received 19 July 1990 (Revised 12 November 1990)

Recurrent relations for orthogonai polynomials, arising in the study of the one-matrix model of two-dimensional gravity, are shown to be equivalent to the equations of the Toda-chain hierarchy supplemented by additional Virasoro constraints. This is the case even before the continuum limit is taken. When the odd times are suppressed, the Volterra hierarchy arises, its continuum limit being the KdV hierarchy. The unitary-matrix model gives rise to a sort of quantum deformation of the Volterra equation. We call it the modified Volterra hierarchy, since in the analogous continuum limit it may turn into the mKdV. For muitimatrix models the Toda lattice hierarchy of the type A~ appears. The time-dependent partition functions are given by r-functions - the sections of the determinant bundle over the infinite-dimensional grassmannian, associated with Riemann surfaces of spectral parameters. However, because of the Virasoro constraints matrix models correspond to a very restricted subset of the grassmannian, intimately related to the W~ Lie subalgebra of UGL(~:). The continuum limit, leading to multicritical behavior is adequately described in terms of peculiar operators, defined as hypergeometric polynomials of the original matrix fields and resembling a sort of handle-gluing operator.

1. Introduction Since the structure of perturbation theory for string models associated with various conformal models is more or less understood on the basis of the free massless field representation [I-4], the most important problems in string theory are (a) the summation of perturbation series for various models (usually implying some "non-perturbative effects"), and (b) the interpolation between different string models, unifying them into a general (and unique?) string theory. Problem (b) is still not too popular (see refs. [5,6] for some suggestions), but recently there has been considerable progress concerning problem (a). This progress 1)550-3213/91/$03.50 ~:3 1991 - Elsevier Science Publishers B.V. (North-Holland)

566

A. Gerasimot" et al. / Matrix models of 2D grat'ity

is due to the development of a new technique of "matrix models" [7-10] or of "the Grothendick esquisse" triangulation of moduli spaces (see ref. [11] for a review). Though the relation of this new formalism to the standard one, associated with determinant bundles over universal moduli space (i.e. the free field calculus) remains obscure, there are no doubts that particular results, mostly for minimal models Mp.q coupled to two-dimensional gravity, are very reasonable. The main one of these results is that the partition function and correlators in certain string models (namely, associated with Mr. q conformal models) considered as functions of the string coupling constant, satisfy certain differential equations, which may be written in the form of [ LI', Lq+] = 1 with the differential operator L p of degree p, related to the theory of integrable equations [12]. This kind of statement looks rather plausible, because from the standard point of view the sum over all ordcrs of perturbation theory in a string model, associated with a conformal model with the action ScH., may" be described as a theory on a Riemann sphere with the action S = S c v x + ~V~I~;, where ~: stands for the string coupling constant and V~G for the "handle-gluing operator", integrated over its moduli. This is because any correlator of free fields on a closed oriented Riemann surface of genus g may be considered as a correlator on the Riemann sphere with g insertions of handle-gluing operators. A handle can be described as an insertion of infinitely many soliton creation operators of the form (in the free field representation) exp[Aij~i~b~(A)]. (For explicit formulas for the simplest handle-gluing operators see refs. [13, 14].) V6, i is a highly non-local operator, and S no longer describes a conformal model. One may hope that for a particularly nice Soy r (say, that of the minimal models Mp.q), it is instead an integrable model. Thus it seems plausible that sums of perturbation series (i.e. integrals over universal moduli space) for (at least some) string models can be written in terms of integrable models on the Riemann sphere as a two-dimensional space-time. (The function ScF v ~ S in the category of two-dimensional field theories seems to be a very interesting object.) There are serious implications [15] that zero-loop (i.e. defined on the Riemann sphere) correlators in integrable models in their turn satisfy certain integrable equations. (The function [equations of motion of original integral model] ~ [equations for the correlators] is also an important thing to think about.) This reasoning explains why something like the implication of matrix models was expected to be true in string theory; however, these arguments are too far from being explicit and concrete. Let us turn now to the discussion of matrix models themselves. Although their equivalence to the standard string models is far from being obvious (see ref. [11] for a discussion of the problems which can arise), this formalism is very interesting by itself. Moreover, it is naturally related to the same theory of Riemann surfaces, universal moduli space and determinant bundles. The only difference is that now these Riemann surfaces have nothing to do with the two-dimensional space-time; they rather appear in the space of string models. (This occurrence of Riemann

A. Gerasimot" et al. / MatrL~ modds of 2D gral~ity

567

surfaces as two-dimensional space-times and as spaces of various string m ~ e l s ties at the basis of the suggestion made in refs. [5, 6] to unify these two spaces into a four-dimensional (two-dimensionally complex) manifold to be treated as configuration space of the entire string theory. The basic structure on this configuration space could be induced by two-loop Lie algebras, like Fairlie's [16] algebra.) We are going to discuss this relation of matrix models to determinants, r-functions, the grassmannian, W~ algebra etc. in a separate detailed publication. This preliminary paper is devoted to a straightforward presentation of the basic connection between orthogonal polynomials and integrable Toda theories. Our main purpose here is to make the subject of matrix models free of unessential technical details and reformulate it directly in familiar terms of conformal and integrable theories. We feel that this kind of ideas should be known to many people. In particular, some closely related observations have recently been described in refs. [17-19].

2. The one-matrix model and the Toda chain. Results and discussion

We shall not go into too much d~tail ab3ut the one-hermitean-matrix model itself, since this subject is very widely known now [7-10]. The main ingredients of the orthogonal polynomials technique [20, 21] are Scalar product:

(A(A),B(A))=~A(A)B*(A)dAexp

_

y~

tt, Ak

(2.1)

In matrix models based on the use of hermitean matrices [7-10] the integral is along the real axis, B*(A)= B(A) (the contour of integration is related to conjugation by the condition that the scalar product is real) and only positive times t k with k >/0 are considered. (Moreover, in most papers only even positive times t2t. are taken into account, which obscures the underlying structure of the Toda equations discussed in this paper.) In models of unitary matrices (see, for example ref. [22]) the integral in eq. (2.1) is over the unit circle and B*(A) = B(1/A). (In general, the • -sign implies something like exchange of the operators J~ and ./! of the Lax representation, cf. sect. 4.) Orthogonal polynomials (in fact, they are almost the Baker-Akhiezer functions for the Toda model in the special points of the grassmannian, see sect. 4) are defined as P , , ( A ) = ,¥' +

n> O,

( P,,, P,,) = e '1',,~1~3 , , , .

(2.2) (2.3)

568

A. (h,rasimot" et aL / MatrLr nlodels of 2D gral"it)'

The physically interesting quantity is the (continuum limit of) {tk}-dependent partition function q~N- the generating functional for correlation functions of palticular string theory, N

e't'x- H e't'".

(2.4)

I1 = 0

We shall prove in sect. 3 that eqs. (2.1)-(2.3) imply the following relations for d', and q~N: [12ff) n =

e,t,,, + I -,t, ....

e,b,,-,h,,_

i

(2

cgt 2i

~

and all other equations of the Toda-chain hierarchy for t k-derivatives of #5,," L ,!,f-"' c ] = 0,

(2.6)

with the generators of the Virasoro algebra given by

Lq~o=

O q O 0 ~ ktk + y~ " k=- ~ Otk +,i . = ) Otk 01,!-k

O L,I =

at.

= - 1.



(3.28)

We will now evaluate the matrix element on the r.h.s, of eq. (3.27). For this purpose let us take the A-derivative of (2.3),

at',,

at,,,+,

P,, + A OA

=

OA

at,,, ~

P" flA

at,,,_. +

R

,

,

~

OA

and make use of eq. (3.24),

p,,= 2

,,~i ] ,,, .



- p,,, P,,, +/¢,,, P,,,_. ]

,,~l (n[Ak-llm > Pit1 m

~ttl/ '

=()

A. Gerasimov et aL / Matr& models o f 2D graHt.r

577

where the expression in solid brackets is merely AP,,,. Thus we obtain, comparing the coefficients in front of/',,-term,

P,,

1=

-

( n - l l n - l )

+

(nln)

= - n + ( n + I )

.

Below we use the short-hand notation ( n l a ~ - 'In - i)

A(,~,- A,,.,,_, -

y" kt k k

(n - i[n - i)

According to (3.21), A(,~) = n. Now, comparing the coefficients in front of P,,, gives

P._,{0 =A,,~,,,_i (,,_ _ A,, (2, +A,,+! ,2)

.

-A"'p,,}

,

i.e.

A(2) ,, + ~ =

A , ,( 2 ) +

H( /On

- Pn-

I ) = rip,, -

p , , - i - p , , _ ,_ -

...

- p~.

(3.29)

Therefore, n a 2 ~01 - n

)

= .-1 ~] (nla2lmSA ('-'")-.__. nl

n(n[A2ln -

1) +A,

(2)

o are non-vanishing is also discussed at the end of the section. 4.1. I N T E G R A B I L I T Y A N D M A T R I X M O D E L S

We begin with a brief preliminary remark to avoid a misunderstanding which could arise from the formulation of our results in sect. 2. We argued that matrix models are intimately related to integrable systems and that this relation arises

586

A. Gerasimot" et aL / Matrix models of 2D grarity

from nothing other than the orthogonality condition (2.3). How can it happen? What has eq. (2.3) to do with integrability? The answer is very simple. The theory of integrable systems is in fact nothing other than the study of Riemann's factorization problem, that is the decomposition of (infinite-dimensional) matrices into products of upper and lower triangular ones. But eq. (2.3) is in fact exactly the problem of this kind. Indeed, if e - ' t ' ' / 2 P,,(A) = E~Tr,,~Ak, then eq. (2.3) turns into 77C7r tr = I,

(4.1)

where I is identity matrix and C,,,,, = (•'"+"). According to eq. (4.1), the construction of the orthogonal basis {Tr,,} is equivalent to decomposing the moment matrix C into the product of upper and lower triangular matrices 7r-~ and [~'-~]tr (note that the r-function is just the determinant of this moment matrix). The constraint that the ~r,, are polynomials is a certain analyticity requirement on the spectral surface of the parameter A. It deserves noting that Riemann's factorization problem is a functional generalization of Gauss decomposition in the case of simple algebras. Gauss decomposition because very useful in conformal theory after an interesting object (namely the Wess-Zumino-Novikov-Witten action) was found, which respects this decomposition (see, for example, ref. [4]). Certainly, analogous important objects should exist in the framework of string theory for the generic factorization problem. This topic however is beyond the scope of the present paper. 4.2. L A X R E P R E S E N T A T I O N A N D B A K E R - A K H I E Z E R

FUNCTIONS

We will now proceed to reformulate the material of sect. 3. First of all we need several new notations for our orthogonal polynomials P,, = A" + o(A"-~). We have already mentioned the orthonormal polynomials ~-,,- e -#'',/2 Pn- Introduce also ~,,(A) = P,,(A)exp{- -~EktkA i k} and qJ,, = e -'t'',/2 ~,, which are orthogonal and orthonormal respectively with respect to a trivial scalar product, dAqJ,,0,,*

6,,,,,

dA~,,~,,, =e't',,6,,,,,

(4.2a,b)

We already mentioned that the system of orthogonal polynomials reduces to the Toda-chain hierarchy. To the experts in integrable theories this is obvious already from eqs. (3.13): If rewritten in terms of orthonormal basis qJ,, they acquire the form of the auxiliary linear problem for the Toda chain (compare, for example, ref. [39], eq. III.2.33), ,~lJ,, = SqJ,, = c,, + iqJ,, + i - P,,~IJ,, + c,,qJ,,_ i, c,, = e ('t'',-'t'',

')/"

= l/R,, ,

(4.3) (4.4)

A. G e r a s i m o c et al. / MatrLr models o f 2 D grm'i~"

587

where the operator S of the Lax representation has m a t t * elcments ~,~., = (nlAk-iln> [ ( n l n ) ] :~ llAk-IIn) lln-

1)

llAk-Jln)

(n-lln-l)

'

eq. (4.21b) turns into

I=

~kt k k

(n + llAk-Iln) (nlAk-Iln -- 1) ] (nln) - ( n - l l n - I) "

(4.24)

In this form it survives after reduction to the Voiterra hierarchy, when all odd times t2k_ t are excluded. As a kind of check of eq. (4.24) let us add to it the same equation with n ~ n - 1. The resulting equation,

2 = ~kt/, k

(n + llAk~ Iln> ( n l A k - I i n - 2) ] (nln) - (n- 21n-2) '

(4.25)

A. Gerasimot" et al. / Matrix" models o f 2D gra~'ity

593

may be obtained alternatively from eq. (3.22):

2=(2n+l)-(2n-l)=

(nlak[n) ~.,ktk

(n + llain> s,r/>~s appears only if the [L] conditions are also taken into account. Therefore, in the two-matrix case there is no simple division into [T] and [L], except for the case, when only a finite number

598

A. Gerasimoc et al. / Matrix" models of 2D gral'ity

of times remains in the potential. In fact, in this statement we imply that an analogous system of equations closes, when times ~>~4+,-, r/>~a+r = 0 on the set of a(/3),,,,, with 0 ~ 0 are supposed to be non-vanishing in the weight function. Let us mention one more problem with the "naive" continuum limit. It does not seem to be the same if one takes the continuum limit of the auxiliary linear problem (of the Lax representation) or the limit of the equations of the hierarchy. While in the first case it is quite clear that the Lax representation for the Volterra hierarchy turns into that of KdV, it is a bit problematic to find a reasonable continuum limit even of the second member of Volterra hierarchy itself, r 4 = (rxxxx + 5rrxx + lO(rx) 2 + lOr3)x.

(7.20)

The problems are similar to those discussed in the previous paragraph. The thing is, that (as it is clear from the example of the Volterra --->KdV transition above) we pick up coefficients in front of different powers of e in the left- and right-hand sides of the equation

O(L,,+ EL, +

E2L2+... )/c)t

--[IL,,+eLt+e -'L2 +. .. ), (L,,+eL,+e 2L 2 +. .. )'],

606

A. Gerasimoc et al. / Matrix models of 2D gral'ity

say OL ~/Ot is equal to the coefficient in front of e- at the r.h.s. The latter clearly contains the contribution with L 0 and L 2. However, this contribution will not appear if we begin from the continuum approximation of the Lax representation itself, (Lo+eL ~+ e 2 L 2 + ...)~(Lo+eL~). As far as we know, the continuum limit of the hierarchy is always understood as the limit of the Lax representation, although the meaning of this notion seems a bit subtle.

7.3. THE BASIC EXAMPLE OF THE PAINLEVt~ EQUATION

Let us turn now to the implications of the Virasoro constraints. (We restrict ourselves to the case when only positive times t k ~ 0 are non-vanishing.) We shall illustrate the main idea with the example of the mKdV hierarchy relevant for the unitary-matrix model, where the automodel condition, implied by the Virasoro constraint, is known (see, for example, ref. [37]) to lead to the Painlev6 II equation. In the case of the KdV hierarchy a more complicated third-order differential equation originates in a similar way. As already done in sect. 4, consider the constraint (3.19) on the r-function of the Toda-chain (or AKNS) hierarchy,

(lo+N2)zN_,=O,

!,,=

EktkO/Ot k,

(7.21)

and transform it into an equation for the Baker-Akhiezer function. Since the vertex operators ~ h k ] exp[ + ~ k - ~ A V+_(A[{,}) - exp[+ E _~tk

-k3/Otk]

satisfy the relation

[(t,,-

t})] = 0,

(7.22)

the Baker-Akhiezer function

1/t"(~1{ t} ) ~---V(

l~ ) T N / T N = V( 1~)II}TN/IIIT N

(7.23)

satisfies the equation ( t,,-

= 0.

(7.24)

(Note that the constraint (7.21) is crucial for this derivation.) On the other hand, the Baker-Akhiezer function satisfies the consistent system of evolution equations:

Ot~.

B

U~k~ lit = (}

(7.25)

A. Gerasbnov et al. / Matrix models of 2D grmi~"

6{~7

with

Ot k

= 0.

U(k) ,

(7.26)

The consistency condition between eqs. (7.24) and (7.25) is Aaqt' = l q , + Y ' . k t k ~ . aa ' atk

(7.27)

In integrable theories the A-dependence of the U~k~ potentials is adjusted in such a way that the A-dependent terms cancel i,_¢~z~ca!ly in eq. (7.27). Therefore it becomes a constraint on tk dependencies of dynamical variables, which is known as the "automodel" constraint. Substitution of such an automodel ansatz into the equation of motion (7.27) gives rise to an integrable subsystem of the original one. In the particular case of the mKdV hierarchy,

Uric=

iA u

u )"

-iA

(7.28)

Eq. (7.27)with 1 = 1 reads ~ll

u + Y'.tk-- =0, tgtk

(7.29)

and can be resolved in terms of a new "automodel" variable F: (7.30)

tt = , ? ' F { t i / , ~ } .

If only t l and t 3 variable is rather:

are

non-vanishing, the conventional choice for the automodel

Y = tl/(3t3)I/3.

u = (3,3)-'/3f(t,/(3t3)'/3),

( 7.31 )

(The form of the resulting equation (7.33), but not its solutions, depend on the choice of variable.) Substituting this ansatz into the ordinary mKdV equation all/at3

= (/ill-

2t/3)i ,

it k = O u / O t k ,

(7.32)

gives rise to (the derivative of) the Painlev6 II equation, [f"(y) -4yf(y)

- 2f3(y)]' =0.

(7.33)

Note that the same automodel constraint (7.29) arises if we start from eq. (7.27) for

608

A. (;erasimor et al. / Ahttr& mo&'ls of 2D grat'io"

the next 1 = 3 member of the mKdV hierarchy, A2u + i A u l / 2 -

_ i a . ~ _ iuea / 2 U{3 )

A2ll

-

-

iAul/2

(Uli -- 2 U 3 ) / 4

• (7.34)

iA3 + ht2A/2

- (u~! - 2 u 3 ) / 4

(The mKdV equation (7.32) is nothing other than eq. (7.26)with ( k , / ) = (1,3).) What we need, however, is the Toda-chain hierarchy. The automodel conditions in this case were derived from the Virasoro conditions in subsect. 4.3. In particular, Op,, Y'~kt k =0, k Otk

OR,, 2 R , , + ~.,kt~. -0. k Otk

(7.35),(7.36)

If only even times are preserved (the Volterra reduction) eq. (7.36) implies that (7.37)

R,, = t 2 'i~,,{t,_k/t~_ } .

In the particular case of Hermite polynomials (i.e. all t2k = 0 except for t2), R,, =n//t 2. This automodel ansatz should be substituted into the equations of the Volterra hierarchy. The first of them is already familiar to us, - 0 ( l o g R , , ) / O t 2 = R,,+~ - R , , _ l ;

(7.38)

the second can be easily derived from (3.6): -0(IogR,,)/at4=R,,+~(R,,+2+R,,+~+R,,)-R,,_~(R,,_

2+R,,_~+R.).

(7.39)

As demonstrated in subsect. 7.2 the appropriate continuum limit of eq. (7.38) is the first KdV (not mKdV!) equation, r 2 = (r,..,.- 3, "2),.

(7.40)

The second member of the KdV hierarchy is r 4 = (r.,..,..,.., + 5rG. ,. + 10(r.,.) 2 + 10r3).,.

(7.41)

As explained in subsect. 7.2, in order to obtain these equations (at least the first one), it is necessary to take R,, ~ R N e '-''¢-' =''' 1 with time-independent R N. This may seem inconsistent with the automodel constraint (7.37). However, in fact RN depends also on N, and N-dependence is not specified by the automodel constraint. When the continuum limit of the Voltcrra equation is taken, N should be adjusted so as to compensate tor the t-dependence of RN.

A. Gerasimcn"et aL / Matr£~m~nh'ls~[ 2D gra~ity

~

7.4. THE NAIVE CONTINUUM LIMIT Let us return to the analysis of the expressions of subsect. 4.3 in the continuum limit. The main ingredient of this limit is the substitution n---,x=En

or

x=n/N,

R,,~R(x), (n-

lla-'k- u[n)

( n - lin- I)

---, Gk{ R( x ) ; E} ;

(7.42~

G k being closely related (but not equal to!) the Geifand-Dikii functions, a I ~-~n,

G2 G3

=

3R'- + E 2 R " + l~E4R .... + o( E6)

•

=10R-~+ez[10RR'+5(R') z] +e4I~"R .... + . . . ] + o ( e

Gk = 2 iC z kk

R k

+o(

E. 2

).

~').

(7.43)

The first term without dr, atives in the r.h.s, of eq. (7.43) is easily derived. Generically, ( m l A k l n ) / ( m l m ) with n - k ~