FORMULAS FOR A_{n} AND B_{n}–SOLUTIONS OF WDVV EQUATIONS

S.M. NATANZON

Abstract The simplest non-trivial solutions of WDVV equations are A_{n} and B_{n}-potentials, which

describe metrics of K.Saito on spaces of versal deformation of A_{n} and B_{n}-singularities. These are some polynomials, which were known for n ⩽ 4. In this paper we find the potentials for all A_{n} and B_{n}-singularities. We give recurrente formula for coefficients of KP and n-KdV hierarchy.

Keywords: WDVV equations hierarchy, singularities.

1991 MSC: 81

1. Introduction

1. WDVV equations appeared in papers of E.Witten [W1], R.Dijkgraaf, E.Verlinde, H.Verlinde [DVV] as equations for primary free energy of two-dimensional topological field theory. The simplest non-trivial solutions of WDVV equations are A_{n} and B_{n}-potentials, which describe metrics of K.Saito on spaces of versal deformation of A_{n} and B_{n}-singula-rities. According to E.Witten [W2] the coefficients of A_{n}–potentials are intersection numbers of Mumford–Morita–Miller classes of a moduli space of spheres with punctures and n–spin structures.

The A_{n}-potential has the following description[D1]. Let us consider the space of polynomials M_{n} = {z^{n+1} + a_{1}z^{n−1} + ... + a_{n}|a_{i} ∈ ℂ}. The tangent space T_{p} the point p = p(z) ∈ M is a set of polinomials Q_{n} = {b_{1}z^{n−1} + ... + b_{n}|b_{i} ∈ ℂ}. Define on T_{p} a structure of a Frobenius algebra, considering that v_{1} ∗ v_{2} = v_{1} · v_{2}(mod dp/dz) is multiplication of the algebra and g(v_{1}, v_{2}) = Res_{z=∞} \frac{v_{1}·v_{2}}{dp/dz} its scalar product. Then M_{n} is a Frobenius manifold. It has the plane coordinats (t^{1}, ..., t^{n}) such that g(\frac{∂}{∂t^{i}} , \frac{∂}{∂t^{j}}) = δ_{i+j,n+1} and E(t^{i}) = \frac{n+2−i}{n+2}t^{i}, where the vector field E is given by E = ∑_{k=1}^{n} \frac{k+1}{n+1}a_{k} \frac{∂}{∂a^{k}} in coordinats \{a_{1}, ..., a_{n}\}. Moreover, there exist a polynom F such that \frac{∂}{∂t^{i}} ∗ \frac{∂}{∂t^{j}} = ∑_{k=1}^{n} \frac{∂^{3}F}{∂t^{i}∂t^{j}∂t^{n+1−k}} \frac{∂}{∂t^{k}}. This polynom is called A_{n}-potential.

According to [DVV], A_{n}-potential is the string solution of n-KdV (Gelfand-Dikii) hierarhy. According to [K], A_{n}-potentialis is the string solution of dispersionless (quasi-classical) limit of n-KdV hierarhy. It is follow directly from our reprezantation that this is the same polynomial. Moreover, we give a recurrent formulas for calculation of all these polynomials. Up to now A_{n} potential was know only for n ⩽ 4 [W2,D1,D2]. We also give recurrence formulas for coefficients of KP and n-KdV hierarchy. Results of J.-B.Zuber [Z] give a reduction of B_{n}–potentials to A_{n}–potentials.

We formulate the main result of the paper in paragraph 2. In paragraph 3 we prove auxiliary combinatorial lemma. It is used in paragraph 4 for representation of KP hierarhy as an equation on a Baker-Akhiezer function. We use a representation of the Baker-Akhiezer function by τ–function and describe KP hierarhy as an infinite system of differential equations on v = − log τ in paragraph 5. We give an analog of this representation for n-KdV hierarhy in paragraph 6 . Using this representation we prove the main theorem in paragraph 7.

Acknowledgement. The final part of this paper was written during the author stay at Max-Planck-Institut für Mathematik in Bonn. I would like to thank this institution for support and hospitality. I would like to thank B.Dubrovin and K.Vaninsky for fruitful discussions.

This work was carried out with the financial support by grants: RFBR 98-01-00612 and INTAS 96-0713.

2. Main theorems

For calculation of the A_{n−1}-potential we use combinatorial coefficients P_{n}(i_{1}...i_{m}), where n and s_{j} are natural numbers.Define

P_{n}(i) = n, P_{n}(s_{1}...s_{m}) = C_{n}^{m} − ∑_{q=1}^{m−1} P_{n}(s_{1}...s_{q})C_{n−q−(s_{1}+···+s_{q})}^{m−q},

where

0! = 1, C_{p}^{t} = \frac{p!}{t!(p − t)!} for p ⩾ t ⩾ 0 and C_{p}^{t} = 0 in another cases.

Let us assume x_{0} = 0. For q > 0 we recurrently define polynomials x_{−q} (x_{1}, ..., x_{n−1}) putting

x_{−q} = −\frac{1}{n} ∑_{m=2}^{∞} ∑ P_{n}(s_{1} … s_{m})x_{n−s_{1}} ⋯ x_{n−s_{m}},

where the second sum is taken over all natural numbers (s_{1}, ..., s_{m}) such that ∑_{i=1}^{m} s_{i} = q + n + 1 − m.

This polynomial has the form

x_{−q} = ∑_{m=1}^{∞} ∑ B_{qi_{1}···i_{m}}x_{i_{1}} ⋯ x_{i_{m}},

where B_{qi_{1},...,i_{m}} are constants and the second sum is taken over numbers (i_{1}, ..., i_{m}) such that 1 ⩽ i_{1} ⩽ ... ⩽ i_{m} ⩽ n − 1.

Define

A_{i_{1}···i_{m}} = −\frac{1}{i_{1}p}B_{i_{1}···i_{m}}

and p is the number of j such that i_{j} = i_{1}.

Consider

F_{A} = ∑_{m=3}^{∞} ∑ A_{i_{1}···i_{m}}x_{i_{1}} ⋯ x_{i_{m}},

F_{B} = ∑_{m=3}^{∞} ∑ A_{2i_{1}−1···2i_{m}−1}x_{i_{1}} ⋯ x_{i_{m}},

where the second sum is taken over all 1 ⩽ i_{1} ⩽ ... ⩽ i_{m} ⩽ n − 1.

Theorem 1. The polynomial F_{A} is a A_{n−1}–potential. The polynomial F_{B} is a B_{n−1}–potential. They satisfy the equations

∑_{γ=1}^{n−1} \frac{∂^{3}F}{∂x_{α}∂x_{β}∂x_{γ}} \frac{∂^{3}F}{∂x_{n−γ}∂x_{η}∂x_{ξ}} = ∑_{γ=1}^{n−1} \frac{∂^{3}F}{∂x_{ξ}∂x_{β}∂x_{γ}} \frac{∂^{3}F}{∂x_{n−γ}∂x_{η}∂x_{α}} , (α, β, η, ξ = 1, ..., n − 1),

\frac{∂^{3}F}{∂x_{1}∂x_{α}∂x_{β}} = δ_{α+β,n}, ∑_{j=1}^{n−1} d_{i} \frac{∂F}{∂x_{i}} = dF,

where d_{i} = 1 + \frac{1−i}{n}, d = 2 + \frac{2}{n} for F = F_{A} and d_{i} = 1 + \frac{1−i}{n−1}, d = 2 + \frac{1}{n−1} for F = F_{B}.

3. Combinatorial lemma

For natural s, i_{1}, ..., i_{n} and integer not negative j_{1}, ..., j_{n} we define P_{s} ( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ) by recurrence formula:

1)P_{s} ( \begin{matrix}i_{1} & … & i_{n} \\ 0 & … & 0 \end{matrix} ) = 0; 2)P_{s} ( \begin{matrix}i \\ j \end{matrix} ) = C_{s}^{j} for j > 0;

3)P_{s} ( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ) = \frac{1}{n!}C_{s}^{j_{1}+···+j_{n}} \frac{(j_{1} + ⋯ + j_{n})!}{j_{1}! ⋯ j_{n}!}−

∑_{q=1}^{n−1} P_{s} ( \begin{matrix}i_{1} & … & i_{q} \\ j_{1} & … & j_{q} \end{matrix} ) \frac{1}{(n − q)!}C_{s−(i_{1}+···+i_{q}+j_{1}+···+j_{q})}^{j_{q+1}+···+j_{n}} \frac{(j_{q+1} + ⋯ + j_{n})!}{j_{q+1}! ⋯ j_{n}!}

for (j_{1}, ..., j_{n}) ≠ (0, ..., 0).

Let [ \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ] be the set of all matrices, which appear from ( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ) by

permutation of columns. Let ∥ \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ∥ be the number of such matrices. Define

P_{s} [ \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ] = ∑ P_{s} ( \begin{matrix}a_{1} & … & a_{n} \\ b_{1} & … & b_{n} \end{matrix} ) ,

where the sum is taken by all

( \begin{matrix}a_{1} & … & a_{n} \\ b_{1} & … & b_{n} \end{matrix} ) ∈ [ \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ]

Lemma 1 Let m > 0, k > 0 and j_{n} ⩾ 1 for n ⩽ m. Then

P_{s} [ \begin{matrix}i_{1} & … & i_{m} & i_{m+1} & … & i_{m+k} \\ j_{1} & … & j_{m} & 0 & … & 0 \end{matrix} ] =

= \{ \begin{matrix}0, if s ⩾ i_{1} + ... + i_{m} + j_{1} + ... + j_{m}, \\ \frac{1}{k!} ∥ \begin{matrix}i_{m+1} & … & i_{m+k} \\ 0 & … & 0 \end{matrix} ∥ P_{s} [ \begin{matrix}i_{1} & … & i_{m} \\ j_{1} & … & j_{m} \end{matrix} ] , if s < i_{1} + ... + i_{m} + j_{1} + ... + j_{m}. \end{matrix}

Proof: Prove at first the lemma for m = 1 using induction by k. For m = k = 1

P_{s} [ \begin{matrix}i_{1} i_{2} \\ j_{1} 0 \end{matrix} ] = P_{s} ( \begin{matrix}i_{1} i_{2} \\ j_{1} 0 \end{matrix} ) + P_{s} ( \begin{matrix}i_{2} i_{1} \\ 0 j_{1} \end{matrix} ) =

\frac{1}{2}C_{s}^{j_{1}} − P_{s} ( \begin{matrix}i_{1} \\ j_{1} \end{matrix} ) · C_{s−(i_{1}+j_{1})}^{0} + \frac{1}{2}C_{s}^{j_{1}} − P_{s} ( \begin{matrix}i_{2} \\ 0 \end{matrix} ) · C_{s−i_{2}}^{0} = C_{s}^{j_{1}} − C_{s}^{j_{1}}C_{s−(i_{1}+j_{1})}^{0} =

= \{ \begin{matrix}0, & if s ⩾ i_{1} + j_{1}, \\ C_{s}^{j_{1}} = P_{s} [ \begin{matrix}i_{1} \\ j_{1} \end{matrix} ] , & if s < i_{1} + j_{1}. \end{matrix}

To prove the lemma for m = 1, k = N, we assume that it is proved for m = 1, k < N. If s ⩾ i_{1} + j_{1}, then

P_{1} [ \begin{matrix}i_{1} & i_{2} & … & i_{k+1} \\ j_{1} & 0 & … & 0 \end{matrix} ] = \frac{1}{k!}C_{s}^{j_{1}} ∥ \begin{matrix}i_{2} & … & i_{k+1} \\ 0 & … & 0 \end{matrix} ∥−P_{s} ( \begin{matrix}i_{1} \\ j_{1} \end{matrix} )·C_{s−(i_{1}+j_{1})}^{0}·\frac{1}{k!} ∥ \begin{matrix}i_{2} & … & i_{k+1} \\ 0 & … & 0 \end{matrix} ∥ = 0.

If s < i_{1} + j_{1} then

P_{1} [ \begin{matrix}i_{1} & i_{2} & … & i_{k+1} \\ j_{1} & 0 & … & 0 \end{matrix} ] = \frac{1}{k!}C_{s}^{j_{1}} ∥ \begin{matrix}i_{2} & … & i_{k+1} \\ 0 & … & 0 \end{matrix} ∥ − A · C_{s−(i_{1}+j_{1})}^{0} =

\frac{1}{k!} ∥ \begin{matrix}i_{2} & … & , i_{k+1} \\ 0 & … & 0 \end{matrix} ∥ P_{s} [ \begin{matrix}i_{1} \\ j_{1} \end{matrix} ] .

Thus the lemma is proved for m = 1.

To prove the lemma for m = N, we assume that it is proved for m < N. Then

P_{s} [ \begin{matrix}i_{1} & … & i_{m} & i_{m+1} & … & i_{m+k} \\ j_{1} & … & j_{m} & 0 & … & 0 \end{matrix} ] =

∑_{( \begin{matrix}α_{1} & … & α_{m} \\ β_{1} & … & β_{m} \end{matrix} )∈[ \begin{matrix}i_{1} & … & i_{m} \\ j_{1} & … & j_{m} \end{matrix} ]}(\frac{1}{(m + k)!}C_{s}^{β_{1}+···+β_{m}} \frac{(β_{1} + ⋯ + β_{m})!}{β_{1}! ⋯ β_{m}!}C_{m+k}^{k}·

∥ \begin{matrix}i_{m+1} & … & i_{m+k} \\ 0 & … & 0 \end{matrix} ∥ −

− ∑_{q=1}^{m} P_{s} ( \begin{matrix}α_{1} & … & α_{q} \\ β_{1} & … & β_{q} \end{matrix} ) \frac{1}{(m + k − q)!}C_{s−(α_{1}+···+α_{q}+β_{1}+···+β_{q})}^{β_{q+1}+···+β_{m}} \frac{(β_{q+1} + ⋯ + β_{m})!}{β_{q+1}! ⋯ β_{m}!}C_{m+k−q}^{k}

∥ \begin{matrix}i_{m+1} & … & i_{m+k} \\ 0 & … & 0 \end{matrix} ∥) = P_{s} [ \begin{matrix}i_{1} & … & i_{m} \\ j_{1} & … & j_{m} \end{matrix} ] \frac{1}{k!} ∥ \begin{matrix}i_{m+1} & … & i_{m+k} \\ 0 & … & 0 \end{matrix} ∥ −

P_{s} [ \begin{matrix}i_{1} & … & i_{m} \\ j_{1} & … & j_{m} \end{matrix} ] C_{s−(i_{1}+···+i_{m}+j_{1}+···+j_{m})}^{0} \frac{1}{k!} ∥ \begin{matrix}i_{m+1} & … & i_{m+k} \\ 0 & … & 0 \end{matrix} ∥ =

= \{ \begin{matrix}0, if s ⩾ i_{1} + ⋯ + i_{m} + j_{1} + ⋯ + j_{m}, \\ P_{s} [ \begin{matrix}i_{1} & … & i_{m} \\ j_{1} & … & j_{m} \end{matrix} ] \frac{1}{k!} ∥ \begin{matrix}i_{m+1} & … & i_{m+k} \\ 0 & … & 0 \end{matrix} ∥ , & if s < i_{1} + ⋯ + i_{m} + j_{1} + ⋯ + j_{m}.□ \end{matrix}

4. Equations for Baker-Akhiezer function

Consider the KP hierarchy. This is a condition of compatibility of an infinite system of differential equations

\frac{∂ψ}{∂x_{n}} = L_{n}ψ,  (1)

where

L_{n} = \frac{∂^{n}}{∂x_{1}^{n}} + ∑_{i=2}^{n} B_{n}^{i}(x)\frac{∂^{n−i}}{∂^{n−i}x_{1}} ,

and ψ is a function of type

ψ(x, k) = exp(∑_{j=1}^{∞} x_{j}k^{j})(1 + ∑_{i=1}^{∞} ξ_{i}k^{−i}),

(here k ∈ ℂ belong to some neighbourhood of ∞ and x = (x_{1}, x_{2}, ...) — is a finite sequence).

We use notation

∂_{i} = \frac{∂}{∂x_{i}} , ∂ = ∂_{1}.

A direct calculation gives

Lemma 2. Conditions of compatibility of (1) are

B_{s}^{t} = − ∑_{i=1}^{t−1} C_{s}^{i}∂^{i}ξ_{t−i} − ∑_{j=2}^{t−1} B_{s}^{j} ∑_{i=0}^{t−j−1} C_{s−j}^{i}∂^{i}ξ_{t−i−j},  (2)

∂_{n}ξ_{i} = ∑_{j=1}^{n+i−1} C_{n}^{j}∂^{j}ξ_{i+n−j} + ∑_{k=2}^{n} B_{n}^{k} ∑_{j=0}^{n−k} C_{n−k}^{j}∂^{j}ξ_{i+n−j−k}. □  (3)

In this case ψ is called a Baker-Akhiezer function.

Consider now the function

ln ψ(x, k) = ∑_{j=1}^{∞} x_{j}k^{j} + ∑_{j=1}^{∞} η_{j}k^{−j},

where

ξ_{j} = ∑_{n=1}^{∞} \frac{1}{n!} ∑_{i_{1}+···+i_{n}=j} η_{i_{1}} ⋯ η_{i_{n}}.

Lemma 3. Let 2 ⩽ t ⩽ s. Then

B_{s}^{t} = − ∑_{n=1}^{∞} ∑ P_{s} ( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ) ∂^{j_{1}}η_{i_{1}} ⋯ ∂^{i_{n}}η_{i_{n}},

where the second sum is taken over all matrices ( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ) such that i_{m} ⩾ 1, j_{m} ⩾ 1 and i_{1} + ⋯ + i_{n} + j_{1} + ⋯ + j_{n} = t.

Proof: By induction on t. For t = 2 according to (2), B_{s}^{2} = −s∂ξ_{1} = −P_{s}\binom{1}{1}∂η_{1}. To prove the lemma for t = N consider that it is proved for t < N. According to (2)

B_{s}^{t} = − ∑_{i=1}^{t−1} C_{s}^{i}∂^{i}( ∑_{n=1}^{∞} \frac{1}{n!} ∑_{i_{1}+···+i_{n}=t−i} η_{i_{1}} ⋯ η_{i_{n}})+

+ ∑_{j=2}^{t−1}(∑_{n=1}^{∞} ∑_{i_{1}+···+j_{n}=j} P_{s}( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} )∂^{j_{1}}η_{i_{1}} ⋯ ∂^{j_{n}}η_{i_{n}})·

( ∑_{i=0}^{t−j−1} C_{s−j}^{i}∂^{j}(∑_{n=1}^{∞} \frac{1}{n!} ∑_{i_{1}+···+i_{n}=t−i−j} η_{i_{1}} ⋯ η_{i_{n}})) =

= − ∑_{n=1}^{∞} ∑(\frac{1}{n!}C_{s}^{j_{1}+···+j_{n}} \frac{(j_{1} + ⋯ + j_{n})!}{j_{1}! ⋯ j_{n}!}−

∑_{q=1}^{n−1} P_{s}( \begin{matrix}i_{1} & … & i_{q} \\ j_{1} & … & j_{q} \end{matrix} )\frac{1}{(n − q)!}C_{s−(i_{1}+···i_{q}+j_{1}+···+j_{q})}^{j_{q+1}···j_{n}} \frac{(j_{q+1} + ⋯ j_{n})!}{j_{q+1}! ⋯ j_{n}!})∂^{j_{1}}η_{i_{1}} ⋯ ∂^{j_{n}}η_{i_{n}} =

= − ∑_{n=1}^{∞} ∑ P_{s}( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} )∂^{j_{1}}η_{i_{1}} ⋯ ∂^{j_{n}}η_{i_{n}},

where the second sums are taken over all matrices ( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ) such that i_{1} + ⋯ + i_{n} + j_{1} + ⋯ + j_{n} = t, i_{m} ⩾ 1, j_{m} ⩾ 0. According to lemma 1 it is possible to consider the last sum with j_{m} > 0 for all m. □

Lemma 4.

∂_{s}η_{r} = ∑_{n=1}^{∞} ∑ P_{s} ( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ) ∂^{j_{1}}η_{i_{1}} ⋯ ∂^{j_{n}}η_{i_{n}},

where the second sum is taken by all matrices ( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ) such that i_{m} ⩾ 1, j_{m} ⩾ 1 and i_{1} + ⋯ + i_{n} + j_{1} + ⋯ + j_{n} = r + s.

Proof: By induction on r. According to (3) and lemma 3 for r = 1

∂_{s}η_{1} = ∂_{s}ξ_{1} = ∑_{j=1}^{∞} C_{s}^{j}∂^{j}ξ_{s+1−j} + ∑_{k=2}^{s} B_{s}^{k} ∑_{j=0}^{s−k} C_{s−k}^{j}∂^{j}ξ_{1+s−j−k} =

∑_{j=1}^{s} C_{s}^{j}∂^{j}ξ_{s+1−j} − ∑_{k=2}^{∞}(∑_{n=1}^{∞} ∑_{i_{1}+···+j_{n}=k} P_{s}( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} )∂^{j_{1}}η_{i_{1}} ⋯ ∂^{j_{n}}η_{i_{n}})·

(∑_{j=0}^{s−k} C_{s−k}^{j}∂^{j}ξ_{1+s−j−k}) = ∑_{n=1}^{∞} ∑ P_{s} ( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ) ∂^{j_{1}}η_{i_{1}} ⋯ ∂^{j_{n}}η_{i_{n}},

where the second sum in the last formula is taken over all matrices ( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ) such that i_{1} + ⋯ + i_{n} + j_{1} + ⋯ + j_{n} = s + 1, i_{m} ⩾ 1, j_{m} ⩾ 0. According to lemma 1 in this sum it is sufficient to consider only matrices, where j_{m} > 0 for all m.

To prove now the lemma for r = N,we assume that it is proved for r < N. According to (3)

∂_{s}(∑_{n=1}^{∞} \frac{1}{n!} ∑_{i_{1}+···+i_{n}=r} η_{i_{1}} ⋯ η_{i_{n}}) = ∑_{j=1}^{s+r−1} C_{s}^{j}∂^{j}ξ_{s+r−j} + ∑_{k=2}^{∞} B_{s}^{k} ∑_{j=0}^{s−k} C_{s−k}^{j}∂^{j}ξ_{r+s−j−k}.

Thus according to lemma 3, lemma 1 and inductive hypothesis,

∂_{s}η_{r} = ∑_{j=1}^{∞} C_{s}^{j}∂^{j}(∑_{n=1}^{∞} \frac{1}{n!} ∑_{i_{1}+···+i_{n}=s+r−j} η_{i_{1}} ⋯ η_{i_{n}})+

+ ∑_{k=2}^{∞}( ∑_{i_{1}+···+j_{n}=k} P_{s}( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} )∂^{j_{1}}η_{i_{1}} ⋯ ∂^{j_{n}}η_{i_{n}})·

∑_{j=0}^{s−k} C_{s−k}^{j}∂^{j}(∑_{n=1}^{∞} \frac{1}{n!} ∑_{i_{1}+···+i_{n}=r+s−j−k} η_{i_{1}} ⋯ η_{i_{n}}) − ∂_{s}(∑_{n=2}^{∞} \frac{1}{n!} ∑_{i_{1}+···+i_{n}=r} η_{i_{1}} ⋯ η_{i_{n}}) =

= ∑_{n=1}^{∞} ∑ P_{s}( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} )∂^{j_{1}}η_{i_{1}} ⋯ ∂^{j_{n}}η_{i_{n}},

where the second sum in the last formula is taken over all matrices ( \begin{matrix}i_{1} & … & i_{n} \\ j_{1} & … & j_{n} \end{matrix} ) such that i_{1} + ⋯ + i_{n} + j_{1} + ⋯ + j_{n} = s + r, i_{m} ⩾ 1, j_{m} ⩾ 1. □

5. KP hierarchy

According to [DKJM] the Baker-Akhiezer function ψ is

ψ(x, k) = exp(∑ x_{j}k^{j})\frac{τ(x_{1} − k^{−1}, x_{2} − \frac{1}{2}k^{−2}, x_{3} − \frac{1}{3}k^{−3}, ...)}{τ(x_{1}, x_{2}, x_{3}, ...)}

for some function τ(x_{1}, x_{2}, ...). This gives a possibility to describe the KP hierarchy as an infinite system of differential equations on v(x, k) = − ln τ(x, k), see[N]. Indeed,

∑_{j=1}^{∞} η_{j}k^{−j} = ln ψ(x, k) − ∑_{j=1}^{∞} x_{j}k^{j} = −v(x_{1} − k^{−1}, x_{2} − \frac{1}{2}k^{−2}, ...) + v(x) =

∑_{n=1}^{∞} ∑_{i_{1}+···+i_{n}=j} \frac{(−1)^{n+1}}{n!i_{1} … i_{n}} ∂_{i_{1}} ⋯ ∂_{i_{n}}v(x)k^{−j}.

Therefore,

η_{r} = ∑_{n=1}^{∞} ∑_{i_{1}+···+i_{n}=r} \frac{(−1)^{n+1}}{n!i_{1} … i_{n}} ∂_{i_{1}} ⋯ ∂_{i_{n}}v.  (4)

Theorem 2. There exist universal rational coefficients

R_{r} ( \begin{matrix}s_{1} & … & s_{n} \\ t_{1} & … & t_{n} \end{matrix} ) , R_{ij} ( \begin{matrix}s_{1} & … & s_{n} \\ t_{1} & … & t_{n} \end{matrix} )

such that

η_{r} = \frac{1}{r}∂_{r}v + ∑_{n=1}^{∞} ∑ R_{r} ( \begin{matrix}s_{1} & … & s_{n} \\ t_{1} & … & t_{n} \end{matrix} ) ∂_{s_{1}}∂^{t_{1}}v ⋯ ∂_{s_{n}}∂^{t_{n}}v,  (5)
∂_{i}∂_{j}v = ∑_{n=1}^{∞} ∑ R_{ij} ( \begin{matrix}s_{1} & … & s_{n} \\ t_{1} & … & t_{n} \end{matrix} ) ∂_{s_{1}}∂^{t_{1}}v ⋯ ∂_{s_{n}}∂^{t_{n}}v,  (6)

where the second sums are taken over all matrices ( \begin{matrix}s_{1} & … & s_{n} \\ t_{1} & … & t_{n} \end{matrix} ) such that s_{m}, t_{m} ⩾ 1, and the sum s_{1} + ⋯ + s_{n} + t_{1} + ⋯ + t_{n} is equal r for (5) and i + j for (6). Moreover,

R_{ij} ( \begin{matrix}s_{1} & … & s_{n} \\ 1 & … & 1 \end{matrix} ) = \frac{j}{s_{1} ⋯ s_{n}} P_{i}(s_{1}...s_{n}).

Proof: By induction on k and i + j. For i + j = 2 the theorem is obvious. For r = 1 it follows from (4). To prove theorem for i+j = N and r = N −1, consider that it is proved for i+j < N and r < N − 1. Later we consider that s_{m}, t_{m} ⩾ 1 and σ_{n} = s_{1} + ⋯ + s_{n} + t_{1} + ⋯ + t_{n}. Then according to (4) and (6)

η_{r} = \frac{1}{r}∂_{r}v + ∑_{n=2}^{∞} ∑_{s_{1}+···+s_{n}=r} \frac{(−1)^{n+1}}{n!s_{1} ⋯ s_{n}} ∂_{s_{1}} ⋯ ∂_{s_{n}}v(x) =

\frac{1}{r}∂_{r}v + ∑_{n=1}^{∞} ∑_{σ_{n}=r} R_{r} ( \begin{matrix}s_{1} & … & s_{n} \\ t_{1} & … & t_{n} \end{matrix} ) ∂_{s_{1}}∂^{t_{1}}v ⋯ ∂_{s_{n}}∂^{t_{n}}v.

Thus according to (5), (6) and lemma 4

\frac{1}{j}∂_{i}∂_{j}v = ∂_{i}η_{j} − ∂_{i}(∑_{n=1}^{∞} ∑_{σ_{n}=j} R_{j} ( \begin{matrix}s_{1} & … & s_{n} \\ t_{1} & … & t_{n} \end{matrix} ) ∂_{s_{1}}∂^{t_{1}}v ⋯ ∂_{s_{n}}∂^{t_{n}}v) =

∑_{n=1}^{∞} ∑_{σ_{n}=i+j} P_{i} ( \begin{matrix}s_{1} & … & s_{n} \\ t_{1} & … & t_{n} \end{matrix} ) ∂^{t_{1}}η_{s_{1}} ⋯ ∂^{t_{n}}η_{s_{n}}−

∂_{i}(∑_{n=1}^{∞} ∑_{σ_{n}=j} R_{j} ( \begin{matrix}s_{1} & … & s_{n} \\ t_{1} & … & t_{n} \end{matrix} ) ∂^{t_{1}}∂_{s_{1}}v ⋯ ∂^{t_{n}}∂_{s_{n}}v) =

∑_{n=1}^{∞} ∑_{s_{1}+···+s_{n}+n=i+j} P_{i} ( \begin{matrix}s_{1} & … & s_{n} \\ 1 & … & 1 \end{matrix} ) ∂(\frac{1}{s_{1}} ∂_{s_{1}}v) ⋯ ∂(\frac{1}{s_{n}} ∂_{s_{n}}v)+

∑_{n=1}^{∞} ∑_{σ_{n}=i+j,t_{1}+···+t_{n}>n} R_{ij} ( \begin{matrix}s_{1} & … & s_{n} \\ t_{1} & … & t_{n} \end{matrix} ) ∂_{s_{1}}∂^{t_{1}}v ⋯ ∂_{s_{n}}∂^{t_{n}}v.

The obvious relation P_{i} ( \begin{matrix}s_{1} & … & s_{n} \\ 1 & … & 1 \end{matrix} ) = P_{i}(s_{1}...s_{n}) concludes the proof. □

Consequence 1. The system (6) is equivalent of KP hierarhy.

Remark. The first equation from (6) is the ”integrated over x_{1}” KP equation

∂_{2}^{2}v = \frac{4}{3}∂_{3}∂_{1}v − \frac{1}{3}∂_{1}^{4}v + 2(∂_{1}^{2}v)^{2}.

The relation (6) was first deduced in [ DN ] from the Hirota hierarchy [ DKJM ]. The method produces another formulas for coefficients R_{ij}. The equality of coefficients provides nontrivial combinatorial identities. □

Remark. From the consequence it follows that any formal solution of KP hierarhy is defined defined up to a constant by an arbitrary infinite set of series of one variable f_{i}(x_{1}) = ∂_{i}v|_{x_{2}=x_{3}=···=0} (i = 1, 2, ...).

6. Gelfand–Dikii hierarchy

According to [S], the set of solution of n–Gelfand–Dikii(n − −KdV) hierarchy bijectively corresponds to the set of solutions of KP hierarchy independent from x_{n}. In this case according to theorem 2

0 = ∂_{m}∂_{n}v = \frac{mn}{m + n − 1}∂_{n+m−1}∂v + ∑_{m=1}^{∞} ∑ R_{mn} ( \begin{matrix}s_{1} & … & s_{m} \\ t_{1} & … & t_{m} \end{matrix} ) ∂_{s_{1}}∂^{t_{1}}v ⋯ ∂_{s_{m}}∂^{t_{m}}v, (7)

where 1 ⩽ s_{j} ⩽ n + m − 2, t_{j} ⩾ 1. This gives recurrence formulas expressing ∂_{k}∂v for k > n via ∂_{r}∂v for r < n. Thus, we have relations

∂∂_{n+r}v = ∑_{m=1}^{∞} ∑ N_{1(n+1)}^{m} ( \begin{matrix}s_{1} & … & s_{m} \\ t_{1} & … & t_{m} \end{matrix} ) ∂_{s_{1}}∂^{t_{1}}v ⋯ ∂_{s_{m}}∂^{t_{m}}v,  (8)

where t_{j} ⩾ 1, s_{j} < n, ∑_{j=1}^{m}(s_{j} + t_{j}) = n + r + 1. Example. For n = 2 the system (8) is reducied to the K∂V hierarchy.

Comparing the systems (8) and (6), we find

Theorem 3. The n–Gelfand–Dikii hierarchy is equivalent to a system of differential equations in the form

∂_{i}∂_{j}v = ∑_{m=1}^{∞} ∑ N_{i_{j}}^{m} ( \begin{matrix}s_{1} & … & s_{m} \\ t_{1} & … & t_{m} \end{matrix} ) ∂_{s_{1}}∂^{t_{1}}v ⋯ ∂_{s_{m}}∂^{t_{m}}v,  (9)

where i, j ⩾ 1, 1 ⩽ s_{α} ⩽ n − 1, t_{α} ⩾ 1, ∑_{i=1}^{m}(s_{α} + t_{α}) = i + j and N_{ij}^{m} ( \begin{matrix}s_{1} & … & s_{m} \\ t_{1} & … & t_{m} \end{matrix} ) are

some universal rational coefficients. Examples. 1. For n = 3 the first equation from the system (9) is the Boussinesq equation

∂_{2}^{2}v = −\frac{1}{3}∂^{4}v + 2(∂^{2}v)^{2}.

2. For n = 4 the first equations of the system (9) are

∂_{2}^{2}v = \frac{4}{3}∂_{3}∂v − \frac{1}{3}∂^{4}v + 2(∂^{2}v)^{2},

∂_{3}∂_{2}v = −\frac{3}{2}∂_{2}∂^{3}v + 3∂_{2}∂v∂^{2}v,

∂_{3}^{2}v = −\frac{1}{4}∂_{3}∂^{3}v + \frac{1}{8}∂^{6}v + \frac{9}{8}(∂_{2}∂v)^{2} − \frac{9}{8}(∂^{3}v)^{2} − \frac{9}{4}∂^{4}v∂^{2}v + 3(∂^{2}v)^{3}.□

Remark The structure of the system (9) demonstrats that any formal solutions of n– Gelfand–Dikii equation is defined up to a constant by an arbitrary set of n − 1 series of one variable f_{i}(x_{1}) = ∂_{i}v|_{x_{2}=x_{3}=···=0} (i = 1, ..., n − 1).

According to [K] the dispersionless (quasiclassical) limit of n–Gelfand–Dikii hierarchy is given by n–Gelfand–Dikii hierarchy with out all monomials

R_{ij} ( \begin{matrix}s_{1} & … & s_{n} \\ t_{1} & … & t_{n} \end{matrix} ) ∂_{s_{1}}∂^{t_{1}}v ⋯ ∂_{s_{n}}∂^{t_{n}}v,

where ∑_{i=1}^{n} t_{j} > n.

Thus, from theorem 2 it follows

Consequence 1. Dispersionless (quasiclassical) limit of n–Gelfand–Dikii hierarchy is given by the infinite system

\frac{1}{j}∂_{i}∂_{j}v = ∑_{m=1}^{∞} ∑_{s_{1}+···+s_{m}=i+j−m} P_{i}(s_{1}...s_{m})\frac{1}{s_{1}} ∂∂_{s_{1}}v ⋯ \frac{1}{s_{m}} ∂∂_{s_{m}}v.  (10)

where i,j=1,2,... , and the equation ∂_{n}v = 0

7. Proof of theorem 1

According to [ K] one of A_{n−1}-potentials is

F̃(x_{1}, ..., x_{n−1}) = v(x_{1}, \frac{x_{2}}{2} , ..., \frac{x_{n−1}}{n − 1}, 0, ...),

where v satisfies the dispersionless n–Gelfand–Dikii hierarchy .

From consequence 1 it follows

0 = ∂_{ℓ}∂_{n}F̃ = ∑_{m=1}^{∞} ∑_{s_{1}+···+s_{m}=ℓ+n−m} P_{n}(s_{1} … s_{m})∂∂_{s_{1}} F̃ ⋯ ∂∂_{s_{m}} F̃ =

n∂∂_{n+ℓ−1}F̃ + ∑_{m=2}^{∞} ∑_{s_{1}+···+s_{m}=ℓ+n−m} P_{n}(s_{1} … s_{m})∂∂_{s_{1}} F̃ ⋯ ∂∂_{s_{m}} F̃.

Thus

∂∂_{n+ℓ−1}F̃ = −\frac{1}{n} ∑_{m=2}^{∞} ∑_{s_{1}+···+s_{m}=ℓ+n−m} P_{n}(s_{1} … s_{m})∂∂_{s_{1}} F̃ ⋯ ∂∂_{s_{m}} F̃.  (8)

According to [W2]

∂∂_{s}F̃ = x_{n−s} and ∂∂_{n+s}F̃ = −s∂_{s}F̃

for s < n. Thus because ∂∂_{n}F̃ = 0 that according to (11) ∂∂_{n+ℓ−1}F̃ = x_{1−ℓ} for all ℓ. Therefore

∂_{s}F̃ = −\frac{1}{s}x_{−s} = ∂_{s}F_{A}

and F_{A} − F̃ — is a linear form. According to [D2, p.16] this form is equal 0. According [Z] F_{B}(x_{1}, ..., x_{n}) = F_{A}(x_{1}, 0, x_{2}, 0, ..., x_{n−1}, 0, x_{n}).□

These are the potentials find from our formulas

(A_{2}) F = \frac{1}{2}x_{1}^{2}x_{2} + \frac{1}{24}x_{2}^{4},

(A_{3}) F = \frac{1}{2}x_{1}^{2}x_{3} + \frac{1}{2}x_{1}x_{2}^{2} + \frac{1}{4}x_{2}^{2}x_{3}^{2} + \frac{1}{60}x_{3}^{5},

(A_{4}) F = \frac{1}{2}x_{1}^{2}x_{4} + x_{1}x_{2}x_{3} + \frac{1}{6}x_{2}^{3} + \frac{1}{12}x_{3}^{4} + \frac{1}{2}x_{2}x_{3}^{2}x_{4} + \frac{1}{4}x_{2}^{2}x_{4}^{2} + \frac{1}{6}x_{3}^{2}x_{4}^{3} + \frac{1}{120}x_{4}^{6},

(A_{5}) F = \frac{1}{2}x_{1}^{2}x_{5} + \frac{1}{2}x_{2}^{2}x_{3} + \frac{1}{4}x_{2}^{2}x_{5}^{2} + x_{2}x_{3}x_{4}x_{5} + x_{1}x_{2}x_{4} + \frac{1}{6}x_{2}x_{4}^{3} + \frac{1}{2}x_{1}x_{3}^{2} + \frac{1}{2}x_{3}^{2}x_{4}^{2}+

+\frac{1}{6}x_{3}^{3}x_{5} + \frac{1}{6}x_{3}^{2}x_{5}^{3} + \frac{1}{2}x_{3}x_{4}^{2}x_{5}^{2} + \frac{1}{6}x_{4}^{4}x_{5} + \frac{1}{8}x_{4}^{2}x_{5}^{4} + \frac{1}{210}x_{5}^{7}.

(B_{2}) F = \frac{1}{2}x_{1}^{2}x_{2} + \frac{1}{60}x_{2}^{5},

(B_{3}) F = \frac{1}{2}x_{1}^{2}x_{3} + \frac{1}{2}x_{1}x_{2}^{2} + \frac{1}{6}x_{2}^{3}x_{3} + \frac{1}{6}x_{2}^{2}x_{3}^{3} + \frac{1}{210}x_{3}^{7}.

References

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[N] S.M.Natanzon, Differentil equations on the Prym theta function. A realness criterion for two-dimensional, finite-zone, potential Schrödinger operators. Functional Anal. Appl. 26:1 (1992), 13-20.

[S] M.Sato, Soliton equations and universal Grassmann manifold. Math. Lect. Notes Ser., Vol.18, Sophia University, Tokyo (1984).

[W1] E.Witten, On the structure of the topological phase of two–dimensional gravity. Nucl. Phys., B340 (1990), 281-332.

[W2] E.Witten, Algebraic geometry associated with matrix models of two dimensional gravity, Topological models in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX 1993, 235-269.

[Z] J.-B.Zuber, On Dubrovin’s topological field theories. Mod. Phys. Lett. A9 (1994) 749-760.

Moscow State University, Moscow, Russia.

Independent University of Moscow, Moscow, Russia.

e-mail: natanzon@mccme.ru