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Supersymmetric Matrix Models and the Meander Problem

We consider matrix-model representations of the meander problem which describes, in particular, combinatorics for foldings of closed polymer chains. We introduce a supersymmetric matrix model for describing the principal meander numbers. This model is of the type proposed by Marinari and Parisi for discretizing a superstring in D=1 while the supersymmetry is realized in D=0 as a rotational symmetry between bosonic and fermionic matrices. Using non-commutative sources, we reformulate the meander problem in a Boltzmannian Fock space whose annihilation and creation operators obey the Cuntz algebra. We discuss also the relation between the matrix models describing the meander problem and the Kazakov-Migdal model on a D-dimensional lattice.


Introduction
Matrix models is a standard tool for describing discretized random surfaces (or, equivalently, strings) [1]. A supersymmetric extension of this construction was first proposed by Marinari and Parisi [2] and studied for the D = 1 dimensional target space.
We introduce in the present paper supersymmetric matrix models in the D = 0 dimensional target space which differ from the Hermitean supermatrix models of Ref. [3] and are a version of the Marinari-Parisi construction in D = 0. Our model deals with the "superfields" W a = (B, F ) ,W a = B † ,F (1.1) where a = 1, 2 while B and F are complex bosonic and fermionic (i.e. Grassmann valued) N × N matrices, respectively.
Since the propagators for both bosonic and fermionic matrices coincide: the supersymmetry reduces in D = 0 simply to rotations between the B-and F -components. The proper transformation reads δB =ǭF , δF = −ǫB , where ǫ andǭ are Grassmann valued.
Any potential, which is symmetrically constructed from the "superfields" (1.1), is supersymmetric so that contributions from the loops of the bosonic and fermionic matrix fields are mutually cancelled which is the key property of the supersymmetry. The simplest Gaussian supersymmetric potential reads We elaborate in this paper the technique for dealing with the supersymmetric matrix models on an example of the one which describes combinatorics of the meander numbers. This challenging combinatorial problem, which is described in Sect. 2, is not yet solved. The associated matrix model, which describes a physical problem of enumerating different ways of foldings of a closed polymer chain, is of a next level of complexity with respect to the Hermitean one-or two-matrix models, the multi-matrix chain and its multi-dimensional extension -the Kazakov-Migdal model [4].
We introduce in Sect. 2 the complex matrix model which is equivalent to the Hermitean one [5] in describing the meander numbers. We construct then the supersymmetric matrix model which describes the principal meanders. We discuss also the relation between the matrix models describing the meander problem and the Kazakov-Migdal model on a D-dimensional lattice.
Using non-commutative sources, we reformulate in Sect. 3 the meander problem as a problem of averaging in a Boltzmannian Fock space whose annihilation and creation operators obey the Cuntz algebra. The averaging expression is represented in the form of a product of two continued fractions.
The Appendix A contains a solution of the combinatorial problem of summing over words built up of unitary matrices, which is equivalent to the Kazakov-Migdal model with the Gaussian potential, via free random variables.
In the Appendix B we demonstrate how the equations of Sect. 3, which are obtained using the matrix-model representations, can be alternatively derived pure combinatorially.
We comment in the Appendix C on a possibility of solving the meander problem via free random variables. We show that this approach does not work for the meander problem since the variables are not free for this case so that the theorem of addition of free random variables is not applicable.

Matrix models for the meander problem
The meander problem is known to people working on Quantum Field Theory since the middle of the eighties from V. Arnold. The problem is to calculate combinatorial numbers associated with the crossings of an infinite river (Meander) and a closed road by 2n bridges. 1 Neither the river nor the road intersects with itself. These principle meander numbers, M n , obviously describe the number of different foldings of a closed strip of 2n stamps or of a closed polymer chain.
One can consider also a generalized problem of the multi-component meander numbers n which are associated with k closed loops of the road so that M n ≡ M (1) n . The results of a computer enumeration of the meander numbers are presented in Refs. [7,6] up to n = 12.

Hermitean matrix model for meanders
Meanders can be described by the following Hermitean matrix model [5]  where the integration goes over the N × N Hermitean matrices W a (a = 1, . . . , m) and φ. The logarithm in Eq. (2.1) leaves only one closed loop of the field φ. The coupling constant c is associated with the (quartic) interaction between W a and φ.
Expanding the generating function (2.1) in c and identifying the diagrams with the ones for the meanders, one relates the large-N limit of F N ×N (c) with the following sum over the meander numbers 2) The N → ∞ limit is needed to keep only planar diagrams as in the meander problem.
The RHS of Eq. (2.1) can be expressed entirely via the Gaussian averages of W 's. This leads to the following representation of the meander numbers: where the average over W 's is calculated with the Gaussian weight -the same as in (2.1). This formula can be proven by calculating the Gaussian integral over φ in Eq. (2.1), expanding the result in c and comparing with the RHS of Eq. (2.2). The factorization at large N is also used.
The principle meander numbers M n are given by Eq. (2.3) as the linear-in-m-terms, i.e. as linear terms of the expansion in m. This looks like the replica trick which suppresses higher loops of the field W .
The ordered but cyclic-symmetric sequence of indices a 1 , a 2 , . . . , a 2n−1 , a 2n is often called a word constructed of m letters. The average on the RHS of Eq. (2.3) is the meaning of a word. Thus, the meander problem is equivalent to summing the squares of all the words with the Gaussian meaning.
The Gaussian averages on the RHS of Eq. (2.2) can be represented, making the Wick pairing, via the Kronecker deltas: (2.4) The "planar permutations" means here that one should sum up over all the permutations of the indices a i 's which are consistent with the planarity. This is standard for the large-N limit.
To calculate the meander numbers, one should sum up the square of the RHS of Eq. (2.4) over a i 's as is prescribed by Eq. (2.3). This is a convenient practical way of calculating the meander numbers.

Complex matrix model for meanders
The combination of deltas on the RHS of Eq. (2.4) can be alternatively represented as the Gaussian average over the complex matrices: The generating function, associated with the representation of the meanders via the complex matrices, reads Here φ 1 and φ 2 are Hermitean while W a (a = 1, . . . , m) are general complex matrices.
Quite similarly to Eq. (2.1) where the Hermitean matrix φ can be represented in a diagonal form φ = diag Λ (1) , . . . , Λ (N ) , the matrices φ 1 and φ 2 in Eqs. (2.8), (2.9) can always be made diagonal: (2.10) This can be shown representing φ 1 and φ 2 as and absorbing the unitary matrices Ω 1 and Ω 2 by the transformation of W a : The measure dW † a dW a does not change under the transformation (2.12) since W a are general complex matrices.
It is convenient to introduce one more generating function (2.13) where only one component of W a , say the first one, enters the averaging expression. Differentiating the generating function (2.8) with respect to c and noting that all m components of W a are on equal footing, we get the relation between the two generating functions.
In order to show how the complex matrix model recovers the meander numbers, let us replace φ ij 1 or φ ij 2 in the numerator of Eq. (2.13) by N −1 ∂/∂φ ji 1 or N −1 ∂/∂φ ji 2 , respectively, and integrate by parts. Repeating this procedure iteratively, we get For m = 1 the formula which is analogous to Eq. (2.5), holds for the complex matrices. This results again in Eq. (2.6).
It is instructive to consider also the case when W is a fermionic Grassmann valued matrixá la Ref. [8] 2 . We shall denote the fermionic matrix as F and its conjugate asF . Then we get [8] 1 N tr FF n Gauss = 0 n = 2p (even) C p n = 2p + 1 (odd) . (2.19) Since each loop of the fermionic field is accompanied by a factor of (−1), we arrive at the sum rule n k=1 (−) k−1 M (k) n = 0 n = 2p (even) C 2 p n = 2p + 1 (odd) .

(2.20)
This is nothing but the second sum rule of Ref. [6].
Note that the trace of the square of a fermionic matrix vanishes because of the anticommutation relation imposed on the components. This is why we did not consider Hermitean fermionic matrices and used first a representation of meanders in terms of complex matrices to discuss fermionic representation of meanders. Fermionic matrix models are a natural representation of the notion of the signature of arch configurations of Ref. [6].

General matrix model for meanders
The consideration of the previous Subsection suggests the following representation of the meander numbers via a general complex matrix model which includes both bosonic and fermionic matrices.
Let us consider a general W a which involves both bosonic (complex) and fermionic (Grassmann) components: and Here m f emerges with the minus sign since fermion loops are always accompanied with the minus sign.
In order to prove Eqs. (2.25), (2.28), let us first note that Eq. (2.17) is extended to the general case of both bosonic and fermionic components as is the signature factor of the component W a .
Expanding the determinant (2.29) on the RHS of Eq. (2.22) in c 2 , we get the following representation where the signs of the fermionic components are transformed at the last step using Therefore, the order of matrices in Eq. (2.25) is chosen in the way to absorb the signature factors for the fermionic components.

Supersymmetric matrix model for principle meander
Having the representation (2.25) of meanders via general complex matrices (either bosonic or fermionic), we can utilize the idea of supersymmetry to kill the loops of the W -field instead of the replica trick. Let us consider the two-component W a whose first component is bosonic while the second one is a fermionic matrix: The generating function (2.22) equals zero for the supersymmetric model since all the loops of the B and F fields are mutually cancelled. One should use alternatively the generating function (2.26) which can be represented for the supersymmetric matrix model as Here S is explicitly given by  where we kept trace of the order of matrices how it appears from Eq. (2.25). The signs, which are essential for the fermionic components, have been transformed using the formula (2.32) with σ(a) being the signature factor of the component W a defined by Eq. (2.30).
Equation (2.37) is a nice representation of the principle meander numbers which looks more natural than the one based on the replica trick. A hope is that it will be simpler to solve the m = 2 supersymmetric model than a pure bosonic one at arbitrary m. How the representation (2.37) reproduces the principle meander numbers is illustrated by the Table 1 up to n = 4.
Alternatively, one can calculate The results are presented in the Table 2.
The analogous results for the pure bosonic case given by Eq. (2.16) at m = 2 are presented in the Table 3. The total number of nonvanishing terms on the RHS of Eq. (2.16) for the pure bosonic case, which we shall denote as # n , is given by the following generating function This formula is derived in the Appendix A using non-commutative free random variables.
These numbers describe the sum of the combinatorial numbers in the third column of the Tables 1 -3.

Relation to the Kazakov-Migdal model
Equation (2.39) is known from the solution of the Kazakov-Migdal model with the Gaussian potential [10]. There is the following reason for that. Suppose that the matrices W a are unitary instead of the general complex ones. Then one has   Here the loops represent the sequences of indices {a 1 , a 2 , . . . , a 2n−1 , a 2n }. The nonvanishing result is only when the loop is closed and encloses a surface of the vanishing minimal area, i.e. each link of the loop is passed at least twice. This is analogous to the so-called local confinement in the Kazakov-Migdal model.
The generating function (2.39) coincides with the following correlator in the Kazakov-Migdal model with the Gaussian potential on an infinite D-dimensional lattice: provided that 2D = m. The integration over the unitary matrices U µ (x) goes over the Haar measure.
The solution of the Kazakov-Migdal model with the Gaussian potential can be completely reformulated as a combinatorial problem of summing over all closed loops of a given length with all possible backtrackings (or foldings) included. Its solution [11] is given by Eq. (2.39).
By virtue of the Eguchi-Kawai reduction [12] 3 , the correlator (2.42) in the Kazakov-Migdal model on the infinite lattice is equivalent to that in the reduced model given by with m = 2D and the reduced action being Haar measure . (2.46) In order to prove the equivalence of Eqs. (2.44) and (2.46), we calculate the integral over dφ 1 and dφ 2 using Eq. (2.17) with W a substituted by U a : The determinants in the numerator and the denominator on the RHS of Eq. (2.46) obviously cancel before the averaging over U a 's. An analogous cancellation happens in Eq. (2.44) as well in spite of the fact that each of them is averaged over its own U a 's. The point is that the determinant (2.47) under the sign of averaging over U behaves at large N as a U-independent constant.
The representation (2.46) looks very similar to the generating function (2.13) of the meander numbers. The difference is that the average is over the unitary matrices in Eq. (2.46) and over the Gaussian complex matrices in Eq. (2.13).
We can interpolate between the two cases by modifying the weight for averaging over W 's along the line of Ref. [14]. Let us introduce Then the averaging over the Gaussian complex matrices is reproduced as α → 0 while the average over the unitary matrices is recovered as α → ∞ since the matrix W α is forced to be unitary as α → ∞.
We see, thus, that the words are the same both for the meander problem and for the Kazakov-Migdal model. The only difference resides in the meaning of nonvanishing words -it equals unity for the unitary matrices.

Representation via non-commutative variables
The set u a , u † a of non-commutative variables obey the Cuntz algebra It is convenient to consider them, respectively, as annihilation and creation operators in a Hilbert space with the vacuum |Ω which satisfies The completeness condition says that There are no more relations between the non-commutative variables.

Bosonic case
Let us construct the generating function for words via the non-commutative sources u a as where the average over a = 1, . . . , m matrices W a is with the Gaussian weight as before.
Then the generating function (2.13) for the meander numbers is given by The generating function G λ (u) obeys the Schwinger-Dyson equation which can be derived in a usual way, by shifting W . The cyclic symmetry of the trace implies Inserting here Eq. (3.6), we get We shall use for G λ (u) and G λ (u † ) the short-hand notation so that Eq. (3.6) can be rewritten as where the summation over repeated indices is implied here and below except when it is specially indicated. Using Eq. (3.1), one alternatively rewrites Eq. (3.10) as If u's were ordinary commutative variables, the solution to the quadratic equation (3.6) would be simple For m = 1 this is nothing but Wigner's semicircle law and one reproduces Eq. (2.18) by expanding in 1/λ.
A formal solution for the non-commutative variables can be obtained representing Eq. (3.6) as Iterations of this equation, as was found by Cvitanović [15], lead in the continued fraction (3.14) Expanding the RHS in 1/λ 2 , one gets where u 2 stands for u a u a . All possible planar combinations of u's appear to next orders (with unit coefficients) while the total number of terms to the order λ −2n−1 equals the Catalan number C n . Technically, it is more simple to derive the expansion of G λ (u) in 1/λ by direct iterations of Eq. (3.6). The substitution into Eq. (3.5) then recovers the lower meander numbers.
Equation (3.6) is in fact well-known for the Gaussian models. It is a consequence of the relation between the generating functionals for all planar graphs G λ (u) and for connected planar graphs which says [16,15] G (u) = W (λuG (u)) (3.17) while the cyclic symmetry gives W (λuG (u)) = W (λG (u) u) .

(3.18)
There is only one connected graph in the Gaussian case so that W (j) is quadratic It would be interesting to apply the theory of non-commutative free random variables [17], whose application to matrix models has been discussed recently in Refs. [18,19], for this problem.

General case
For a general complex matrix model when some components of W a are bosonic and some are fermionic, we define the generating functions u a 1 u a 2 · · · u a 2n−1 u a 2n 1 N tr W a 1W a 2 · · · W a 2n−1W a 2n Gauss (3.20) where the non-commutative sources u a are the same as before.

Equations (3.24) and (3.25) and the cyclic symmetry of the trace result in the equations
and The solution to Eqs. and with the solutionḠ and In order to find a formal solution to Eqs. (3.26) and (3.27) in the form of a continued fraction, let us rewrite them as whereū a is defined by Eq. (3.23). Iterations of this equations lead to the following analog of (3.14) (3.39) Here u andū interchange in the consequent lines of the continued fractions.
The quantities on the LHS's of Eqs. (3.30) and (3.31) read explicitly (3.40) and (3.41) The generating function (2.26) is determined by where m b and m f are the numbers of bosonic and fermionic components of W a , respectively, and λ 2 = 1/c. This formula follows from the fact that the LHS reproduces the contraction of indices as in Eq. (2.25) using Eqs. (3.1) and (3.2).

Supersymmetric case
or alternatively The equality sign between the two expressions on the RHS of Eq. (3.44) or Eq. (3.45) is due to the supersymmetry. This can be shown using the completeness condition (3.3) which takes in the supersymmetric case the form (3.46) Inserting this, say, between G λ andḠ † λ , we get Taking the vacuum expectation value of this formula and remembering that due to the supersymmetry, we prove the statement.
This representation of the generating function for the principle meander numbers is convenient for an iterative procedure.
To perform calculations order by order in c = 1/λ 2 , it looks reasonable to try to use the fact that there are only two non-commutative variables and to expand in v. This is a standard trick of dealing with two non-commutative variables in the Hermitean twomatrix model [20,21].
We introduce, therefore, the expansions where (−1) n in the expansion ofḠ λ is due to Eqs. (3.22) (3.23). We have λ involves v exactly 2n times. As we shall see below, for n odd , for n ≥ 2 even .
At each step one has to solve the equation with some A n whose solution is given by where p ≥ 1. Both formulas can be rewritten in a unique way via the contour integral We shall also introduce the obvious short-hand notation for RHS's of Eqs. (3.56) or (3.57) as the brackets G The functions A n can be found recursively from Eq. (3.54). Few lower ones read explicitly They obviously satisfy the equation and analogously which can be obtained substituting the expansion (3.51) into Eq. (3.30) (or Eq. (3.31)) and using the fact that G (n) due to Eq. (3.56).
Using Eq. (3.58), we get from Eq. (3.59) for G where we have used the fact that G (0) λ commutes with a bracket but not with v. There are no problems here with the order of brackets which act like in TeX: they immediately contract each other when it is possible. This contraction is associated with the summation (3.56) (or the integration (3.57)). The general G (n) λ contains C n (the Catalan number given by Eq. (2.5)) terms of this kind with alternating signs. Some rules for representing a general term can be formulated which resemble the Wick pairing of bilinear combinations of v's.
It is worth noting that most of the above formulas would look quite similar for the case of just 2 bosonic fields, i.e. of m = 2, while there will be no alternative signs for 2 bosons. Due to these minus signs, the supersymmetric case is somewhat simpler than the m = 2 one. Say, the leading-order terms v 4 are cancelled in Eq. (3.59) for A 2 and in Eq. (3.63) for G (2) λ . This is a reflection of the general property (2.19) of Gaussian averages of the fermionic matrices. It is the reason why the leading order term vanishes in Eq. (3.53) for even n. In addition some relations are imposed by the supersymmetry. The simplest one follows from the substitution of the expansion (3.51) into Eq. (3.48) which yields Only the diagonal terms survive here due to the definitions (3.1), (3.2).
In order to calculate M n , we substitute the expansion (3.51) into Eq. (3.45) which gives We can also rewrite the RHS as The first term on the RHS recovers M 1 = 1 and M 2 = 2 while the contribution of the next terms are controlled by Eq. (3.53).
Let us demonstrate of how this iterative procedure works by the explicit calculation up to n = 3 when at most G (3) λ is essential. We get from Eq. (3.63) We have introduced a new representation of the meander numbers via the general complex matrix model. This allowed us to consider the supersymmetric matrix model which includes both bosonic and fermionic matrices and describes the principal meanders. This model looks simpler than a pure bosonic model and one can try to solve it by using supersymmetry Ward identities.
Using non-commutative sources, we have reformulated the meander problem as that of averaging in a Boltzmannian Fock space whose annihilation and creation operators obey the Cuntz algebra and have shown the equivalence with the combinatorial approach based on the arch statistics. The averaging expression is represented in the form of a product of two continued fractions but further progress requires their better understanding as functions of the non-commutative variables. One of the possible approaches can be based on the standard procedure of disentangling via the path integral.
We have discussed also the relation between the matrix models describing the meander problem and the Kazakov-Migdal model on a D-dimensional lattice. The words are the same both for the meander problem and for the Kazakov-Migdal model while the only difference resides in the meaning of nonvanishing words. This relation could give a hint on how to solve the meander problem.
We have demonstrated how the solution of the Kazakov-Migdal model with the quadratic potential can be obtained using the theorem of addition of free random variables. We have shown that this approach does not work for the meander problem even within the interaction representation since the variables are not free for this case so that the theorem of addition is not applicable. This observation casts doubts that a solution of large-N QCD can be obtained in the language of a masterfield given by free random variables.
The supersymmetric matrix models of the type discussed in this paper for the meander problem are novel ones. It is worth studying them in connection with other physical applications, in particular, with a discretization of super-Riemann surfaces and superstrings.
where the matrices A i are Hermitean. The simplest nontrivial example is that of m = 2, when the LHS has for n = 2 six nonvanishing terms of the type of The RHS has six nonvanishing terms of the type of A 4 , A 2 B 2 , AB 2 A, B 4 , B 2 A 2 , BA 2 B, where we have used the notations A 1 ≡ A, A 2 ≡ B -the same as in the Table 3.
If we now introduce free random variables A i , such that since the vacuum average of each nonvanishing term equals one. The resolvent for each A i reads The inverse function to (A.4) is given by Then the master-field operator can explicitly be constructed as [18,19] A i (a, a † ) = a + 1 + 4(a † ) 2 − 1 where the RHS is understood as the expansion around zero.
The theorem of addition of free random variables states that if A i 's are free random variables and so they are representable as where f i 's are functions analytic around zero, then Using this theorem, we get for the m matrices The resolvent is given by the inverse function Equation (2.39) can be reproduced substituting z = 1/c and dividing by c.
As far as the Kazakov-Migdal model is concerned, Eq. (2.46) results in the following representation of the correlator (2.42) via the words built up of the m = 2D unitary matrices: Haar measure

.(A.11)
Since the meaning of each word is either one or zero, the square on the RHS is not essential. Equation (2.39) with m = 2D then yields which reproduces the solution of Ref. [10].
We discuss in the Appendix C that the approach, which is described here for the Kazakov-Migdal model, does not work for the meander problem since the random variables are not free in the latter case. The "multiplication" • can naturally be represented in the operator language as follows. Let us introduce m non-commuting variables u a (a = 1, . . . , m), one for each color in the arch configurations. Let us associate with each arch configuration a word made up of u a 's in a way, which is evident from the following example: ✓✏ ✓✏ ←→ u 1 u 1 u 2 u 2 , if u 1 ↔ and u 2 ↔ .
We have from Eq. (B.2) the following equation in terms of u a 's: The squares on the RHS of Eq. (C.6) are due to analogous contributions from the operators with tildes.
The nonvanishing RHS of Eq. (C.6) means that the variables A i Ã i are not free. This is, as is already mentioned, a consequence of the direct product of matrices which results in the product of A i and Ã i and leads, in turn, to the squares in Eq. (C.6). If there are no squares, the axiom of freeness is obviously satisfied. The same is true for the supersymmetric case as well.
The example of this Appendix shows that random variables are not necessarily free in (multi-) matrix models with complicated interaction even within the framework of the interaction representation. The simplest average for the meander problem which violates the axiom of freeness is that (C.5). For this reason, the theorem of addition with where C n are given by Eq. (2.5), can be used, quite similarly to the Appendix A, to describe the meander numbers only up to n = 3 but fails to reproduce M n for n ≥ 4.