[7]G^ #1,#2_#3,#4(#5 #6— #7)

# d-orthogonal polynomials, Toda Lattice and Virasoro symmetries

Emil Horozov  Department of Mathematics and Informatics
Sofia University
5 J. Bourchier Blvd.
Sofia 1126
Bulgaria, and
Institute of Mathematics and Informatics,
Bulg. Acad. of Sci., Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria
<EMAIL_ADDRESS>
###### Abstract.

The subject of this paper is a connection between d-orthogonal polynomials and
the Toda lattice hierarchy. In more details we consider some polynomial
systems similar to Hermite polynomials, but satisfying $d+2$-term recurrence
relation, $d>1$. Any such polynomial system defines a solution of the Toda
lattice hierarchy. However we impose also the condition that the polynomials
are also eigenfunctions of a differential operator, i.e. a bispectral problem.
This leads to a solution of the Toda lattice hierarchy, enjoying a number of
special properties. In particular the corresponding tau-functions $\tau_{m}$
satisfy the Virasoro constraints. The most spectacular feature of these tau-
functions is that all of them are partition functions of matrix models. Some
of them are well known matrix models - e.g. Kontsevich model, Kontsevich-
Penner models, $r$-spin models, etc. A remarkable phenomenon is that the
solution corresponding to $d=2$ contains two famous tau functions describing
the intersection numbers on moduli spaces of compact Riemann surfaces and of
open Riemann surfaces.

###### Key words and phrases:

d-orthogonal polynomials, bispectral problem, Toda lattice, Virasoro
constraints, matrix models

###### 2010 Mathematics Subject Classification:

34L20 (Primary); 30C15, 33E05 (Secondary)

##  1\. Introduction

In 1975 J. Moser published the paper [34], in which he made explicit the
connection between orthogonal polynomials and the Toda lattice. Namely,
starting with any system of orthogonal polynomials he pointed out that the
Toda flow of the coefficients of the 3-term recurrence relation can be
expressed in terms of the Stieltjes function (the generating function of the
moments of the corresponding measure) of the polynomial system. Later this
research was continued both for orthogonal polynomials and for their
generalizations (like multiple orthogonal polynomials), see e.g. [3, 7, 35].

The present paper also deals with integrable systems of Toda type and their
connections with polynomial systems which are generalizations of orthogonal
polynomials. Before explaining the main results we give a brief account of the
needed notions.

In 1986 Duistermaat and Grünbaum [21] introduced the notion of bispectral
operators. The operators $L(x)$ in the variable $x$ and $\Lambda(z)$ in $z$
are called bispectral if there exists a joint eigen-function $\Psi(x,z)$ such
that

$\begin{split}L(x)\Psi(x,z)=f(z)\Psi(x,z)\\\
\Lambda(z)\Psi(x,z)=\theta(x)\Psi(x,z).\end{split}$

Here $f(z)$ and $\theta(x)$ are some nonconstant functions. I don’t specify
the type of the operators, neither the variables as they can be of different
types. Some examples will be more helpful to clarify the situation.

1) $\psi(x,z)=e^{xz}$, $L=\partial_{x}$, $\Lambda=\partial_{z}$,
$x,z\in{\mathbb{C}}$.

$Le^{xz}=ze^{xz},\;\;\Lambda e^{xz}=xe^{xz}.$

2) Airy operator

$\begin{split}(\partial_{x}^{2}-x)Ai(x+z)=zAi(x+z)\\\
(\partial_{z}^{2}-z)Ai(x+z)=xAi(x+z).\end{split}$

3) Let T be the shift operator in $n\in{\mathbb{Z}}$; $T^{\pm 1}f(n)=f(n\pm
1)$ and $L=x\partial_{x}$, $x\in{\mathbb{C}}$. Then

$Lx^{n}=nx^{n},Tx^{n}=x\cdot x^{n}.$

4) Let $H_{n}(x),\;n=0,1,\ldots$ be the Hermite polynomials. Then

$(-\partial_{x}^{2}+x\partial_{x})H_{n}(x)=nH_{n}(x),\;(T-nT^{-1})H_{n}(x)=xH_{n}(x).$

The bispectral operators have a number of connections with other fields of
research such as integrable systems (KP hierarchy, Sato’s Grassmannian and the
Calogero-Moser systems, orthogonal polynomials, representation theory of
$W_{1+\infty}$ and Virasoro algebras, ideal structure and automorphisms of the
first Weyl algebra, etc. See [8, 9, 10, 43, 11]

In fact the examples 2) Airy functions and 4) Hermite polynomials are obtained
via automorphisms of the Weyl algebra from 1) and respectively from 3). This
will be demonstrated in the next section in more general context.

The main objects in the paper are the polynomial systems $P_{n}(x)$, called
$d$-orthogonal polynomials with some extra structure - they enjoy the
generalized Bochner property, see below. $d$-orthogonal polynomials are
orthogonal with respect to $d$ measures (see for details Definition 2.1)
rather than one. Then using these polynomials we are going to find special
solutions of the Toda lattice hierarchy which lead to important matrix models
as explained in Section 8.

As in the case of orthogonal polynomials this property has purely algebraic
expression. Due to P. Maroni [32], we can use as a definition of
$d$-orthogonality the following one: the polynomials satisfy a $d+2$-term
recursion relation of the form

$x\cdot P_{n}(x)=P_{n+1}(x)+\sum_{j=0}^{d}\gamma_{j}(n)P_{n-j}(x).$

with constants $\gamma_{j}(n)$, independent of $x$, $\gamma_{d}(n)\neq 0$ for
$n\geq d$.

Generalized Bochner problem (GBP). Find systems of polynomials
$P_{n}(x),\;n=0,1,\ldots$ that are eigenfunctions of a differential operator
$L$ of order $m$ with eigenvalues $\lambda(n)$ depending on the discrete
variable $n$ (the index):

$LP_{n}(x)=\lambda(n)P_{n}(x)$

and which at the same time are eigenfunctions of a difference operator, i.e.
that satisfy a finite-term (of fixed length $d+2$), recursion relation of the
form

$x\cdot P_{n}(x)=P_{n+1}(x)+\sum_{j=0}^{d}\gamma_{j}(n)P_{n-j}(x).$

In the case of classical orthogonal polynomials Bochner’s theorem shows that
all their properties follow from the fact that they are eigenfunctions of a
differential operator. The above generalization produces $d$-orthogonal
polynomials analogs of the classical orthogonal polynomials, which have very
similar properties.

We found a large class of d-orthogonal polynomials with this property
(conjecturally all). See Section 2 or [26] for details.

The construction goes as follows. We start with Weyl algebra $W_{1}$, i.e. the
algebra of differential operators in one variable $x$ with polynomial
coefficients. It has a natural representation in the space of all polynomials
${\mathbb{C}}[x]$.

Consider the trivial polynomial system $\\{x^{n}\\},\;n=0,1,2,\ldots$. The
polynomials $x^{n}$ are eigenfunctions of the differential operator
$H=x\partial_{x}$ with eigenvalues $n$. They also satisfy the trivial
recurrence relation

$x\cdot x^{n}=x^{n+1}=Tx^{n},$

where $T$ is the shift operator $Tf(n)=f(n+1)$. Also notice the identity
$\partial_{x}x^{n}=nx^{n-1}$. We see that the operators $H$ and $T$ are
bispectral. Now we will explain how to construct non-trivial solutions of the
bispectral problem.

It is known (and easy to see), cf. [20], that given a polynomial
$q(\partial_{x})$ without constant term, one can construct an automorphism of
$W_{1}$ by the formula

$\sigma
A:=e^{{\mathrm{ad}}_{q(\partial_{x})}}A=\sum_{j=0}^{\infty}\frac{{\mathrm{ad}}_{q(\partial_{x})}^{j}A}{j!},\;\;A\in
W_{1},$

where ${\mathrm{ad}}_{B}A=[B,A]$ and
${\mathrm{ad}}_{B}^{j}A=[B,{\mathrm{ad}}^{j-1}_{B}A]$ . Then following [8] we
can use $\sigma$ to obtain a new pair of bispectral operators with non-trivial
polynomial system $P_{n}(x)$, being the common eigenfuction. Namely we put

$P_{n}(x)=\sum_{j=0}^{\infty}\frac{q(\partial_{x})^{j}x^{n}}{j!}.$

It is obvious that the above sum is finite and $P_{n}(x)$ is a polynomial of
degree $n$. Also introduce the operators

$L=\sigma H,\;\;D=\sigma^{-1}x.$

For example if $q(\partial_{x})=-\partial_{x}^{2}/2$ we obtain that

$L=-\partial_{x}^{2}+x\partial_{x}\;\;\sigma\partial_{x}=\partial_{x}\;\;\text{and}\;\;\sigma^{-1}x=x+\partial_{x}^{2}.$

Then

$\begin{split}LP_{n}(x)&=nP_{n}(x)\\\
xP_{n}(x)&=[T+n(n-1)T^{-1}]P_{n}(x)\end{split}$

For the last identity, see [26] or the next section for details. Thus we
obtain the classical Hermite polynomials. The case of general polynomial $q$
is treated in the same manner. In this paper we consider the special case of
$q(\partial_{x})=\partial_{x}^{d+1}/(d+1)$, $d\in{\mathbb{N}},\;d\geq 1$.
Notice that we obtain simultaneously all ingredients - the polynomial system,
the differential operator and the recurrence relation.

Our next step is to construct the measures or which is equivalent - their
weights. This is done in Section 3. It turns out that they play the same role
as the polynomials in the construction of solutions of Toda hierarchy. This
can be seen from their integral representations, which are of the same class
as the polynomials:

$\nu(n,x)=\int_{C}z^{-n-1}\exp(\frac{-z^{d+1}}{d+1}+xz)\mathrm{d}z$

with an appropriate contour $C$. The polynomials correspond to closed contour
around $z=0$ and $n=0,1,\ldots$. For the weights we have to choose a contour,
going to infinity in both directions, on which the integrand tends to zero and
$n=-1,-2,\ldots$. However the above integrals make sense for all values of
$n\in{\mathbb{Z}}$ when we chose the contour as for the weights. Moreover the
set of functions $\nu(n,x),\;n\in{\mathbb{Z}}$ satisfy both the recurrence
relations and the differential equation, i.e. they give another solution of
the bispectral problem, this time the range of the variable $n$ is
${\mathbb{Z}}$. Let us formulate the corresponding result, proven in Section
4.

######  Lemma 1.1.

(i) The functions $\nu(n,x)$ satisfy the equation

(1.1) $-\partial_{x}^{d+1}\nu(n,x)+x\partial_{x}\nu(n,x)=n\nu(n,x).$

(ii) They satisfy the recurrence relation

(1.2) $x\nu(n,x)=(n+1)\nu(n+1,x)+\nu(n-d,x).$

(iii) The lowering operator $\partial_{x}$ acts on the solutions $\nu(n,x)$ as

(1.3) $\partial_{x}\nu(n,x)=\nu(n-1,x).$

In the theory of classical orthogonal polynomials functions with such
properties are called functions of the second kind [37]. Each contour $C$ for
which the integral is convergent defines such type of functions.

This gives a bi-infinite system of functions, from which we can construct a
solution of the the Toda hierarchy.

We know (after Sato), see [36, 17] that both Toda and KP hierarchy of
equations can be written as evolution equations in a certain infinite-
dimensional Grassmannian - Sato Grassmannian. Each point of the Sato’s
Grassmannian corresponds to a plane $W$, which consists of functions in an
infinite-dimensional linear space. We will be interested in planes possessing
certain symmetries.

Let us recall the idea of these symmetries on a simple example. In the studies
of generalized Kontsevich models it turned out that a very important
ingredient is the notion of Kac-Schwarz [30] operators $A_{j}$. Roughly
speaking they act on a solution of Airy equations, spanning the entire plane
$W$ and leaving it invariant: $A_{j}W\subset W$.

The above constructed functions $\nu(n,x)$ span a flag $F=\ldots W_{n}\subset
W_{n+1}\subset\ldots$ of planes $W_{n}=(\nu(n,x),\nu(n-1,x),\ldots)$. Each of
these planes gives rise to a tau function $\tau(n,t)$, where
$t=(t_{1},t_{2},\ldots)$ via the boson-fermion correspondence.

In the simplest case of $d=2$ the differential operator is

$L=-\partial_{x}^{3}+x\partial_{x}.$

It is easy to check that the corresponding function $\nu(-1,x)$ is the Airy
function $Ai(x)$. It is the first weight function, defining the measure for
the $d$-orthogonal polynomials $P_{n}(x)$, $d=2$, which are eigenfunctions of
the differential operator. The rest of the weights are given by the
derivatives of $Ai(x)$.

The next result in the paper shows that all the tau-functions satisfy the
Virasoro constraints. Virasoro algebra is an algebra spanned by operators
$L_{j},j\in{\mathbb{Z}}$ and the central element $c$ and satisfying the
commutator relations

$[L_{k},L_{m}]=(k-m)L_{k+m}\frac{k^{3}-k}{12}\delta_{k,-m}c.$

In the bosonic representation the operators $L_{j}$ are differential operators
of order one or two in infinite number of variables $t_{1},t_{2},\ldots$. The
Virasoro constraints are equations of the type

$L_{j}\tau(n,t)=0,\;\text{for}\;j=-1,1,2,\ldots$

while the function $\tau(n,t)$ is an eigenvector of the operator $L_{0}$. For
the case of $d=3$ and $n=-1$ the result belongs to Witten and Kontsevich [44,
31] (although in different terminology and notations) For $d=3$ and $n=0$ the
above result belongs to Alexandrov [5].

The corresponding tau-function $\tau(-1,t)$ has a deep and beautiful algebro-
geometric meaning as conjectured by Witten [44] and proved by Kontsevich [31].
Namely it gives the intersection theory of the moduli spaces of compact
Riemann surfaces.

The function $\tau(0,t)$ also has algebro-geometric meaning as proved by
Alexandrov, Buryak, Tessler [5, 15]. It describes the intersection theory of
moduli spaces of open Riemann surfaces. See the cited papers and the
references therein.

The above tau-functions have appeared in the cited references as solutions of
other integrable hierarchies - KP or extended KP. Here they appear as one
solution  of the Toda lattice $\tau(n,t)$.

For the other values of $d$ at least one of the tau-functions also has
algebro-geometric meaning. For $d>2$ the function $\tau(-1,t)$ is connected
with the so called $r$-spin structures, introduced by Witten [44], see also
[22].

Let us point that the connection of the bispectral problem with the solutions
of integrable systems exhibiting Kac-Scwartz symmetry has been discussed in
many different places. The first documented one that I know is in [2]. Here we
present other instances of such connections.

### Acknowledgements

This research has been partially supported by the Grant No DN 02-5 of the
Bulgarian Fund ”Scientific research”.

##  2\. Preliminaries

###  2.1. $d$-orthogonal polynomials

######  Definition 2.1.

Let $\\{P_{n}(x),\;n=0,1,\ldots\\}$ be a family of monic polynomials such that
$\deg P_{n}=n$. The polynomials are d-orthogonal iff there exist $d$
functionals $\mathcal{L}_{j},j=0,\ldots,d-1$ on the space of all polynomials
${\mathbb{C}}[x]$ such that

$\begin{cases}\mathcal{L}_{j}(P_{n}P_{m})=0,m>nd+j,n\geq 0,\\\
\mathcal{L}_{j}(P_{n}P_{nd+j})\neq 0,n\geq 0,\end{cases}$

for each $j\in N_{d+1}:=\\{0,\ldots,d-1\\}$. When $d=1$ this is the ordinary
notion of orthogonal polynomials.

Notice that $\mathcal{L}_{j}(P_{j})\neq 0,\;j=0,\ldots,d-1$. The notion of
$d$-orthogonal polynomials was introduced by J. Van Iseghem, [41].

Let $v(x)$ be the weight (function) on a subset $U\subset{\mathbb{R}}$, which
defines the functional

$\mathcal{L}(P)=\int_{U}v(x)P(x)\mathrm{d}x.$

Instead of $\mathcal{L}(P)$ we are going to use the notation

$\left\langle v(x),P(x)\right\rangle,$

which is very convenient for algebraic manipulations.

One can show easily that for each $n$ there exist polynomials
$A_{0}^{(n)}(x),\ldots A_{d-1}^{(n)}$ such that the weights
$v_{n}=\sum_{i=0}^{d-1}v_{j}A_{j}^{(n)}$ have the property

$\left\langle v_{j}(x),P_{m}(x)\right\rangle=\delta_{j,m}c_{j}\;c_{j}\neq 0.$

The orthogonality connected with $d$ functionals rather than with only one,
gives the name of d-orthogonal polynomials.

Norming suitably the weights $v_{j}$ we see that they define the bilinear
forms with the properties

$\left\langle v_{j}(x),P_{m}(x)\right\rangle=\delta_{j,m}.$

A very important theorem of P. Maroni [32] is the following one.

######  Theorem 2.2.

A polynomial system $\\{P_{n}(x)\\},\;n\geq 0$, is d-orthogonal if and only if
the polynomials satisfy a $d+2$-term recurrence relation of the form

(2.4) $xP_{n}(x)=P_{n+1}+\sum_{j=0}^{d}\gamma_{j}(n)P_{n-j}(x)$

with constants $\gamma_{j}(n)$, independent of $x$, $\gamma_{d}(n)\neq 0,n\geq
d$.

###  2.2. Bochner’s property

In this subsection we briefly recall some notions and results from [26, 27]
about d-orthogonal polynomials, which are eigenfunctions of a differential
operator. We have called this property ”Bochner’s property” in analogy with
the classical theorem by Bochner [13].

The differential operators with polynomial coefficients form the Weyl algebra
$W_{1}$. It is spanned by the two generators $x,\partial_{x}$.

For any polynomial
$q(\partial_{x})=\sum_{j=1}^{d+1}a_{j}\partial_{x}^{j}\in{\mathbb{C}}[\partial_{x}]$
(notice that the coefficients $a_{j}$ are constants) we define the
automorphism of $W_{1}$

$\sigma_{q}(A)=e^{{\mathrm{ad}}_{q(\partial_{x})}}(A)=e^{q}Ae^{-q},\;\;A\in
W_{1}.$

Here ${\mathrm{ad}}_{A}(B)=[A,B]=AB-BA$, $A,B\in W_{1}$.

For any polynomial $q(\partial_{x})=\sum_{j=1}^{d+1}a_{j}\partial_{x}^{j}$
without a constant term111The constant term contributes only to multiplication
of the polynomials by a constant. On the other hand without it the formulas
and the arguments are simpler. we defined the automorphism
$\sigma_{q}=e^{q(\partial_{x})}$ of $W_{1}$.

Then we found the images of the generators $x,\partial_{x}$ of $W_{1}$ under
the automorphism $\sigma_{q}$:

######  Lemma 2.3.

(2.5) $\begin{cases}\sigma_{q}(\partial_{x})=\partial_{x}\\\
\sigma_{q}(x)=x+q^{\prime}(\partial_{x}).\end{cases}$

∎

Let us introduce an auxiliary algebra ${\mathcal{R}}_{2}$ defined over
${\mathcal{F}}$ with generators $T,T^{-1},\hat{n}$ subject to the relations

$T\cdot T^{-1}=T^{-1}\cdot T=1,\;[T,\hat{n}]=T,\;[T^{-1},\hat{n}]=-T^{-1}.$

One easily sees that $[T,\hat{n}T^{-1}]=1$. Hence the operators
$T,\hat{n}T^{-1}$ define a realization of the Weyl algebra. We can introduce
another algebra ${\mathcal{B}}_{2}$ as follows.

First we define an anti-homomorphism $b$, i.e. a map $b\colon
W_{1}\to{\mathcal{R}}_{2}$ satisfying

$b(m_{1}\cdot m_{2})=b(m_{2})\cdot b(m_{1}),$

for each $m_{1},\;m_{2}\in W_{1}$ by:

$\begin{cases}b(x)=T\\\ b(\partial_{x})=\hat{n}T^{-1}\\\ \end{cases}$

The algebra ${\mathcal{B}}_{2}$ will be the image $b(W_{1})$ of $W_{1}$. Then
$b\colon W_{1}\to{\mathcal{B}}_{2}$ is an anti-isomorphism and in particular
$b^{-1}\colon{\mathcal{B}}_{2}\to W_{1}$ is well defined.

We start with the polynomial system $\psi(x,n)=x^{n}$. Then the corresponding
operators $\partial_{x},\hat{x}$ act on ${\mathbb{C}}[x]$. In the same way we
represent the algebra ${\mathcal{B}}_{2}$ in ${\mathbb{C}}[x]$ by realizing
$T$ and $T^{-1}$ as the shift operators acting on functions $f(n)$ by
$T^{\pm}f(n)=f(n\pm 1)$. Finally $\hat{n}$ denotes the operator of
multiplication by $n$. With this notation we have

######  Lemma 2.4.

$\begin{cases}\partial_{x}\psi(x,n)=\hat{n}T^{-1}\psi(x,n)\\\
x\psi(x,n)=T\psi(x,n)\\\ x\partial_{x}\psi(x,n)=n\psi(x,n).\end{cases}$

∎

Using $q(\partial_{x})$ we define another polynomial system $P_{n}(x)$ by

$P_{n}(x)=e^{q(\partial_{x})}\psi(x,n)=\sum_{j=0}^{\infty}\frac{q(\partial_{x})^{j}\psi(x,n)}{j!}.$

Notice that the above series is in fact finite as the operator
$q(\partial_{x})$ reduces the degree of any polynomial. Let us denote the
operator $\sigma_{q}(x\partial_{x})$ by $L$. We proved in [26] among other
statements the following

######  Theorem 2.5.

The polynomials $P_{n}(x)$ have the following properties:

(i) They are eigenfunctions of the differential operator

(2.6) $L:=q^{{}^{\prime}}(\partial_{x})\partial+x\partial$

with eigenvalues $\lambda(n)=n$.

(ii) They have the lowering operator $\partial_{x}$, i.e.

(2.7) $\partial_{x}P_{n}(x)=nP_{n-1}.$

(iii) They satisfy the recurrence relation

(2.8) $xP_{n}(x)=P_{n+1}+\sum_{j=0}^{d}ja_{j}n(n-1)\ldots(n-j+1)P_{n-j}.$

###### Proof.

We repeat the proof from [26] as here it is simpler.

$(i)\;\;LP_{n}(x)=(e^{q(\partial)}\cdot x\partial\cdot
e^{-q(\partial)})\cdot(e^{q(\partial)}x^{n})=e^{q(\partial)}x\partial
x^{n}=nP_{n}(x)$

(ii) is obvious.

(iii) This requires more computations but it is similar to (i). One has to
compute $\sigma^{-1}x=e^{-q(\partial)}\cdot x\cdot e^{q(\partial)}$ and then
use the obvious $x\cdot x^{n}=x^{n+1}$. Then the expression $\sigma^{-1}x\cdot
x^{n}$ has to be written only in terms of $T$ and $nT^{-1}$, using that
$x\cdot x^{n}=x^{n+1}$ and $\partial x^{n}=nx^{n-1}$. In details the
computation is:

$\begin{split}&x\cdot P_{n}(x)=x\cdot
e^{q(\partial)}x^{n}=e^{q(\partial)}e^{-q(\partial)}\cdot x\cdot
e^{q(\partial)}x^{n}\\\ &=e^{q(\partial)}(\sigma^{-1}x)\cdot
x^{n}=e^{q(\partial)}\big{[}x+\sum_{j=0}^{d}\gamma_{j}\partial^{j}\big{]}\cdot
x^{n}\\\
&e^{q(\partial)}\cdot\big{[}x^{n+1}+\sum_{j=0}^{d}\gamma_{j}n(n-1)\ldots(n-j+1)x^{n-j}\big{]}.\end{split}$

∎

######  Example 2.6.

$q(\partial_{x})=-\partial_{x}^{d+1}/(d+1)$ corresponds to Gould-Hopper
polynomials [24]. When $d=1$ these are the Hermite polynomials. The operator
$L$ reads

$L=x\partial_{x}-\partial_{x}^{d+1}.$

For the recurrence relation we need to compute $\sigma^{-1}x$. Similarly to
the above we obtain that

$\sigma^{-1}x=x+n(n-1)\ldots(n-d)\partial^{n-d},$

which gives

$xP_{n}=P_{n+1}+n(n-1)\ldots(n-d)P_{n-d}.$

##  3\. Weights

###  3.1. Pearson equations

In what follows with an abuse of language we will not distinguish between a
functional and its weight $v(x)$. Let $\\{P_{j}(x)\\},\;j=0,1,\ldots$ be a
system of polynomials $\deg P_{j}(x)=j$. We introduce the dual system of
functionals $v_{j}(x)$ on ${\mathbb{C}}[x]$, such that

$\left\langle v_{j},P_{n}(x)\right\rangle=\delta_{jn},\;\;j,n=0,1\ldots.$

For each functional $v$ we can define new functionals $xv$ and $\partial_{x}v$
by

$\left\langle xv,P_{n}(x)\right\rangle=\left\langle
v,xP_{n}(x)\right\rangle,\;\left\langle\partial_{x}v,P_{n}(x)\right\rangle=\left\langle
v,-\partial_{x}P_{n}(x)\right\rangle.$

Below we will determine explicitly the dual system $v_{j}(x)$ for the
polynomials $\\{P_{j}(x)\\}$, defined by $q(\partial_{x})$.

######  Lemma 3.1.

(i) The functional $v_{0}(x)$ is a solution of the differential equation

(3.9) $\big{(}q^{\prime}(-\partial_{x})+x\big{)}v_{0}=0.$

(ii) The functionals $v_{j}(x)$ are given by

$v_{j}(x)=\frac{(-1)^{j}}{j!}\partial^{j}v_{0}(x).$

###### Proof.

From the properties (1.3) of the polynomials $P_{n}(x)$ we see that

$\frac{1}{n+1}\partial_{x}P_{n+1}(x)=P_{n}(x).$

(i) We start with the functional $v_{0}$. It satisfies

$\left\langle v_{0},P_{n}(x)\right\rangle=\delta_{0n}.$

Let us use the differential operator $L$. We have $LP_{n}(x)=nP_{n}(x)$. We
will compute the action of $v_{0}$ on the system $P_{n}(x)$.

Denote by $L^{*}$ the formal adjoint to $L$, i.e. $L^{*}$ satisfies the
identity

$\left\langle f,LQ(x)\right\rangle=\left\langle L^{*}f,Q(x)\right\rangle$

for any polynomials $Q(x)$ and $f(x)$. Using that $LP_{n}(x)=nP_{n}(x)$ we
obtain

$\left\langle v_{0},LP_{n}(x)\right\rangle=\left\langle
v_{0},nP_{n}(x)\right\rangle=0,\;\;\text{for all}\;\;n\geq 0.$

Hence

$0=\left\langle L^{*}v_{0},P_{n}(x)\right\rangle=0,\;\;\text{for all}\;\;n\geq
0,$

where the operator $L^{*}$ explicitly reads

$L^{*}=-\partial_{x}q^{\prime}(-\partial_{x})-\partial_{x}x.$

We see that the functional $L^{*}v_{0}\equiv 0$. Integrating once the function
$\partial_{x}[q^{\prime}(-\partial_{x})+x]v_{0}(x)=0$ we obtain that

$\big{(}q^{\prime}(-\partial_{x})+x\big{)}v_{0}=c,\;\;c\in{\mathbb{C}}.$

We will compute the constant by applying the functional
$\big{(}q^{\prime}(-\partial_{x}+x\big{)}v_{0}$ to the system $P_{n}(x)$,
$n=0,1,\ldots$. We have

$\begin{split}&\left\langle c,P_{n}(x)\right\rangle=\\\
&=\left\langle\big{(}q^{\prime}(-\partial_{x})+x\big{)}v_{0},P_{n}(x)\right\rangle\\\
&=\left\langle\big{(}q^{\prime}(-\partial_{x})+x\big{)}v_{0},\frac{1}{n+1}\partial_{x}P_{n+1}(x)\right\rangle\\\
&=\left\langle v_{0},\frac{1}{n+1}LP_{n+1}(x)\right\rangle\\\ &=\left\langle
v_{0},\frac{n+1}{n+1}P_{n+1}(x)\right\rangle=\left\langle
v_{0},P_{n+1}(x)\right\rangle=0.\end{split}$

Hence $c=0$ and we see that $v_{0}(x)$ satisfies the equation (3.9).

Next consider

$1=\delta_{00}=\left\langle v_{0},P_{0}(x)\right\rangle=\left\langle
v_{0},\partial_{x}P_{1}(x)\right\rangle=\left\langle-\partial_{x}v_{0},P_{1}(x)\right\rangle,$

i.e.

$\left\langle-\partial_{x}v_{0},P_{1}(x)\right\rangle=1.$

Using that

$\delta_{0n}=\left\langle
v_{0},P_{n}(x)\right\rangle=\left\langle-\partial_{x}v_{0},P_{n+1}/(n+1)(x)\right\rangle=0$

for $n>0$ we obtain that $v_{1}=-\partial_{x}v_{0}$. We continue by induction
to see

$v_{j}(x)=\frac{(-1)^{j}}{j!}\partial^{j}v_{0}(x).$

∎

The functions $v_{j}(x)$ are similar to Airy function and also have explicit
representations in terms of definite integrals. We shall derive them in the
next section.

Using the bi-orthogonality property and the recurrence relation for the
polynomials it is easy to show that the functions $v_{j}$ also satisfy
recurrence relation of the form:

$xv_{m}(x)=v_{m-1}(x)+\sum_{j=0}^{d-1}b_{j}(m)v_{m+j}(x),$

see e.g. [18, 38] for details and more general results.

##  4\. Integral representations

###  4.1. Integral representations for the weights

We will find integral representations of the weights of the polynomials
corresponding to $q(\partial_{x})=-\partial_{x}^{d+1}/(d+1)$. As we know, it
is enough to find one for $v_{0}(x)$.

######  Lemma 4.1.

The weight $v_{0}(x)$ is given by the formula

$v_{0}(x)=\int_{C}\exp(\frac{-z^{d+1}}{d+1}+xz)\mathrm{d}z.$

with an appropriate contour $C$, see Fig. 1.

###### Proof.

Denote by $\hat{v}_{0}(z)$ the Laplace transform of $v_{0}(x)$, i.e.

$\hat{v}_{0}(z)=\int_{C}e^{-zx}v_{0}(x)\mathrm{d}x.$

Then performing the Laplace transform on the equation (3.9), where
$q(z)=\frac{-z^{d+1}}{d+1}$, we obtain

$[-z^{d}-\partial_{z}]\hat{v}_{0}(z)=0.$

Solving the differential equation we obtain (up to a multiplicative constant)

$\hat{v}_{0}(z)=\exp(\frac{-z^{d+1}}{d+1}).$

Finally the inverse Laplace transform applied to $\hat{v}_{0}(z)$ gives

$v_{0}(x)=\int_{C}\exp(\frac{-z^{d+1}}{d+1}+xz)\mathrm{d}z,$

where the contour $C$ is chosen so that the integral to be convergent. All
such contours give solutions. For example on fig. 1 we have taken the union of
two rays $R_{1}\cup R_{-1}$, where

$R_{k}=R\exp(\pm 2\pi i/(d+1)),\;k=\pm 1\;\;\text{and}\;\;R\in[0,\infty).$

We see that on the rays $R_{k}$ the function
${\mathrm{Re}}\big{[}-\frac{z^{d+1}}{d+1}+xz\big{]}=-R^{d+1}(1/(d+1)+\mathcal{O}(R^{-1}))$,
as a function of $z$. Hence the integral is convergent.

The rest of the weights are given by

(4.10)
$v_{j}(x)=\frac{(-1)^{j}}{j!}\int_{C}z^{j}\exp(\frac{-z^{d+1}}{d+1}+xz)\mathrm{d}z,$

which follows from Lemma 3.1 (ii).

∎

x axisy axis Figure 1. The integration contour $C$.

###  4.2. Integral representations for the polynomials

Consider the following scalar integrals with suitable contours $C$

(4.11)
$w_{n}(x)=\int_{C}z^{-n-1}\exp\big(\frac{-z^{d+1}}{d+1}+xz\big{missing})\mathrm{d}z,\;\;n\in{\mathbb{Z}}_{+}.$

######  Lemma 4.2.

The function $w_{n}(x)$ satisfies the equation

(4.12) $\big{(}-\partial_{x}^{d+1}+x\partial_{x}\big{)}w_{n}(x)=nw_{n}(x).$

###### Proof.

It is an easy computation using integration by parts. On one hand we have that

$-\partial_{x}^{d+1}w_{n}(x)=\int_{C}z^{-n-1}(-z^{d+1})\exp(\frac{-z^{d+1}}{d+1}+zx)\mathrm{d}z.$

On the other hand integrate by parts the expression

$\begin{split}x\partial_{x}w_{n}(x)&=\int_{C}z^{-n}\exp(\frac{-z^{d+1}}{d+1}+zx)\mathrm{d}(xz).\\\
\end{split}$

This gives

$\begin{split}&=\int_{C}z^{-n}\exp(\frac{-z^{d+1}}{d+1})\mathrm{d}\exp(zx)\\\
&=-\int_{C}[(-n)z^{-n-1}-z^{d}z^{-n}]\exp(\frac{-z^{d+1}}{d+1}+zx)\mathrm{d}z\\\
&=\int_{C}[nz^{-n-1}+z^{d+1}z^{-n-1}]\exp(\frac{-z^{d+1}}{d+1}+zx)\mathrm{d}z\\\
\end{split}.$

Hence

$(-\partial_{x}^{d+1}+x\partial_{x})w_{n}(x)=nw_{n}(x).$

∎

In fact the different solutions of this differential equation are given by
appropriately choosing the contour. This is done in a similar way to the case
of the weights. The only thing we need here is to avoid the point $z=0$ as the
integrals are not convergent in $z=0$ when $n+1>0$. It can be achieved by
appropriately deforming the contour $C$ to avoid the singularity.

If we want to obtain representation for the polynomials we need to go around
the point $z=0$.

######  Corollary 4.3.

The monic polynomials $\\{P_{n}(x)\\}$ have the following integral
representation

$P_{n}(x)=\frac{n!}{2\pi
i}\int_{|z|=\varepsilon}z^{-n-1}\exp(\frac{-z^{d+1}}{d+1}+zx)\mathrm{d}z.$

###### Proof.

The polynomial $P_{n}(x)$ is the residue of the integral. By expanding the
exponent we see that the highest degree $x^{n}$ of the coefficient at $z^{-1}$
is given by $x^{n}/n!$ which gives the needed result.

∎

###  4.3. Functions of the second kind

The rest of the solutions of the differential equations for the classical
orthogonal polynomials share a lot of properties with the polynomial
solutions. See, e.g. [28], sect. 3.6 and [37], 4.23. Here we shall give
analogs of these results for the cases of interest. We are going to prove
slightly more general results, taking any number $s\in{\mathbb{C}}$ instead of
non-negative integer $n$ . Let us introduce the functions

(4.13) $\nu(s,x)=\int_{C}z^{-s-1}\exp(-z^{d+1}/(d+1)+zx)\mathrm{d}z,$

where $C$ is a contour along which
${\mathrm{Re}}\big{[}(-z)^{d+1}/(d+1)-xz\big{]}\to-\infty$ and which does not
pass through the point $z=0$.

######  Lemma 4.4.

(i) The functions $\nu(s,x)$ satisfy the equation

(4.14) $-\partial_{x}^{d+1}\nu(s,x)+x\partial_{x}\nu(s,x)=s\nu(s,x).$

(ii) They satisfy the recurrence relation

(4.15) $x\nu(s,x)=(s+1)\nu(s+1,x)+\nu(s-d,x).$

(iii) The lowering operator $\partial_{x}$ acts on the solutions $\nu(s,x)$ as

(4.16) $\partial_{x}\nu(s,x)=\nu(s-1,x).$

###### Proof.

(i) The proof repeats the polynomial case.

(ii) Integrating by parts the expression $x\nu(s,x)$ we obtain

$\begin{split}&\int_{C}z^{-s-1}\exp(\frac{-z^{d+1}}{d+1})\mathrm{d}\exp(+xz)=\\\
&=-\int_{C}[(-s-1)z^{-s-2}-z^{-s-1+d}]\exp(-\frac{z^{d+1}}{d+1}+xz)\mathrm{d}z\\\
&=\int_{C}(s+1)z^{-s-2}\exp(-\frac{z^{d+1}}{d+1}+xz)\mathrm{d}z\\\
&+\int_{C}z^{-s+d-1}\exp(-\frac{z^{d+1}}{d+1}+xz)\mathrm{d}z=\\\
&=(s+1)\nu(s,x)+\nu(s-d,x).\end{split}$

(iii) Differentiating the integral gives

$\partial_{x}\nu(s,x)=\int_{C}z^{-s}\exp(-\frac{z^{d+1}}{d+1}+zx)\mathrm{d}z=\nu(s-1,x).\\\
$

∎

######  Remark 4.5.

The functions $\nu(n,x)$, where $n$ is a negative integer correspond to the
weights, while $n\geq 0$ correspond to the polynomials (when we take a closed
contour around $0$) or to the functions of the second kind. In this way
formula (4.13) gives integral representations of the same kind both for the
weights and the functions of the second kind (the polynomials).

We can rescale $\nu(s,x)$ introducing $u(s,x)$ by $\nu(s,x)=a(s)u(s,x)$ so
that the recurrence (4.15) takes the original form

$xu(s,x)=u(s+1,x)+b(s-d)u(s-d,x).$

In this way we will obtain the recurrence for the monic polynomials
$P_{n}(x)$. However we will work primarily with (1.2).

##  5\. Preliminaries on infinite dimensional Lie algebras

###  5.1. The Lie algebra $\mathfrak{gl}_{\infty}$ and its representations

By $\mathfrak{gl}_{\infty}$ we denote the algebra of doubly infinite matrices
with only finite number of nonzero elements. We will need also the algebra
$\bar{a}_{\infty}$ defined by the condition that the nonzero diagonals are
finitely many:

$\bar{a}_{\infty}=\\{\big{(}a_{ij}\big{)}|a_{ij}=0\;\;\text{for}\;\;|i-j|>>0\\}.$

The fermionic Fock space is defined as follows. Introduce a vector space
${\mathbb{V}}$ with a fixed basis $v_{j},\;j\in{\mathbb{Z}}$:

${\mathbb{V}}=\mathop{\oplus}_{j\in{\mathbb{Z}}}{\mathbb{C}}v_{j}.$

The algebra $\mathfrak{gl}_{\infty}$ acts on ${\mathbb{V}}$ in the standard
way.

Consider the vectors of the form

$v_{i_{0}}\wedge v_{i_{-1}}\wedge v_{i_{-2}}\wedge\ldots$

such that $i_{0}>i_{-1}>i_{-2}\ldots$ and $i_{-k}=-k$ for $k$ big enough. We
consider the space $F^{(0)}$ spanned by the above vectors with coefficients in
${\mathbb{C}}$. The element

$\ket{0}=v_{0}\wedge v_{-1}\wedge v_{-2}\wedge\ldots$

is called the $0$-vacuum.

Exactly in the same way we define the $m$-th vacuum by

$\ket{m}=v_{m}\wedge v_{m-1}\wedge v_{m-2}\wedge\ldots$

and the corresponding space $F^{(m)}$. The direct sum

$F=\mathop{\oplus}_{m\in{\mathbb{Z}}}F^{(m)}.$

will be called fermionic Fock space.

Next we define an action of $\mathfrak{gl}_{\infty}$ on $F$ as follows. Let
$a\in\mathfrak{gl}_{\infty}$. Take a vector $v_{i_{0}}\wedge
v_{i_{1}}\wedge\ldots$ and define the map $r(a)\colon F^{(m)}\to F^{(m+1)}$ as

$r(a)(v_{i_{0}}\wedge v_{i_{1}}\wedge\ldots)=av_{i_{0}}\wedge
v_{i_{1}}\wedge\ldots+v_{i_{0}}\wedge av_{i_{1}}\wedge\ldots+\ldots$

and by linearity.

We also will need the group-like operator $R(\exp(a))\colon F^{m}\to F^{(m)}$,
where $a\in\mathfrak{gl}_{\infty}$. It is defined by the action

$R(\exp(a)(v_{i_{0}})\wedge
v_{i_{1}}\wedge\ldots)=\exp(a)v_{i_{0}}\wedge\exp(a)v_{i_{0}}\wedge\exp(a)v_{i_{1}}\wedge\ldots$

We will need slightly more general group $G_{\infty}$ of matrices $g$ with
$\det g\neq 0$, which have only finite number of diagonals below the main one
but any number above it. It is obvious that the above formula holds for them.
Next we define the bosonic Fock spaces $B^{(m)}$ for $m\in{\mathbb{Z}}$.
Introduce the formal variable $Q$ and put

(5.17) $B^{(m)}:={\mathbb{C}}[Q^{m};t_{1},t_{2},\ldots].$

The element $Q^{m}\cdot 1\in B^{(m)}$ is called the $m$-th vacuum.

Introduce the oscillator algebra $\mathcal{A}$ spanned by operators
$a_{j},\;j\in{\mathbb{Z}}$ and an operator $c$, called central charge subject
to the identities

$[a_{i},a_{j}]=i\delta_{i,-j}\cdot c,\;\;[c,a_{j}]=0.$

This algebra can be represented in both the bosonic and the fermionic Fock
spaces.

If we want to represent the oscillator algebra in the fermionic Fock space in
terms of the operators $r(a)$ we see that the operator $r(I)$ (here $I$ is the
identity matrix) is not well defined. So we need a regularization. We will
recall its definition (cf. [29]) for the algebra $\bar{a}_{\infty}$.

Let $E_{ij}\in\mathfrak{gl}_{\infty}$ be the matrix with $1$ on the
intersection of the $i$-th row and the $j$-th column. Put
$\Lambda_{j}=\sum_{-\infty}^{+\infty}E_{i,i+j}$.

Define the operator $\hat{r}(E_{ij})$ by

$\hat{r}_{m}(E_{ij})=\begin{cases}r_{m}(E_{ij}),\;\;\text{if}\;\;i\neq
j\;\text{or}\;\;i=j<0\\\ r_{m}(E_{ii})-id,\;\;\text{if}\;\;i\geq
0.\end{cases}$

and for the entire algebra $\bar{a}_{\infty}$ continue by linearity.

Notice that the elements
$\hat{r}(\Lambda_{j})\in\bar{a}_{\infty},\;j\in{\mathbb{Z}}$ together with the
central element $c\in\bar{a}_{\infty}$ satisfy the commutation relation

$[\hat{r}(\Lambda_{i}),\hat{r}(\Lambda_{j})]=i\delta_{i,-j}\cdot c.$

We obtain a highest weight representation of $\mathcal{A}$ in the fermionic
Fock space noticing that the operators $\hat{r}(\Lambda_{j})$ act on the
vacuum as

$\hat{r}_{m}(\Lambda_{j})\ket{m}=0,\;j=1,2,\ldots$

On the other hand the elements
$\hat{r}_{m}(\Lambda_{-j_{1}}),\ldots\hat{r}_{m}(\Lambda_{-j_{k}})\ket{m}$
span $F^{(m)}$ so we have a highest weight representation of the oscillator
algebra.

Similarly we can define a representation of $\mathcal{A}$ in the bosonic
spaces (5.17), using the operators

$J_{k}=\partial_{t_{k}}\;\text{and}\;\;J_{-k}=k{t_{k}}\;\;\text{for}\;\;k\geq
0.$

Notice that the operators $J_{k},\;k>0$ act on the vacuum $Q^{m}\cdot 1$ by
$J_{k}Q^{m}\cdot 1=0$ and the operators $J_{-k},\;k>0$ acting on it span the
entire bosonic space $B^{(m)}$.

It is known that the two representations of $\mathcal{A}$ in the bosonic and
in the fermionic Fock spaces are equivalent, see [29]. The map between
$F^{(m)}$ and $B^{(m)}$ is called boson-fermion correspondence Let us denote
it by $\sigma_{m}$.

Our next goal is to introduce the Virasoro algebra $Vir$. It is the algebra
spanned by a set of operators $L_{k},\;k\in{\mathbb{Z}}$ and $c$ (the central
element) with the following commutation relation

(5.18) $[L_{k},L_{m}]=(k-m)L_{k+m}+\frac{k^{3}-k}{12}\delta_{k,-m}c.$

We aim at defining two representations of this algebra. First we introduce the
bosonic field

$J(z)=\sum_{m\in{\mathbb{Z}}}J_{m}z^{-1-m}$

and define the operators $L_{k}$ by

$\frac{1}{2}\colon J(z)^{2}\colon=\sum_{m\in{\mathbb{Z}}}L_{m}z^{-m-2},$

where

$\colon J_{k}J_{l}\colon=\begin{cases}J_{k}J_{l},\;\text{if }\;k<0\\\
J_{l}J_{k},\;\text{if }\;k>0\end{cases}.$

Explicitly the operators $L_{m}$ are given by

$L_{m}=\sum_{l+k=-m}lkt_{l}t_{k}+\sum_{k=1}^{\infty}kt_{k}\partial_{t_{k+m}}+\frac{1}{2}\sum_{k+l=m}\frac{\partial^{2}}{\partial{t_{k}}\partial{t_{l}}}.$

One can show that they satisfy the commutation relation of the Virasoro
algebra.

We can define a fermionic representation of $Vir$. Let us take the space
${\mathbb{V}}$ to be spanned by functions
$v_{k}(z)=z^{k}+\mathcal{O}(z^{k-1}),\;\;k\in{\mathbb{Z}}$. We see that the
operators $d_{k}=-z^{k+1}\partial_{z}$ act on it and satisfy the commutation
relation

$[d_{k},d_{m}]=(k-m)d_{k+m},$

which defines centerless algebra. Then we can define action on the
corresponding fermionic Fock space $F^{(m)}$. It is easy to show that the
operators $\hat{r}(d_{k}),\;k\in{\mathbb{Z}}$ and $c=c_{m}$, (where $c_{m}$ is
not important here) define a representation of the Virasoro algebra.

One can show that the above described representations of the Virasoro algebra
in $F^{(m)}$ and $B^{(m)}$ are also equivalent, cf. [29].

Below we will present the formula for this equivalence in an appropriate
basis. Let us take instead of $d_{k}$ the elements
$b_{k}=-\partial_{z}z^{k+1}$. Then

(5.19)
$Y_{b_{k}}=\Res_{z}\Big{(}z^{k+1}\frac{\colon(J(z)+\partial_{z})J(z)\colon}{2}\Big{)}.$

######  Remark 5.1.

We point out that there are subalgebras of Vir, which are isomorphic to Vir,
cf. [29]. Namely let $p\in{\mathbb{N}}$ be fixed. Consider the elements
$L_{pj},\;j\in{\mathbb{Z}}$. These elements satisfy the commutation relations

$[L_{pk},L_{pn}]=p(k-n)L_{pk+pn}+\frac{(pk)^{3}-pk}{12}c\delta_{k,-n}.$

If we put

$\tilde{c}=pc,\;\tilde{L}_{j}=\frac{1}{p}L_{pj}\;\text{for}\;j\neq
0,\;\text{and}\;\tilde{L}_{0}=\frac{1}{p}\big{(}L_{0}+\frac{p^{2}c-c}{24}\big{)}$

we see that the new operators $\tilde{L}_{j}$ and $\tilde{c}$ satisfy the
Virasoro commutation relations (5.18). In what follows we are going to use
this subalgebra.

##  6\. Toda flows

In this section we briefly recall the formalism of the 2D-Toda hierarchy
following [40, 39]. It is formulated in terms of difference operators.

A pseudo-difference operator is a finite linear combination of the form

$A=\sum_{n=-\infty}^{N}a_{n}(s)\Lambda^{n}\;\;\text{(operator of
}(-\infty,N]\;\text{type})$

and

$A=\sum_{n=M}^{\infty}a_{n}(s)\Lambda^{n}\;\;\text{(operator of
}[M,+\infty)\;\text{type}).$

These operators are analogs of the pseudo-differential operators for the
theory of KP-hierarchy, [1, 17, 42, 19]. We briefly describe the construction.

Let the pseudo-difference operator $A$ have the form

$A=\Lambda+\sum_{n=-\infty}^{0}a_{n}(s)\Lambda^{n}.$

The operator $A$ can be conjugated to $\Lambda$ by some operator $W$ of the
form

$W=1+\sum_{j=0}^{\infty}\gamma_{j}\Lambda^{-j},$

called wave operator.

For an operator

$A=\sum_{-\infty}^{N}\gamma_{j}\Lambda^{j}$

as usually we define $A_{+}$ to be the part of $A$ containing the nonnegative
powers of $\Lambda$:

$A_{+}=\sum_{j=0}^{N}\gamma_{j}\Lambda^{j}$

and $A_{-}=A-A_{+}$. Then the Toda flows are given by

$A_{t_{k}}=[(A^{k})_{+},A].$

These flows act on the wave operator $W$ as

$W_{t_{k}}=-(A^{k})_{-}W.$

Consider a recurrence relation of the form

$xw_{n}(x)=w_{n+1}(x)+\sum_{j=0}^{d}\gamma_{j}(n)w_{n-j}(n).$

It defines a difference operator $D$

$D=\Lambda+\sum_{j=0}^{d}\gamma_{j}\Lambda^{-j},$

such that $Dw=xw$ with $w=(\ldots,w_{n-1},w_{n},w_{n+1},\ldots)^{T}$. $D$ can
be conjugated to $\Lambda$ using an operator of the form
$\overline{W}=1+\sum_{j=1}^{\infty}\gamma_{j}\Lambda^{-j}$ or to
$\Lambda^{-d}$ by using $W=\sum_{j=0}^{\infty}\delta_{j}\Lambda^{j}$,
$\delta_{0}\neq 0$.

$D=\overline{W}\Lambda\overline{W}^{-1}=W\Lambda^{-d}W^{-1}.$

One can easily show that the definition of $D^{\frac{1}{d}}$ is correct.

The 2D-Toda hierarchy is the system of equations

$\begin{split}\partial_{\bar{t}_{k}}D=[(D^{k})_{+},D]\\\
\partial_{{t}_{k}}D=[(D^{\frac{k}{d}})_{+},D].\end{split}$

Notice that in our case we can assume that $\bar{t}_{k}=t_{kd}$. For this
reason from now on we will work only with the set of variables $t_{k}$.

We will apply this in the situation from Lemma 4.4. It shows that we can
construct the vector

$\nu=(\ldots\nu(s+n-1,x),\nu(s,n,x),\nu(s+n+1,x),\ldots)^{T},\;n\in{\mathbb{Z}}.$

According to (1.2) we have

$x\nu(s,x)=(s+1)\nu(s+1,x)+\nu(s-d,x),$

which can be written as

$x\nu=D\nu$

where the difference operator is given by

$D=A\Lambda+\Lambda^{-d},A={\mathrm{diag}}(\ldots,s-1,s,s+1,\ldots).$

Next we put $x=y^{d}$, which gives the formula

(6.20) $y^{d}\nu(s)=(s+1)\nu(s+1)+\nu(s-d).$

There exists a tau-function $\tau(s,t,x)$ such that the corresponding solution
of the hierarchy (i.e. the coefficients of $D$) is given in terms of this
function. While we don’t need the explicit form of $D$ we do need the
connection of $\tau(s,t,x)$ with the vector $w$. Namely the boson-fermion
correspondence maps a plane of Sato’s Grassmannian (or wedge product) to the
tau-function. We explain the construction more precisely. Each vacuum
$\ket{m}$ corresponds to the vector $(w_{m},w_{m-1},\ldots)$. Then the boson-
fermion correspondence maps $\ket{m}$ to $\tau(m,t)$.

##  7\. Virasoro constraints

We denote by $L_{m}$ the operators of the Virasoro algebra. Our approach will
be a standard one - to obtain them from differential operators $A_{j}$ acting
on the functions $w(s,y)$. It is enough to do this for $j=-1,0,1,2$ and use
the identity $[L_{k},L_{m}]=(k-m)L_{k+m}$ to make induction. Then we transport
their action on the corresponding fermionic Fock space.

Let us first define the fermionic Fock space. Put $v_{j}=w(j,y)$ and

${\mathbb{V}}=\mathop{\oplus}_{j\in{\mathbb{Z}}}{\mathbb{C}}v_{j}.$

Let $m$ be an integer. The fermionic Fock space $F^{(m)}$ is spanned by the
wedge products

$v_{i_{m}}\wedge v_{i_{m-1}}\wedge\ldots,$

where $i_{m}>i_{m-1}>\ldots$ and $i_{k}=k$ for $k<<m$.

Define the operators $A_{-1},A_{0},A_{1}$ that act on $w(s,y)$ as follows

$\begin{split}&A_{-1}w(s,y)=-\frac{1}{d}y^{-d+1}\partial_{y}w(s,y)\\\
&A_{0}w(s,y)=-\frac{1}{d}y\partial_{y}w(s,y)\\\
&A_{1}w(s,y)=-\frac{1}{d}y^{d}[y\partial_{y}-ds]w(s,y).\end{split}$

Simple computations show that these operators satisfy the following
commutation relations

(7.21) $\begin{split}&[A_{0},A_{-1}]=A_{-1}\\\ &[A_{1},A_{0}]=A_{1}\\\
&[A_{1},A_{-1}]=2A_{0}.\end{split}$

Our goal will be to present the differential operators $A_{j},\;j=-1,0,1$ as
linear operators (eliminating the differentiation) in the basis $v_{k}$. Then
we will transport them to the fermionic Fock space $F^{(m)}$ to obtain the
Virasoro operators $L_{j}$. However we will see that with $L_{2}$ the
situation is more subtle - it does not come from any differential operator.

First we write $A_{j},\;j=-1,0,1$ in the space spanned by
$w(n,y),\;n\in{\mathbb{Z}}$.

######  Lemma 7.1.

The following identities hold

(7.22) $\begin{split}(i)\;A_{-1}w(s,y)&=-w(s-1,y)\\\
(ii)\;\;A_{0}w(s,y)&=-[sw(s,y)+w(s-d-1,y)]\\\
(iii)A_{1}w(s,y)&=-[(s-d)w(s-d)+w(s-2d-1)].\end{split}$

###### Proof.

Using that

$dx\partial_{x}=y\partial_{y}.$

we obtain

$A_{-1}w(s,y)=-d^{-1}y^{1-d}\partial_{y}w(s,y)=-w(s-1,y).$

The second identity follows from the first one by applying the recurrence
relation once:

$\begin{split}A_{0}w(s,y)&=y^{d}A_{-1}w(s,y)=-w(s-1,y)\\\
&=-[sw(s,y)+w(s-d-1)].\end{split}$

Repeat the above procedure to obtain the third relation:

$\begin{split}A_{1}w(s,y)&=-y^{d}w(s-d-1)\\\
&=-[(s-d)w(s-d)+w(s-2d-1)].\end{split}$

∎

######  Corollary 7.2.

In the basis $v_{j}$ the operators $A_{k}$ are given by

(7.23) $\begin{split}(i)\;A_{-1}v_{j}&=-v_{j-1}\\\
(ii)\;\;A_{0}v_{j}&=-[jv_{j}+v_{j-d-1}]\\\
(iii)\;A_{1}v_{j}&=-(j-d)v_{j-d}-v_{j-2d-1}.\end{split}$

We want to transport the action of the operators $A_{k}$ on the vacuum

$\ket{m}=v_{m}\wedge v_{m-1}\wedge\ldots$

######  Lemma 7.3.

We have

(7.24) $\begin{split}&\hat{r}_{m}(A_{-1})\ket{m}=0\\\
&\hat{r}_{m}(A_{0})\ket{m}=\frac{(m+1)m}{2}\ket{m}\\\
&\hat{r}_{m}(A_{1})\ket{m}=0.\\\ \end{split}$

###### Proof.

The first and the third identities are obvious. For the second one we need to
apply the regularization procedure.

∎

The operator corresponding to Virasoro $L_{2}$ will be obtained following an
idea from [25] (see also [5]). It will be constructed as a sum of two
operators, acting on the fermionic spaces $F^{(m)}$. The first one will be the
image of some differential operator $\tilde{A}_{2}$. The second one will be
$\beta[\hat{r}(y^{d})]^{2}$, where $\beta$ is some constant, which will be
determined later. Notice that $\beta[\hat{r}(y^{d})]^{2}$ is not an operator
that comes from differential one.

We define the operator $\tilde{A}_{2}$ as

$\tilde{A}_{2}=y^{2d}\big{[}A_{0}+\alpha\big{]},$

where $\alpha\in{\mathbb{C}}$ will be determined later. Now put

$L_{2}^{F}=\hat{r}_{m}(\tilde{A}_{2})-[\hat{r}_{m}(y^{d})]^{2}.$

######  Lemma 7.4.

If $\beta=-1,\;\alpha=2$ then

$L_{2}^{F}\ket{m}=0.$

###### Proof.

First let us compute action of $\tilde{A}_{2}$ on the vectors $v_{j}$. We use
the recurrence relation.

$\begin{split}&\tilde{A}_{2}v_{j}=y^{2d}[A_{0}+\alpha]v_{j}=\\\
&=-y^{2d}\big{[}(j-\alpha)v_{j}+v_{j-d-1}\big{]}.\\\ \end{split}$

Then using the recurrence relation twice we obtain

$\begin{split}\tilde{A}_{2}v_{j}&=-y^{d}\big{[}(j-\alpha)[(j+1)v_{j+1}+\mathcal{L}(j-d)\big{]}\\\
&=-(j-\alpha)(j+1)v_{j+2}+\mathcal{L}(j-d+1)\big{]},\end{split}$

where the symbol $\mathcal{L}(n)$ denotes terms $v_{n}$ with indexes $n$ and
lower.

Then for $\hat{r}_{m}(\tilde{A}_{2})\ket{m}$ we obtain

$\begin{split}&\hat{r}_{m}(\tilde{A}_{2})\ket{m}\\\
&=(m-\alpha)(m+1)v_{m+2}\wedge v_{m-1}\wedge\ldots\\\
&-(m-1-\alpha)mv_{m+1}\wedge v_{m}\wedge v_{m-2}\wedge\ldots\end{split}$

For the computation of $(\hat{r}_{m}(y^{d}))^{2}$ we will use the following
formula (see, e.g. [4])

$\begin{split}&[\hat{r}(b)]^{2}(v_{m}\wedge v_{m-1}\wedge\ldots)=\\\
&=\sum_{-\infty}^{m}(v_{m}\wedge v_{m-1}\wedge\ldots\wedge
b^{2}v_{l}\wedge\ldots)+2\sum_{k>l}(v_{m}\wedge v_{m-1}\wedge\ldots\wedge
bv_{k}\ldots\wedge bv_{l}\wedge\ldots).\end{split}$

Using this formula we obtain

$\begin{split}&(\hat{r}_{n}(y^{d}))^{2}[v_{m}\wedge v_{m-1}\wedge\ldots]\\\
&=(m+2)(m+1)v_{m+2}\wedge v_{m-1}\wedge v_{m-2}\ldots+m(m+1)v_{m}\wedge
v_{m+1}\wedge v_{m-2}\ldots\\\ &+2(m+1)mv_{m+1}\wedge v_{m}\wedge
v_{m-2}\ldots\\\ &=(m+2)(m+1)v_{m+2}\wedge v_{m-1}\wedge
v_{m-2}\ldots+m(m+1)v_{m+1}\wedge v_{m}\wedge v_{m-2}\ldots.\end{split}$

Finally for $\hat{r}(\tilde{A}_{2})+\beta(\hat{r}_{m}(y^{d}))^{2}$ we obtain

$\begin{split}&[\hat{r}(\tilde{A}_{2})+\beta(\hat{r}_{m}(y^{d}))^{2}]\ket{m}=\\\
&=\big{[}(m-\alpha)(m+1)+\beta(m+2)(m+1)\big{]}v_{m+2}\wedge
v_{m-1}\wedge\ldots\\\ &+\big{[}(m-1-\alpha)m+\beta m(m+1)\big{]}v_{m+1}\wedge
v_{m}\wedge\ldots\\\ \end{split}$

If we want that the above terms to cancel we have to put

$\beta=-1,\;\alpha=-2.$

∎

######  Theorem 7.5.

The bosonic representation of the operators $A_{j},\;j=0,\pm 1$ are as follows

$\begin{split}Y_{A_{-1}}&=\frac{1}{d}\big{[}\sum_{j=d}^{\infty}jt_{j}\partial_{t_{j-d}}+\frac{1}{2}\sum_{j=1}^{d-1}jt_{j}(d-j)t_{d-j}\big{]}\\\
Y_{A_{0}}&=\frac{1}{d}\sum_{j=1}^{\infty}jt_{j+d}\partial_{t_{j}}\\\
Y_{A_{1}}&=\frac{1}{d}\big{[}\sum_{j=1}^{\infty}jt_{j}\partial_{t_{j+d}}-\frac{1}{2}\sum_{j=1}^{d-1}\partial_{t_{j}}\partial_{t_{d-j}}\big{]}\end{split}$

The Virasoro operators $L_{j}$ for $j=0,\pm 1$ are given by $L_{j}=Y_{A_{j}}$.

###### Proof.

The boson-fermion correspondence (5.19) gives

$\begin{split}Y_{A_{-1}}&=\frac{1}{d}\Res_{y}\Big{(}y^{1-d}\frac{\colon(J(y)+\partial_{y})J(y)\colon}{2}\Big{)}\\\
&=\frac{1}{d}\big{[}\sum_{j=d}^{\infty}jt_{j}\partial_{t_{j-d}}+\frac{1}{2}\sum_{j=1}^{d-1}jt_{j}(d-j)t_{d-j}\big{]}.\end{split}$

Next consider

$Y_{A_{0}}=-\frac{1}{d}\Res_{y}\Big{(}y\frac{\colon(J(y)+\partial_{y})J(y)\colon}{2}\Big{)}=-\sum_{j=1}^{\infty}jt_{j+d}\partial_{t_{j}}.$

For the computation of $Y_{A_{1}}$ we need the formula

(7.25)
$\begin{split}Y_{y^{d}}&=-\Res_{y}\Big{(}y^{d}J(y)\Big{)}=\partial_{t_{d}}.\end{split}$

Then we get

$\begin{split}Y_{A_{1}}&=-\frac{1}{d}\Res_{y}\Big{(}y^{d+1}\frac{\colon(J(y)+\partial_{y})J(y)\colon}{2}\Big{)}-m\Res_{y}\Big{(}y^{d}J(y)\Big{)}\\\
&=-\sum_{j=1}^{\infty}jt_{j}\partial_{t_{j+d}}+\frac{1}{2}\sum_{j=1}^{d-1}\partial_{t_{j}}\partial_{t_{d-j}}+m\partial_{t_{d}}.\end{split}$

From the above formulas we obtain the explicit expressions for
$Y_{A_{j}},j=0,\pm 1$.

The last statement follows from the commutation relations for the operators
$A_{j}$.

∎

Now we turn to $L_{2}^{F}$.

######  Lemma 7.6.

The bosonic represenation of $L_{2}^{F}$ is given by

$L_{2}=\frac{1}{d}\big{[}\sum_{j=1}^{\infty}jt_{j}\partial_{t_{j+2d}}+\frac{1}{2}\sum_{j=1}^{2d-1}\partial_{t_{j}}\partial_{t_{2d-j}}+\partial_{t_{d}}^{2}\big{]}.$

###### Proof.

The same computation as above gives

$\begin{split}Y_{A_{2}}&=\frac{1}{d}\Res_{y}\Big{(}y^{2d+1}\frac{\colon(J(y)+\partial_{y})J(y)\colon}{2}+2y^{2d}J(y)\Big{)}\\\
&=\frac{1}{d}\big{[}\sum_{j=1}^{\infty}jt_{j}\partial_{t_{j+2d}}+\frac{1}{2}\sum_{j=1}^{2d-1}\partial_{t_{j}}\partial_{t_{2d-j}}+2\partial_{t_{2d}}\big{]}.\end{split}$

From this and from (7.25) we find

$L_{2}=\frac{1}{d}\big{[}\sum_{j=1}^{\infty}jt_{j}\partial_{t_{j+2d}}+\frac{1}{2}\sum_{j=1}^{2d-1}\partial_{t_{j}}\partial_{t_{2d-j}}+\partial_{t_{d}}^{2}\big{]}.$

∎

We sum up the obtained results in the next theorem.

######  Theorem 7.7.

The tau functions $\tau_{n}$ satisfy the Virasoro constraints

$\begin{split}&L_{-1}\tau_{m}=0\\\ &L_{0}\tau_{m}=\frac{m(m+1)}{2}\tau_{m}\\\
&L_{j}\tau_{m}=0,\;j=1,2,\ldots\end{split}$

##  8\. Discussion

The goal of this section is to review well known facts about some of the above
tau-functions and to put all of them in more general context. Following [31,
2, 33, 5] etc. one can show that all of them can be represented as matrix
integrals of the form

(8.26)
$Z(M)=\det(M)C\int\exp(-\Tr(\frac{\Phi^{d+1}}{d+1}+M\Phi+(m+1)\log(\Phi)))[\mathrm{d}\Phi].$

Here $M$ is a diagonal matrix
$M={\mathrm{diag}}(\mu_{1},\mu_{2},\ldots,\mu_{N})$, $\det M\neq 0$, $C\neq 0$
is some constant, which is irrelevant for the tau-function. The integration is
taken on the space of Hermitian $N\times N$ matrices. The connection with the
tau function $\tau_{m}(t_{1},t_{2},\ldots)$ is the following. Let us define
the parameters $t_{1},t_{2},\ldots$ by the well known Miwa parametrization:

$t_{j}=\frac{1}{k}\Tr M^{-j}.$

Then it is known that the function $\tau(t)=Z(M)$ up to irrelevant
multiplicative constant. The parametrization for any fixed number of variables
$t_{1},t_{2},\ldots$ does not depend on the size $N$ of the matrices provided
it is large enough. For more details see [33].

To be precise, it is proved for few cases. However they all can be done more
or less following the same pattern.

Some of these matrix integrals describe beautiful algebraic geometry -
intersection numbers on different moduli spaces. Namely introduce the function
(it is rather asymptotic expansion)

$F=\log(Z(t))=\sum_{\alpha}B_{\alpha}t^{\alpha}.$

Then the coefficients $B_{\alpha}$ give intersection numbers in different
situations.

The case of $d=2$ is the most important. When $m=-1$ it corresponds to the
Kontsevich integral [31, 44], which describes the intersection theory on all
moduli spaces $\mathcal{M}_{g,n}$ of compact Riemann surfaces of genus $g$ and
$n$ marked points. The case of $m=0$ describes the intersection theory of open
Riemann surfaces. These are Riemann surfaces from which a finite number of
discs have been deleted. For more details of this very recent theory see [5,
14] and the references therein.

Other cases of interest include arbitrary $d$ and $m=-1$. These are the
partition function for $r$-spin structures of type $A$ (here $r=d+1$). These
were introduced by Witten [44] who also formulated the conjecture that the
corresponding intersection theory is governed by the model (8.26) with $d>2$
and $m=-1$. The conjecture was proved in [22].

Recently there was a suggestion [12] that the case of $m=0$ will produce the
open analog for the $r$-spin structures.

It is natural to ask if the rest of the tau-functions have some algebro-
geometric meaning. This is not clear even for the case $d=2$.

Finally let us recall that instead of integer $m$ we can take any complex
number (see (4.13)). Then we will obtain the extended Toda hierarchy, see
[16]. It contains other flows. They deserve further studies.

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