Chains of KP, Semi-infinite 1-Toda Lattice Hierarchy and Kontsevich Integral

There are well-known constructions of integrable systems which are chains of infinitely many copies of the equations of the KP hierarchy ``glued'' together with some additional variables, e.g., the modified KP hierarchy. Another interpretation of the latter, in terms of infinite matrices, is called the 1-Toda lattice hierarchy. One way infinite reduction of this hierarchy has all solutions in the form of sequences of expanding Wronskians. We define another chain of the KP equations, also with solutions of the Wronsksian type, which is characterized by the property to stabilize with respect to a gradation. Under some constraints imposed, the tau functions of the chain are the tau functions associated with the Kontsevich integrals.


Introduction.
This paper was motivated by the following arguments. There are well-known chains of infinitely many copies of the equations of the KP hierarchy "glued" together with some variables, like, e.g., modified KP (see Eqs(1a,b,c) below). The latter is a sequence of dressing operators of the KP hierarchy {ŵ N } along with "gluing" variables {u N }. All these variables make a large integrable system. The chain (1a,b,c) has another interpretation, in terms of infinite matrices, which is called the 1-Toda lattice hierarchy (see [1][2][3]). There exist different reductions of this chain, e.g., modified KdV, or another reduction, a semi-infinite chain for which allŵ N with negative N are trivial,ŵ N = 1, andŵ N with positive N are P N ∂ −N where P N is an Nth order differential operator (the corresponding matrices of the 1-Toda lattice hierarchy also are semi-infinite). It can be shown (see below) that all the solutions are sequences of well-known Wronskian solutions to KP, eachŵ N being represented by a determinant of Nth order. Every next determinant is obtained from the preceding one by an extension of the Wronskian when a new function is added to the existing ones.
There is another situation where one deals with a sequence of Wronskian solutions of increasing order. This time the Wronskians are not obtained by a successive extension. The rule is more complicated. We talk about the so-called Kontsevich integral [4][5][6] which has its origin in quantum physics. This is an integral over the group U(N) which is a function of a matrix, invariant with respect to the matrix conjugation, i.e., a function of eigenvalues λ i of the matrix. The main fact about the Kontsevich integral is that it is a tau function of the KP hierarchy of the Wronskian type in variables t i = k λ −i k . The dimension of the Wronskian is N. The sequence of Wronskians has, in a sense, a limit when N → ∞. More precisely, this is a stable limit. There is some grading and the terms of a fixed weight stabilize when N → ∞: they become independent of N when N is large enough. The stable limit belongs to the nth reduction of KP (nth GD) and, besides, satisfies the string equation. The question we try to answer here is whether the sequence of Kontsevich tau functions is interesting by itself, not only by its limit. Is it possible to complete it with "gluing" variables to obtain a chain of related KP equations similar to (1a,b,c)? The answer is positive (see Sect. 2.1, Eqs(a,b,c,d)). Unfortunately, we do not know a matrix version of this chain like that of the 1-Toda lattice hierarchy.
Thus, in this paper we define the "stabilizing" chain of KP, study its solutions and demonstrate that they are exactly those which are represented by the Kontsevich integral. In Appendix we briefly, skipping all the calculations, show the way from the Wronskian solutions to the Kontsevich integral. This is actually the conversion of Itzykson and Zuber's [5] reasoning, and the reader can find the skipped detail there 2 I am thankful to H. Aratyn discussions with whom were very helpful.

1.1.
Recall some basic facts about the modified KP and 1-Toda lattice hierarchy (see [3]). The modified KP hierarchy is a collection of the following objects (∂ k = ∂/∂t k , ∂ = ∂ 1 ): and {u N }, N ∈ Z and relations: Notice that multiplying Eq.(1.c) by (∂ + u N ) −1 on the right and taking the residue we get an equivalent form of this equation: Let us construct a both way infinite matrix W with elements Then a proposition holds: Proposition (see [3]  where L = WΛW −1 , Λ is the matrix of the shift: (Λ) ij = δ i,j−1 and the subscript "−" symbolizes the strictly lower triangular part of a matrix.
Suppose we have a special solution such thatŵ N = 1 and u N −1 = 0 when N ≤ 0. Let P N =ŵ N ∂ N . Then P 0 = 1, Eq.(1.a) implies (∂ + u N )P N = P N +1 and therefore P N is an Nth order differential monic operator when N > 0.
The matrix W is a direct sum of two semi-infinite blocks, one is the unity and one is (W ij ) with i and j ≥ 0. We shall show that all the solutions are simply the well-known Wronskian solutions of KP. Proof. Suppose, y i 's are already constructed for i < N −1 (if N = 1, nothing is supposed). Then P N −1 y i = 0 and P N y i = 0 for i = 0, ..., N − 2 since (∂ + u N −1 )P N −1 = P N . The kernel of the operator P N is N-dimensional, therefore there is one more function y N −1 independent of y 0 , ..., y N −2 such that P N y N −1 = 0.
First of all, let us prove that if a function, in this case y N −1 but this is a general fact, belongs to the kernel of P N then so does (∂ k − ∂ k )y N −1 . We have The middle term vanishes since P N y N −1 = 0 and (L k N ) + is a differential operator. Thus, The coefficients A ki do not depend on t 1 . We have and, by virtue of the linear independence of y i as functions of t 1 , (1.

2.1)
This is the compatibility condition of the equations We can find m i , with m N −1 = 1, and then Proof. In one way, the proposition is almost proved by the preceding analysis. It only remains to notice that P N =ŵ N ∂ N whereŵ N is given by Eq.(1.3.1) is the monic differential operator with the kernel spanned by y 0 , ..., y N −1 , that (∂ + u N )P N y N = 0 and P N y The converse follows from the fact thatŵ N given by (1.3.1), as it is well known, is a dressing operator of the KP hierarchy, the operators (∂ + u N )P N where u N = − ln(W N +1 /W N ) and P N +1 have the same kernel spanned by y 0 , ..., y N and, therefore, coincide.
Subtracting the second row divided by z from the first one, then the third divided by z from the second one etc, we obtain zeros in the last column except the last element which is 1. In the ith row there will be elements y we see that all the above example have a form y i (t) = c ik p k (t). Conversely, any series of this form has the property (∂ k − ∂ k )y i = 0 since the Schur polynomials have it, as it is easy to see. Apparently, this is, in a sense, the most general form of such functions.
2.1. Definition. The stabilizing chain is a collection of the following objects: and u N , v N , N = 1, 2, 3, ... , and relations: The equations (c) and (d) also can be written as It can happen that starting from some term u N = v N +1 and allŵ N are equal, the chain stabilizes. Then w N N = 0. Also it can happen that the chain contains constant segments and after that again begins to change. These segments can be just skipped. We shall assume that all w N N = 0. Nevertheless, some tendency to stabilization remains, as we shall see below. A grading will be introduced so that all quantities will be sums of terms of different weights. We shall see that terms of a given weight stabilize, the greater is the weight the later the stabilization occurs. The stabilization is actually the most important feature of this chain allowing one to consider the stable limits when N → ∞. This is used, e.g., in the Kontsevich integral.
The chain is well defined if one proves that 1) the right-hand side of (b) is a ΨDO of the form N 1 a j ∂ −i , 2) vector fields ∂ k defined by (b), (c) and (d) respect the relation (a), 3) vector fields ∂ k commute.
Eq. (b) defines a copy of the KP hierarchy for each n. It is clear that the operator in the Now, one has to prove that The last two terms are transformed as: Taking into account (a) and the sum of all the terms is zero. Before we prove 3), the relations (c) and (d) will be presented in a different form.

Lemma. The following remarkable formula holds
where w is a function. Proof. Corollary.
Notice that the relation (a) implies (∂ + v N +1 )w N +1,N +1 = 0 where (∂ + v N +1 )w N +1,N +1 is understood as a result of action of the operator (∂ + v N +1 ) on the function w N +1,N +1 (not a product!). Indeed, this is the coefficient in ∂ −N −1 of the expression in the r.-h.s. while the l.-h.s. does not contain this term. Thus, Subtracting (c ′ ) and (d ′ ) we also have These two equations are equivalent to (c) and (d).
An alternative way to get (2.2.2) is the following. Eq.
3. Solutions to the chain. Let y 0N , ..., y N −1N be a basis of the kernel of the differential operator P N =ŵ N ∂ N :

3.1.
Lemma. Passing if needed to linear combinations of y iN with coefficients depending only on t 2 , t 3 , ..., one can always achieve which belong to the kernel of (∂ + u N )P N are linearly independent, otherwise there would be a linear combination of y i,N +1 belonging to the kernel of P N +1 which is constant (with respect to t 1 = x) while we know that P N +1 1 = w N +1,N +1 = 0 by assumption. Hence, at least one of these functions does not belong to ker P N , let it be y ′ N,N +1 : P N y ′ N,N +1 = 0. Since all P N y ′ i,N +1 belong to the 1-dimensional kernel of ∂ + u N , there must be constants a i such that P N (y ′ i,N +1 − a i y ′ N,N +1 ) = 0. (When we speak about constants, we mean constants with respect to t 1 = x depending, maybe, on higher times). Thus (y i,N +1 − a i y N,N +1 ) ′ form a basis of the kernel of P N . There exist their linear combinations (y (1) i,N +1 ) ′ coinciding with y iN : (y (1) i,N +1 ) ′ = y iN where i = 0, ..., N − 1. This yields ∂(∂ k − ∂ k )y (1) i,N +1 = 0 and (∂ k − ∂ k )y (1) i,N +1 = c ki =const. As in the Lemma 1.2, we can prove that (∂ k − ∂ k )y (1) i,N +1 ∈Ker P N +1 , therefore c ki = 0 since constants do not belong to the kernel. It remains to consider y N,N +1 . Since (∂ k − ∂ k )y N,N +1 = A i y i,N +1 , the same reasoning as in the Lemma of Sec.

will do the rest.
3.2. Proposition. All solutions to the chain (a-d) have the following structure. Let y iN where N = 1, 2, ... and i = 0, ..., N − 1 be arbitrary functions of variables t 1 = x, t 2 , ... satisfying the relations ∂ k y iN = ∂ k y iN (i) and (3.

2.3)
Proof. In one way, the proposition follows from the analysis of the preceding subsection. Indeed, P N =ŵ N ∂ N given by Eq.(3.2.1) is the unique differential monic operator having the kernel spanned by y 0N , ..., y N −1N , the latter always can be assumed satisfying the relations (i) and (ii). Further, u N = −∂ ln P N ∂y N,N +1 and v N +1,N +1 = −∂ ln P N +1 1 which easily yields Eqs (

.2) and (3.2.3) present a solution to the chain (a − d).
Indeed, KP equations (b) can be obtained in a standard way: differentiatingŵ N ∂ N y iN = 0 with respect to t k one gets The operator (L k N ) −ŵN ∂ N has an order less than N. On the other hand, this is a differential operator since it is equal toŵ N ∂ N − (L k N ) +ŵN ∂ N = P N − (L k N ) + P N . Thus, the differential operator (∂ kŵN )∂ N + (L k N ) −ŵN ∂ N of order less than N has an N-dimensional kernel and must vanish.
The operators (∂ + u N )P N ∂ and (∂ + v N +1 )P N +1 have the same kernels spanned by y 0,N +1 , ..., y N,N +1 , 1, therefore they coincide. We have the equation (a). From Finally, applying the operator ∂ k to (a) and equating terms of zero degree in ∂, we obtain The last two equations are equivalent to (c) and (d) which completes the proof.

Solutions in the form of series in Schur polynomials . Stabilization.
Recall that the Schur polynomial are defined by the equation A grading can be introduced being prescribed that the variable t i has the weight i, the weight of z is −1. Then the polynomial p k (t) is of weight k. It is easy to verify that the Schur polynomials have properties which can be obtained from the definition. Let  Proof. The diagonal terms of the determinant are p m 0 , p m 1 , ..., p m N−1 . All the terms of the determinant are of equal weights, namely, m 0 + m 1 + ... + m N −1 = l. Let us consider a determinant of weight l and prove that all m i with i ≥ l vanish unless the determinant vanishes. Suppose that there is some i ≥ l such that m i = 0. The elements of determinant which are located in the ith column above p m i have the following subscripts: m 0 + i, m 1 + i − 1, ..., m i−1 + 1. Together with m i , there are i + 1 non-zero integers with a sum m 0 + m 1 + ... Appendix. From the stabilizing chain to the Kontsevich integral, overview.
A1. It is well-known that the so-called Kontsevich integral which originates in quantum field theory is a tau function of the type (3.3.3) with some special coefficients c (i) m . We briefly sketch here the way from the general solution (3.3.3) to the Kontsevich integral if two additional requirements are imposed: the stable limit of τ N must belong to an nth restriction of KP hierarchy (the nth GD) and satisfy the string equation. All the skipped detail of calculations can be found in the article by Itzykson and Zuber [5] on which we base our presentation. The only difference is that we do this in reverse order: not from matrix integrals to tau functions (3.3.3) but vice versa. It is interesting to see what kind of reasoning and motivation could lead one from integrable systems to matrix integrals of the type studied by Kontsevich and to make sure that nothing essential is lost. Thus, contrary to a tradition, the Kontsevich integral appears only in the very last lines of the article, v kontse, which is Russian for "at the end".
First, we need to make the stable limit of τ N belonging to the nth restriction of KP which is equivalent to independence of the nth time, t n .
Usually, when one wishes to make a tau function (3.3.2) independent of t n , one requires that ∂ n y iN = α iN y iN where α iN are some numbers. Then y iN = exp(α iN t n )y iN (0) where y iN (0) does not depend on t n , and τ N = exp(a 0N + ... + a N 1 ,N )t n · τ N (0) where τ N (0) does not depend on t n . The exponential factor can be dropped since a tau function is determined up to a multiplication by an exponential of any linear combination of time variables. Now, in the problem we are talking about, we do not necessarily wish to make all τ N independent of t n , only their stable limit. Then it suffices instead of the "horizontal quasiperiodicity" ∂ n y iN = y (n) iN = α in y iN to require the "vertical periodicity" ∂ n y iN = y i+n,N . In terms of the series (3.3.1), this means that The proof is in [5]. The idea is clear: when a row which is not one of n last rows is differentiated then the resulting determinant has two equal rows. If we consider only terms of a fixed weight, then they depend only on a minor of a fixed size in the upper left corner for all N large enough, and these term vanish though the whole determinant does not.
A.2. Looking at the determinant in (3.3.3) one can recognize primitive characters of the group GL(n) or U(n) where and ǫ k are the eigenvalues of a matrix for which this character is evaluated (see [7]); it is supposed that m 0 ≥ m 1 ≥ ... ≥ m N −1 . The latter can always be achieved by a permutation and relabeling of indices. It is not easy to understand why τ -functions happen to be related to characters (a good explanation of this fact can lead to new profound theories). However, we can extract lessons from this relationship. First of all, the τ -function is given as an expansion in a series in characters. Hence it can be considered as a function on the unitary or the general linear group invariant with respect to the conjugation. In the end we will have an explicit formula giving this function which is the Kontsevich integral.
Secondly, the benefit of the usage of variables ǫ k instead of t i is obvious: the elaborated techniques of the theory of characters can be applied to the τ -function as well. We shall use also an inverse matrix with the eigenvalues λ i = ǫ −1 i . Thus, we have This change of variables is called the Miwa transformation.
It is easy to show that p l (t) = l 1 +...+l N =l This is the Newton formula expressing complete symmetric functions of variables ǫ k in terms of sums of their powers, t i . Introduce a notation |g 0 (λ), ..., g N −1 (λ)| = det(g j (λ i )).
Then it can be proven that A.3. Up to this point there were no restrictions imposed on the coefficients c (k) m except the periodicity (A.1.1). Now we try to satisfy the string equation (see, e.g., [9] or [10][11]). The string equation is closely related to the so-called additional symmetries of the KP hierarchy (each equation of the hierarchy provides a symmetry for all other equations, they are the main symmetries, while an additional symmetry is not contained in the hierarchy itself). The string equation is equivalent to the fact that τ does not depend on the additional variable t * −n+1,1 . However, it is known that the derivative ∂ * −n+1,1 is defined not uniquely: there is still a possibility for gauge transformations and for a shift of variables t i → t i + a i .
First of all, the operator ∂ * −n+1,1 acts on τ , as the operator This has to be expressed in terms of new variables, λ's or ǫ's. The last two terms are not important: an arbitrary linear term in t i and a constant can be added, this is precisely a gauge transformation. The result is If we recall that f 0 = f n then for the first term we have if and only if j a j jλ j−n = λ, i.e., j a j λ j = λ n+1 /(n + 1).
It is possible to perform some scaling transformation λ → aλ, D → a −1 D. This is not so important, but just in order that our formulas coincide with those in [5] we take a = n 1/(n+1) . Then D = λ + n − 1 2nλ n + where n.l.(Z + Λ) n+1 symbolizes all terms of degree higher than 1 in the expansion of (Z + Λ) n+1 in powers of Z while quad.(Z + Λ) n+1 stands for the quadratic term; Λ is a matrix with eigenvalues λ k . The expression (A.3.7) is called the Kontsevich integral (more precisely, its generalization from n = 2 to any n).