NBI-HE-92-89

PAR–LPTHE–92-49

FTUAM 92-44

November 1992

Matrix model calculations beyond the spherical limit

J. Ambjørn

The Niels Bohr Institute

Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark

L. Chekhov^{1}^{1}1
Permanent address: Steklov Mathematical Institute,
Vavilov st.42, GSP-1 117966 Moscow, Russia

L.P.T.H.E., Universitè Pierre et Marie Curie

4, pl. Jussieu, 75252, Paris Cedex 05, France

C. F. Kristjansen

The Niels Bohr Institute

Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark

Yu. Makeenko

Institute of Theoretical and Experimental Physics

B.Cheremushkinskaya 25, 117259 Moscow, Russia

Abstract

We propose an improved iterative scheme for calculating higher genus contributions to the multi-loop (or multi-point) correlators and the partition function of the hermitian one matrix model. We present explicit results up to genus two. We develop a version which gives directly the result in the double scaling limit and present explicit results up to genus four. Using the latter version we prove that the hermitian and the complex matrix model are equivalent in the double scaling limit and that in this limit they are both equivalent to the Kontsevich model. We discuss how our results away from the double scaling limit are related to the structure of moduli space.

## 1 Introduction

The hermitian one matrix model is interesting for many reasons. The matrices involved being , the model can be solved in the limit when , as was first demonstrated in Ref. [1]. Furthermore the model admits a -expansion. This expansion is the simplest example of the ’t Hooft topological (or genus) expansion [2] — in the given case of a dimensional quantum field theory. If the leading order (planar or genus zero approximation) result for some quantity, say the vacuum energy, is normalized to be of order , the genus contribution is of order . Even though the hermitian matrix model is often considered a toy model, its perturbative expansion reveals the combinatorial problems one will encounter in the multi-dimensional case: The sum of graphs contributing to a given order of the genus expansion is convergent, while the attempt of summing over all genera leads to a divergence. This was one of the original motivations for studying the genus expansion of the hermitian matrix model.

One more application of the hermitian matrix model is due to the fact that it
describes [3] discretized 2-dimensional random surfaces
and hence 2D quantum gravity.
In this language a diagram of genus is equivalent to
a (piecewise linear) surface of genus .
The hermitian one matrix model was extensively studied from this point of
view after the observation [4] that it might not only describe
pure 2D gravity but also some minimal conformal field theories coupled
to 2D quantum gravity^{2}^{2}2Although the relation conjectured in
[4] turned out to be wrong, the paper was nevertheless
instrumental for the later developments.. The relation given in
[4] for genus zero
was extended to any order of the genus
expansion in the so-called double scaling limit,
where the coupling constants
approach some critical values in a -dependent way [5].

In the double scaling limit many interesting approaches have been developed in order to deal with higher genus contributions. The technique of exactly integrable systems (KdV and KP hierarchies) was applied to the model in Ref. [6]. It provides an algorithm for genus by genus calculations in the double scaling limit. Another approach proposed by Witten [7] is based on the interpretation of the double scaling limit of the matrix model as a topological theory. The problem is then reduced to calculating intersection indices on moduli space. In this approach it was possible to obtain explicit results in the double scaling limit up to genus four [8]. A further development of this approach was the representation [9] of the generating function for the intersection indices as a hermitian one matrix model in an external field (the Kontsevich model).

For the hermitian matrix model away from the double scaling limit, the original genus zero solution [1] was obtained by solving the large-N integral saddle point equation for the spectral density. To extend the results to higher genera, the orthogonal polynomial technique was applied in Ref. [10]. While this method works well for the partition function in the case of simple potentials and lower genera, it is difficult to implement if one wants an explicit calculation of multi-point correlators and if the potential is “generic”.

Another method of solving the hermitian matrix model is based on the loop equations [11] which were first written down for the hermitian matrix model in Ref. [12]. In this approach one solves the genus zero equation and uses the solution in an iterative genus by genus procedure. In the scheme originally proposed in Ref. [11] and elaborated in Ref. [13], one determines the higher genera contributions by solving algebraically the higher loop equations and imposing on all correlators the same analyticity structure as that of their genus zero versions. Following this line of action genus one correlators for and potentials were calculated in ref. [14]. However, this method involves the entire chain of loop equations and is in practise only applicable when one only wants to calculate a few terms in the genus expansion for a potential with a small number of terms.

An alternative iterative procedure for calculating higher genus
contributions to the correlators of the
the hermitian matrix model has been proposed recently [15]. It is based only on the
first in the chain of loop equations, is applicable for
an arbitrary potential and can be pursued order by order of the genus
expansion without any problems.
A key point in this scheme is a change of variables to the so called
moments of
the potential. The scheme is close in spirit to the iterative
solution of the unitary and hermitian (with cubic potential) one matrix
models in an external field [16]^{3}^{3}3 For the Kontsevich model similar moments has been used in
Ref. [17]..
The genus one
partition function and correlators were calculated for an even
potential in Ref. [18] and for an arbitrary potential
in Ref. [15], where it was also
proven that genus contribution to the -loop correlator depends
on at most lower moments. An analogous algorithm for
the complex matrix model was developed in Ref. [19] and the genus
and correlators were explicitly calculated for an arbitrary potential.

In the present paper we propose an improved algorithm for an iterative calculation of higher genus contributions to the multi-loop correlators and to the partition function of the hermitian one matrix model. We perform calculations up to and including genus two. Our algorithm differs from that of Ref. [15] by a redefinition of the moments, which has the advantage that it allows us to develop a version which gives directly the result in the double scaling limit where we perform explicit calculations up to and including genus four. We compare the analysis to that of the complex matrix model and prove in particular that the hermitian and complex matrix models are equivalent in the double scaling limit. We formulate a limiting procedure which allows us to obtain the Kontsevich from the hermitian matrix model and use it to prove in a very direct manner the equivalence between the double scaling limit of the hermitian matrix model and the Kontsevich model.

We discuss a possible interpretation of our results for the hermitian matrix model as representing a particular discretization of moduli space which preserves basic geometrical properties such as intersection indices, and which allows for a detailed study of the boundary of moduli space [20]. We conjecture that the moment-technique developed in the present paper might be an efficient tool in the study of this “fine structure” of moduli space.

The paper is organized as follows: In Section 2 we introduce the basic ingredients of the iterative procedure, the loop insertion operator, the loop equation and the new variables — the moments. In Section 3 we describe the iterative procedure and present the results for the average of loop operators and the partition function for the hermitian matrix model up to genus two. In Section 4 we demonstrate in detail how our iterative procedure works in the double scaling limit and calculate the correlators and the partition function up to genus four. In Section 5 we show that the Kontsevich model can be obtained as a certain limit of the hermitian matrix model, prove the equivalence of the double scaling limit of the hermitian matrix model and the Kontsevich model and discuss the possible connection between our results for the hermitian matrix model away from the double scaling limit and the structure of the discretized moduli space. Finally Section 6 contains a short discussion of future perspectives and in the Appendix we develop a double scaling version of the iterative procedure for the complex matrix model proposed in [19], and prove that the double scaling limits of the complex and hermitian models coincide.

## 2 Main definitions

The iterative procedure we are going to describe is based on three main ingredients, the loop insertion operator, the first in the chain of loop equations of our model and a suitable change of variables. It allows us to calculate the genus contribution to the -loop correlator for (in principle) any and any and (also in practise) for any potential. The possibility of going to an arbitrarily high genus is provided by the loop equation while the possibility of considering an arbitrarily high number of loops is provided by the loop insertion operator. The change of variables allows us to consider an arbitrary potential.

### 2.1 Loop insertion operator

The hermitian one matrix model is defined by the partition function

(2.1) |

where the integration is over hermitian matrices and

(2.2) |

Expectation values (or averages) are defined in the usual way as

(2.3) |

We introduce the generating functional (the -loop average)

(2.4) |

and the –loop correlator

(2.5) |

where refers to the connected part. One can rewrite the last two equations as follows

(2.6) |

As is explained in Ref. [4], these quantities are associated with the (Laplace images of the) sum over discretized open surfaces with boundaries.

The correlators can be obtained from the free energy, , by application of the loop insertion operator, :

(2.7) |

where

(2.8) |

Notice that Eq. (2.7) can be rewritten as

(2.9) |

which shows that if the -loop correlator is known for an arbitrary potential, all multi-loop correlators can be calculated. This is why it is named above as the generating functional.

With the normalization chosen above, the genus expansion of the correlators reads

(2.10) |

Similarly we have

(2.11) |

for the genus expansion of the free energy.

### 2.2 The loop equation

The first in the chain of loop equations for the hermitian matrix model can conveniently be written as [21]

(2.12) |

where and is a curve which encloses the singularities of and not the point . This contour integration acts as a projector picking up the coefficients of . Due to Eq. (2.9) the second term on the right hand side of the loop equation (2.12) is expressed via , so that (2.12) is a closed equation which determines this quantity.

To leading order in one can ignore the last term in Eq. (2.12) and assuming that the singularities of consist of only one cut on the real axis and that behaves as as one finds [11]

(2.13) |

where and are determined by the matrix model potential in the following way

(2.14) |

This genus zero solution will be used below in the iterative procedure of solving the loop equation.

### 2.3 The new variables

To characterize the matrix model potential we introduce instead of the couplings the moments and defined by

(2.17) | |||||

(2.18) |

These moments depend on the coupling constants ’s both explicitly and via and which are determined by Eq. (2.14):

(2.19) | |||||

(2.20) |

Notice that and depend explicitly only on with .

There are several motivations for introducing these new variables. First, as we shall see below, for each term in the genus expansion of the partition function and the correlators, the dependence on the infinite set of coupling constants will arrange into a simple function of a finte number of the moments. Moreover, these new variables reflect more directly than the coupling constants the possible critical behaviour of the matrix model. Let us briefly describe how this comes about.

Performing the contour integral in (2.13) by taking residuals at and one finds

(2.21) |

where is a polynomial in of degree two less than . As already discussed in the Introduction, can also be determined by an analysis of the matrix model in the eigenvalue picture [1]. Requiring that is analytic in the complex plane except for a branch cut on the real axis and behaves as for corresponds to requiring that the eigenvalue density, , has support only on the interval and is normalized to 1. The eigenvalue density is in this situation given by

(2.22) |

This function vanishes under normal circumstances as a square root at both ends of its support. Critical behaviour arises when some of the roots of approach or . For a non-symmetric potential the so called multi-critical point is reached when extra zeros accumulate at either or . Comparing (2.13) and (2.21) it appears that

(2.23) |

so the condition for being at the multi-critical point is simply and if the extra zeros accumulate at and , if the extra zeros accumulate at . For the symmetric potential () the eigenvalues will be distributed symmetrically around zero and hence confined to an interval of the type in the one arc case. In a situation like this the multi-critical point is characterized by the eigenvalue density having extra zeros at both and . Reexamining (2.17) and (2.18) it appears that for the symmetric potential

(2.24) |

Hence in this case we have only one set of moments and the condition for being at the multi-critical point is the vanishing of the first of these.

This formalism obviously allows for a treatment of a more general situation where extra zeros accumulate at and extra zeros accumulate at . Such multi-critical models have been studied in Ref. [22]. We will however restrict ourselves to the traditional models.

The superiority of the moments defined in (2.17) and (2.18) as compared to the coupling constants will become even more clear when we consider the double scaling limit in Section 4.1. By then it will also become evident why these new moments are more convenient than those originally introduced in Ref. [15].

## 3 The iterative procedure

Our iterative solution of the loop equation results in a certain representation of the free energy and the correlators in terms of the moments. In this section we describe the structure of and and show that the iterative procedure can be conveniently formulated by referring to the eigenvectors of a linear operator, . We prove that the genus contribution to the -loop correlator depends at most on lower moments ( for the partition function) for . We perform explicit calculations up to genus two.

### 3.1 The structure of and

Our iterative procedure of solving the loop equation results in the following representation of the genus contribution to the free energy

(3.1) |

where is the distance between the endpoints of the support of the eigenvalue distribution, the brackets denote rational numbers and , and are non-negative integers. The indices take values in the interval and the summation is over sets of indices. In particular depends on at most moments. Furthermore, since nothing allows us to distinguish between and , must be invariant under the interchange of the two. Hence one gets

(3.2) |

There are certain restrictions on the integers which enter Eq. (3.1). Let us denote by and the total powers of ’s and ’s, respectively, i.e.

(3.3) |

Then it holds that

(3.4) |

and

(3.5) | |||||

(3.6) |

The relation (3.5) follows from the fact that the partition function is invariant under simultaneous rescalings of and the eigenvalue density, ; , . The relation (3.6) follows from the invariance of under rescalings of the type , . Finally the following inequality must be fulfilled:

(3.7) |

This requirement becomes more transparent when we consider the double scaling limit in Section 4.1. In combination with Eq. (3.6) it gives

(3.8) |

To explain the structure of , let us introduce the basis vectors and :

(3.9) | |||||

(3.10) |

where

(3.11) | |||||

(3.12) |

It is easy to show for the operator defined by Eq. (2.16) that

(3.13) | |||||

(3.14) |

and that the kernel of is spanned by .

Since
can be obtained from according to Eq. (2.7),
the representation (3.1) implies^{4}^{4}4 A similar formula is proven in Ref. [15] for a different
definition of the moments and of the basis vectors.

(3.15) |

We do not add any multiple of or . Doing so would contradict the boundary condition for since this behaviour was already obtained for genus zero. We note that this structure of is in agreement with the assumption [13] that is analytic in the complex plane except for a branch cut on the real axis.

The coefficients are of the same structure as and the relation (3.4) still holds. However in this case the indices take values in the interval . Hence depends on at most moments. The invariance of the partition function under the rescalings described above has the following implications for the structure of :

(3.16) | |||||

(3.17) |

We also have an analogue of (3.7) for . It reads

(3.18) |

Again we refer to Section 4.1 for further explanations. However, we note that combining (3.17) and (3.18) one gets again the bound (3.8) on .

As was the case for , must be invariant under the interchange of and . This means that must appear from by the replacements , . (We note that we do not have a relation like (3.2) for the ’s.) Furthermore, must be an odd function of for a symmetric potential. This implies that .

That the structure of and actually is as described in this section can be proven by induction. We will not go through the proof here. Instead we refer to Refs. [15, 19]. In the latter reference a formula somewhat similar to (3.15) was proven for the complex matrix model. However the strategy of the proof will be evident from Sections 3.2 and 3.3 where we describe the iterative procedures which allow us to calculate and for any starting from .

### 3.2 The iterative procedure for determining

According to Eq. (2.12), we need to calculate in order to start the iterative procedure. To do this we write the loop insertion operator as

(3.19) |

where

(3.20) |

The derivatives and can be obtained from (2.14) and read

(3.21) |

Using the relation

(3.22) |

one finds [23]

(3.23) |

This enables us to find and we see that it is of the form (3.15) with

(3.24) | |||||

(3.25) |

Carrying on the iteration process is straightforward. In each step one must calculate the right hand side of the loop equation (2.12). Decomposing the result obtained into fractions of the type , allows one to identify immediately the coefficients and .

To calculate it is convenient to write the loop insertion operator as

(3.26) |

where

(3.27) | |||||

The derivatives and are given by (3.21) and the function was defined in (3.11). Of course just appears from by the replacements and . We note that there is no simplification of the algorithm in the case of the symmetric potential. We can only put at the end of the calculation. This complication stems of course from the fact that we have to keep the odd coupling constants in the loop insertion operator until all differentiations have been performed. Only hereafter they can be put equal to zero. The same is not true in the double scaling limit however. We will come back to this later.

By taking a closer look at the loop insertion operator (3.26) and bearing in mind the results (3.23), (3.24) and (3.25), it is easy to convince oneself that and depend only on and via and have the structure shown in equation (3.1). Furthermore it appears that depends on at most parameters. The results for obtained with the aid of Mathematica read

(3.28) |

The genus two contribution to is now determined by Eq. (3.15).

### 3.3 The iterative procedure for

In this section we present an algorithm which allows one to determine , as soon as the result for is known. The strategy consists in writing the basis vectors and as derivatives with respect to . It is easy to verify that the following relations hold

(3.29) | |||||

(3.30) | |||||

(3.31) | |||||

(3.32) |

Combining this with the results (3.24) and (3.25) one immediately finds

(3.33) |

which coincides with the expression of Ref. [15].

In the general case things are not quite as simple. The basis vectors can not be written as total derivatives. This is of course in accordance with the fact that the and coefficients now have a more complicated dependence on the potential (cf. Section 3.2 ). However, a rewriting of the basis vectors allows one to identify relatively simply as a total derivative. In the case of this rewriting reads

(3.34) | |||||

where should be written as

(3.35) | |||||

(3.36) |

The basis vector should of course still be written as in (3.29). The rewriting of the ’s is analogous to that of the ’s. It can be obtained by performing the replacements and in the formulas above.

By means of these rewritings we have been able to determine . The result reads

(3.37) | |||||

It is obvious from the formulas above that depends for a non-symmetric potential on at most moments. Furthermore, for a symmetric potential, is a sum of two identical terms and depends on only at most moments. A consequence of this doubling for the double scaling limit will be discussed in Section 4.3.

## 4 The double scaling limit

It is easy to determine which terms in the explicit expressions for and determined in the previous section that contribute in the double scaling limit. However it is rather time consuming to determine and away from the double scaling limit. In this section we develop an algorithm which gives us directly the result in the double scaling limit. Using this algorithm we calculate the correlators and the partition function explicitly up to genus four and describe their general structure.

### 4.1 Multi-critical points

Let us consider first the case of the symmetric potential. We hence have and for all values of . As is mentioned earlier, the multi-critical point is characterized by the eigenvalue density having extra zeros accumulating at both and , and the condition for being at this point is the vanishing of the first moments. For the multi-critical model the double scaling limit of the correlators is obtained by fixing the ratio of any given coupling and, say , to its critical value and setting

(4.1) | |||||

(4.2) |

The moments then scale as

(4.3) |

Furthermore, it is well known that the genus () contribution to the free energy has the following scaling behaviour

(4.4) |

Bearing in mind that the structure of is as shown in Eq. (3.1), one finds that the following relation must hold

(4.5) |

Since the free energy should look the same for all multi-critical models, we have

(4.6) |

(4.7) |

We already know from the analysis in Section 3.1 that the equality sign must hold in (4.6). Only terms for which and for which the equality sign holds also in (4.7) will contribute in the double scaling limit. From Eq. (3.6) it follows that these terms will have

(4.8) |

A similar analysis can be carried out for the generating functional. Here it is known that the genus contribution to has the following scaling behaviour