CRM-3xxx(2006)
Fermionic approach to the evaluation of integrals of
rational symmetric functions 1
J. Harnad†‡2 and A. Yu. Orlov⋆3
† Centre de recherches mathématiques, Université de Montréal
C. P. 6128, succ. centre ville, Montréal, Québec, Canada H3C 3J7
‡ Department of Mathematics and Statistics, Concordia University
7141 Sherbrooke W., Montréal, Québec, Canada H4B 1R6
⋆ Nonlinear Wave Processes Laboratory,
Oceanology Institute, 36 Nakhimovskii Prospect
Moscow 117851, Russia
Abstract
We use the fermionic construction of two-matrix model partition functions to evaluate inte-
grals over rational symmetric functions. This approach is complementary to the one used in the
paper “Integrals of Rational Symmetric Functions, Two-Matrix Models and Biorthogonal Poly-
nomials” [1], where these integrals were evaluated by a direct method. Using Wick’s theorem,
we obtain the same determinantal expressions in terms of biorthogonal polynomials as in [1].
1 Introduction
In this work, we consider the following integral
N (ξ, ζ, η, µ) :=
dµ(x1, y1) . . .
dµ(xN , yN)∆N(x)∆N (y)
1Work of (J.H.) supported in part by the Natural Sciences and Engineering Research Council of
Canada (NSERC) and the Fonds FCAR du Québec; that of (A.O.) by the Russian Academy of Science
program “Mathematical Methods in Nonlinear Dynamics” and RFBR grant No 05-01-00498.
2harnad@crm.umontreal.ca
3orlovs@wave.sio.rssi.ru
http://arxiv.org/abs/0704.1150v1
α=1(ξα − xa)
β=1(ζα − ya)
j=1(ηj − xa)
k=1(µk − ya)
(1.1)
where
∆N(x) :=
(xi − xj), ∆N (y) :=
(yi − yj) (1.2)
are Vandermonde determinants, and
dµ(x1, y1) . . .
dµ(xN , yN)∆N (x)∆N(y), (1.3)
is a normalization constant, which in the context of two-matrix models is interpreted as
the partition function. Here dµ(x, y) is a measure (in general, complex), supported on a
finite set of products of curves in the complex x and y planes.
It is known (see for instance [5]) that this integral can be presented in the form of a
determinant of an N ×N matrix; namely,
N (ξ, ζ, η, µ) =
α=1(ξα − x)
β=1(ζβ − y)
j=1(ηj − x)
k=1(µk − y)
dµ(x, y)(x, y)xjyk
0≤j,k≤N−1
(1.4)
However for some purposes, it is more useful to express it as the determinant of a matrix
whose size does not depend on N .
The problem of evaluation of integrals of such symmetric functions of N variables is
of importance in the context of matrix models [4], where N is the number of eigenvalues.
Expressing integrals that determine correlation functions and expectation values as de-
terminants of matrices whose size is independent of N is of importance e.g., in the study
of N → ∞ limits. For one-matrix models such integrals were studied in [3], [6], [7], [8]
and [9]. In the case of complex, normal or two-matrix models [10] the problem was
considered in [8], [2], [1].
Remark 1.1. In order to compare with the results in [8] and [2] on complex integrals the
variables {xa, ya}a=1,...N must be replaced by complex conjugate pairs {za, z̄a}a=1,...N , and the
integration domains taken as N copies of the complex plane.
The present paper is complementary to ref. [1], where such integrals were evaluated
by a “direct” method based on partial fraction expansions and the Cauchy-Binet identity.
Ref. [1] and the present work are intended to provide concise presentations and new
derivations of previously known as well as new results, using two distinct methods.
The approach used here is based on constructing a fermionic representation of integrals
of such rational symmetric functions, introduced in [19], using two–component fermions.
This representation is different from the ones used previously in the context of matrix
models in [16], [17] and [18], which were based on one-component fermions.
Remark 1.2. Although there is a close connection between matrix integrals, orthogonal poly-
nomials and the spectral transform approach to the theory of integrable systems, in the present
work we do not develop this latter aspect, which concerns the deformation theory of the mea-
sures involved. Our starting point however is similar to the previous paper [19], where the
deformation theory is addressed, and the fermionic approach to integrable systems developed
in [14] is utilized. The relation of these results to the objects appearing in the spectral transform
approach is, however, briefly explained in the Appendices, where the biorthogonal polynomials
and their Hilbert transforms are interpreted as Baker functions and adjoint Baker functions.
The language of free two-component fermions used here is borrowed from [14]. The
integrals that we are interested in are expressed as vacuum state expectation values of
operator products formed from free fermion generators. Besides successive application
of Wick’s theorem, the main computational device used consists of applying canonical
(“dressing”) transformations to pass from free fermions fields, whose Laurent coefficients
are the free Fermi creation and annihilation operators, to “dressed’ ones, where the mono-
mials are replaced by biorthogonal polynomials. Wick’s theorem then serves to express the
same operator product vacuum state expectation value (VEV) both as a multiple ratio-
nal integral and as the determinant of a matrix formed from elementary factors involving
orthogonal polynomials and their Hilbert transforms.
Let dµ(x, y) be a measure (in general, complex), supported on a finite set of products
of curves in the complex x and y planes, for which the semi-infinite matrix of bimoments
is finite:
Bjk :=
dµ(x, y)xjyk <∞, 0, ∀j, k ∈ N. (1.5)
The integrals are understood to be evaluated on a specified linear combination of products
of the support curves. Assuming that, for all N ≥ 1, the N×N submatrix (Bjk)0≤j,k,≤N−1
is nonsingular, the Gram-Schmidt process may be used to construct an infinite sequence
of pairs of biorthogonal polynomials {Pj(x), Sj(y)}j=0...∞, unique up to signs, satisfying
dµ(x, y)Pj(x)Sk(y) = δjk, (1.6)
and normalized to have leading coefficients that are equal:
Pj(x) =
+O(xj−1), Sj(x) =
+O(yj−1). (1.7)
We will also assume that the Hilbert transforms of these biorthogonal polynomials,
P̃n(µ) :=
dµ(x, y)
Pm(x)
, S̃n(η) :=
dµ(x, y)
Sm(x)
η − x
, (1.8)
exist for all n ∈ N.
The following expression for Z
N in terms of the leading term normalization factors
hn is then easily shown to hold (see, e.g., [4]).
N = N !det (Bjk) |j,k=1,...,N = N !
hn (1.9)
Defining
N1 := N + L1 −M1, N2 := N + L2 −M2, (1.10)
we consider different cases. In each case the answer is written in form of the determinant
of a matrix G which is different for different cases. Results are given
(1) N2 ≥ N1 ≥ 0: by formulae (5.9)-(5.15). G is a (L2 +M1)× (L2 +M1) matrix
(2) N1 ≤ 0 ≤ N2: by formulae (5.34)-(5.35). G is a (L2 +M1)× (L2 +M1) matrix
(3) N1 ≤ N2 ≤ 0: by formulae (5.44)-(5.45). G is a (M1 +M2 −N)× (M1 +M2 −N)
matrix
The cases N1 ≥ N2 ≥ 0, N2 ≤ 0 ≤ N1 and N2 ≤ N1 ≤ 0 are related to previous ones
by interchanging quantities L1 ↔ L2, M1 ↔ M2, N1 ↔ N2, ξ ↔ ζ , µ ↔ η, Pn ↔ Sn and
P̃n ↔ S̃n in the final formulae. (See respectively (5.9)-(5.15) vs. (5.22)-(5.26), (5.34)-
(5.35) vs. (5.36)-(5.37) and (5.44)-(5.45) vs. (5.46)-(5.47).)
For instance, for the case N1 ≥ N2 ≥ 0 the answer is
N (ξ, ζ, η, µ) = (−1)
(M1+M2)(M1+M2−1)(−1)L1M2
N+L2−M2−1
N+L1−M1−1
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG,
(1.11)
where G is a (L1+M2)×(L1+M2) matrix whose entries are expressed in terms of the values
of Pn, Sn, P̃n and S̃n for n = 0, . . . ,max(N1, N2) evaluated at the points {ξα, ζβ, ηj , µk}.
Remark 1.3. It is assumed that the support curves for the integrals involved do not pass
through the points {ηj , µk}j=1,...M1,k=1,...M2 . The definition of the integrals may be extended
throughout the complex plane of these parameters by analytic continuation, but might then take
on multiple values. However, if the measures dµ(x, y) vanish with sufficient rapidity as these
points approach the support curves, the integrals considered in the present paper become single
valued. If the replacements {xa, ya}a=1,...N→{za, z̄}a=1,...N , are made, in order to compare with
the results of [8], [2], it must be assumed that the corresponding measures dµ(z, z̄ in the complex
plane vanish sufficiently rapidly at the points {z = ηj , z̄ = µk} for the integrals to converge.
In the following sections, we present the integral (1.1) as a vacuum state expectation
value of a certain fermionic expression, using two-component fermions (see (3.9) below).
To evaluate this expectation value we use Wick’s theorem, which expresses the vacuum
state expectation value of products of linear combinations of fermions as a determinant
of the matrix formed by evaluating the pair-wise VEV’s. Since the expression (3.9) is not
as yet of the form of a vacuum expectation value of a product of fermions, we use a set of
tricks (namely, canonical transformations of fermions and re-writing of charged vacuum
vectors in a suitable form) which serve also to reduce the final number of fermions inside
the vacuum expectation value, and therefore to reduce as much as possible the size of the
matrix G.
The structure of the paper is the following. First we recall some facts about free
fermions, including Wick’s theorem. Then we equal I
N (ξ, ζ, η, µ) to a certain vacuum
expectation value, see (3.12). After certain preliminaries, via (4.25), we introduce dressed
fermions bα, b̄α to adapt formula (3.12) for a usage of Wick’s theorem in its determinantal
form. Then we consider each of the numbered cases separately. For the case (1) (which
is the most involved case) we introduce fermionic operators aα, āα, obtained from bα, b̄α
via a sort of conjugation (see (5.4)-(5.7)), and whose vacuum expectation evaluated via
Wick’s theorem yields the final result given by formulae (5.9)-(5.15) and are illustrated by
six examples. For the case (2), instead of aα, āα, we similarly use operators bα, b̄α. At last,
for the case (3), instead of aα, āα, we exploit operators cα, c̄α introduced in (5.39)-(5.42).
Remark 1.4. One can obtain the corresponding results for N -fold integrals arising in one-
matrix models by specifying that the measure be proportional to Dirac delta function, say
δ(x − y). As this specialization will not be consider detailed here we shall write IN instead of
2 Summary of free fermions
The following is a summary regarding the one and two-component free fermion algebra
based on the introductory section of [19]. The reader may refer to [19], or [15], [14] for
further details.
2.1 One component fermions
In the following, A denotes the complex Clifford algebra over C generated by charged free
fermions {fi, f̄i}i∈Z, satisfying the anticommutation relations
[fi, fj]+ = [f̄i, f̄j]+ = 0, [fi, f̄j ]+ = δij . (2.1)
where [, ]+ denotes the anticommutator.
Elements of the linear part
W := (⊕m∈ZCfm)⊕
⊕m∈ZCf̄m
(2.2)
will be referred to as a free fermions. The fermionic free fields
f(x) :=
k, f̄(y) :=
−k−1, (2.3)
may be viewed as generating functions for the fj , f̄j’s.
This Clifford algebra has a standard Fock space representation F and dual space
F̄ (see [19]) which contain unique vacuum states |0〉 and 〈0| respectively satisfying the
properties
fm|0〉 = 0 (m < 0), f̄m|0〉 = 0 (m ≥ 0),
〈0|fm = 0 (m ≥ 0), 〈0|f̄m = 0 (m < 0). (2.4)
The Fock spaces F and F̄ are mutually dual, with the hermitian pairing defined via the
linear form 〈0||0〉 on A called the vacuum expectation value. This satisfies
〈0|1|0〉 = 1; 〈0|fmf̄m|0〉 = 1, m < 0; 〈0|f̄mfm|0〉 = 1, m ≥ 0, (2.5)
〈0|fn|0〉 = 〈0|f̄n|0〉 = 〈0|fmfn|0〉 = 〈0|f̄mf̄n|0〉 = 0; 〈0|fmf̄n|0〉 = 0, m 6= n, . (2.6)
Wick’s theorem implies that for any finite set of elements {wk ∈ W}, we have
〈0|w1 · · ·w2n+1|0〉 = 0,
〈0|w1 · · ·w2n|0〉 =
σ∈S2n
sgnσ〈0|wσ(1)wσ(2)|0〉 · · · 〈0|wσ(2n−1)wσ(2n)|0〉. (2.7)
Here σ runs over permutations for which σ(1) < σ(2), . . . , σ(2n− 1) < σ(2n) and σ(1) <
σ(3) < · · · < σ(2n− 1).
If {wi}i=1,...,N , are linear combinations of the fj ’s only, j ∈ Z, and {w̄i}i=1,...,N linear
combinations of the f̄j ’s, j ∈ Z, then(2.7) implies
〈0|w1 · · ·wN w̄N · · · w̄1|0〉 = det (〈0|wiw̄j|0〉) |i,j=1,...,N (2.8)
Following [14], [15], for all N ∈ Z, we also introduce the states
〈N | := 〈0|CN (2.9)
where
CN := f̄0 · · · f̄N−1 if N > 0 (2.10)
CN := f−1 · · · fN if N < 0 (2.11)
CN := 1 if N = 0 (2.12)
|N〉 := C̄N |0〉 (2.13)
where
C̄N := fN−1 · · · f0 if N > 0 (2.14)
C̄N := f̄N · · · f̄−1 if N < 0 (2.15)
C̄N := 1 if N = 0 (2.16)
The states (2.9) and (2.13) are referred to as the left and right charged vacuum vectors,
respectively, with charge N .
From the relations
〈0|f̄N−kf(xi)|0〉 = xN−ki , 〈0|f−N+k−1f̄(yi)|0〉 = yN−ki , k = 1, 2, . . .N, (2.17)
and (2.8), it follows that
〈N |f(xn) · · ·f(x1)|0〉 = δn,N∆N(x), N ∈ Z, (2.18)
〈−N |f̄(yn) · · · f̄(y1)|0〉 = δn,N∆N(y), N ∈ Z. (2.19)
For free fermion generators with |x| 6= |y|,
〈0|f(x)f̄(y)|0〉 = 1
(2.20)
Note that the expression on the right hand side is actually defined, by (2.5), as the
infinite series
n=0 y
nx−n−1 which converges only inside |x| < |y|. However one can
consider expression (2.20) for the whole region of x and y (when |x| 6= |y|) in the sense of
analytical continuation.
From Wick’s theorem it follows that
〈n−m|f(xn) · · ·f(x1)f̄(ym) · · · f̄(y1)|0〉 =
∆n(x)∆m(y)
i=1,...,n
j=1,...,m
(xi − yj)
(2.21)
2.2 Two-component fermions
The two-component fermion formalism is obtained by relabelling the above as follows.
f (α)n := f2n+α−1 , f̄
n := f̄2n+α−1 , (2.22)
(α)(z) :=
k , f̄
(α)(z) :=
k , (2.23)
where α = 1, 2. Then (2.1) is equivalent to
[f (α)n , f
m ]+ = [f̄
n , f̄
m ]+ = 0, [f
n , f̄
m ]+ = δα,βδnm. (2.24)
We denote the right and left vacuum vectors respectively as
|0, 0〉 := |0〉, 〈0, 0| := 〈0|. (2.25)
Relations (2.4) then become, for α = 1, 2,
f (α)m |0, 0〉 = 0 (m < 0), f̄ (α)m |0, 0〉 = 0 (m ≥ 0), (2.26)
〈0, 0|f (α)m = 0 (m ≥ 0), 〈0, 0|f̄ (α)m = 0 (m < 0). (2.27)
For n,m ∈ Z, i, j = 1, 2 it follows from (2.5)-(2.6) that
〈0, 0|f (i)n f (j)m |0, 0〉 = 〈0, 0|f̄ (i)n f̄ (j)m |0, 0〉 = 0, (2.28)
〈0, 0|f (i)n f̄ (j)m |0, 0〉 = δijδnm, n < 0 (2.29)
Following [14], [15], we also introduce the states
〈n(1), n(2)| := 〈0, 0|Cn(1)Cn(2), (2.30)
where
Cn(α) := f̄
0 · · · f̄
n(α)−1
if n(α) > 0 (2.31)
Cn(α) := f
−1 · · · f
if n(α) < 0 (2.32)
Cn(α) := 1 if n
(α) = 0 (2.33)
|n(1), n(2)〉 := C̄n(2)C̄n(1)|0, 0〉 (2.34)
where
C̄n(α) := f
n(α)−1
· · ·f (α)0 if n(α) > 0 (2.35)
C̄n(α) := f̄
· · · f̄ (α)−1 if n(α) < 0 (2.36)
C̄n(α) := 1 if n
(α) = 0 (2.37)
The states (2.30) and (2.34) will be referred to, respectively, as left and right charged
vacuum vectors with charges (n(1), n(2)).
It follows that
f (1)m |n, ∗〉 = 0 (m < n), f̄ (1)m |n, ∗〉 = 0 (m ≥ n), (2.38)
〈n, ∗|f (1)m = 0 (m ≥ n), 〈n, ∗|f̄ (1)m = 0 (m < n), (2.39)
f (2)m |∗, n〉 = 0 (m < n), f̄ (2)m |∗, n〉 = 0 (m ≥ n), (2.40)
〈∗, n|f (2)m = 0 (m ≥ n), 〈∗, n|f̄ (2)m = 0 (m < n). (2.41)
Remark 2.1. Note that if we shift the charges of the vacuum vectors
〈∗, ∗| → 〈∗, ∗+ n|, |∗, ∗〉 → |∗, ∗ + n〉 (2.42)
and, at the same time, re-label
i → f
i+n, f̄
i → f̄
i+n, i, n ∈ Z, (2.43)
then the vacuum expectation values remain invariant.
Remark 2.2. Wick’s theorem will be used in the form (2.8) in the following. In the two
component notation, there are two possible ways to do this; either
(1) Use formula (2.8), remembering that 2-component fermions consist just of the usual even
and odd ones (see (2.22)), or
(2) Use formula (2.8) separately for each component. To calculate the vacuum state expec-
tation value of an operator O, we first present it in the form
i . (2.44)
〈0, 0|O|0, 0〉 =
〈0, 0|O(1)i O
i |0, 0〉 =
〈0, 0|O(1)i |0, 0〉〈0, 0|O
i |0, 0〉 (2.45)
where Wick’s theorem in the form (2.8) may be applied to each factor 〈0, 0|O(α)i |0, 0〉
The version (2) is used in the Section 3, and the version (1) is used in the Section 5.
Now define
F (j)(x(j)) := f (j)(x(j)nj ) · · ·f
(j)(x
F̄ (j)(y(j)) := f̄ (j)(y(j)mj ) · · · f̄
(j)(y
1 ), j = 1, 2 (2.46)
Combining (2.21) and (2.45) we obtain
〈n1 −m1, n2 −m2|F (2)(x(2))F (1)(x(1))F̄ (1)(y(1))F̄ (2)(y(2))|0, 0〉
= (−1)m2(m1+n1) ∆n1(x
(1))∆m1(y
(1))∆n2(x
(2))∆m2(y
i=1,...,n1
j=1,...,m1
i − y
i=1,...,n2
j=1,...,m2
i − y
. (2.47)
expectation value
3 The integral of rational symmetric functions as a
certain expectation value
We shall use the following notations
η = (η1, . . . , ηM1), ξ = (ξ1, . . . , ξL1), µ = (µ1, . . . , µM2), ζ = (ζ1, . . . , ζL2) (3.1)
Nj = N + Lj −Mj, j = 1, 2, (3.2)
dµ(x,y)(·) =
dµ(x1, y1) . . .
dµ(xN , yN)(·) (3.3)
F (1)(ξ) := f (1)(ξL1) · · ·f (1)(ξ1), (3.4)
F (2)(µ) := f (2)(µM2) · · ·f (2)(µ1), (3.5)
F̄ (1)(η) := f̄ (1)(ηM1) · · · f̄ (1)(η1), (3.6)
F̄ (2)(ζ) := f̄ (2)(ζL2) · · · f̄ (2)(ζ1), (3.7)
g := eA, A :=
f (1)(x)f̄ (2)(y)dµ(x, y), (3.8)
and consider the expression
JN(ξ, ζ, η, µ) := RN 〈N1,−N2|F (2)(µ)F (1)(ξ) g F̄ (1)(η)F̄ (2)(ζ)|0, 0〉 (3.9)
where
RN = RN (ξ, ζ, η, µ) :=
s(L1, L2,M1,M2)
n=0 hn
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
(3.10)
and s(L1, L2,M1,M2) is the sign factor
s(L1, L2,M1,M2)) = (−1)
N(N+1)+L2(L1+M1)+N(L1+M1)+M2L2 (3.11)
Remark 3.1. Because of the form (3.8), g commutes with both F (1)(ξ) and F̄ (2)(µ). Moreover
he conditions given in Remark (1.3) also imply that g commutes with F̄ (1)(η) and F (2)(ζ).
The main result of this subsection is the equality
IN(ξ, ζ, η, µ) = JN(ξ, ζ, η, µ) (3.12)
Proof. Inserting
into (3.9), we note that only the N -th power contributes, giving
JN(ξ, ζ, η, µ) =
〈N1,−N2|F (2)(µ)F (1)(ξ)AN F̄ (1)(η)F̄ (2)(ζ)|0, 0〉 (3.13)
Collecting the fermion terms of the same types (which gives rise to a sign factor), the
right hand side of (3.13) may be expressed as
N(N−1)+NM1
dµ(x,y)〈N1,−N2|F (2)(µ)F (1)(ξ, x)F̄ (1)(η)F̄ (2)(y, ζ)|0, 0〉
where (ξ, x) = (ξ1, . . . , ξL1, x1, . . . , xN ) and (y, ζ) = (y1, . . . , yN , ζ1, . . . , ζL2). Using (2.47)
to evaluate the expectation value in the integrand we get
JN(ξ, ζ, η, µ) =
dµ(x,y)∆N(x)∆N (y)
α=1(ξα − xa)
β=1(ζα − ya)
j=1(ηj − xa)
k=1(µk − ya)
= IN(ξ, ζ, η, µ).
Remark 3.2. From (3.9) and (3.12) in the absence of the F (1), F (2), F̄ (1), F̄ (2) terms, we obtain
the representation which we used in [19]
= (−1)
N(N+1)N !〈N,−N |g|0, 0〉.
Having the representation (3.12) we evaluate IN(ξ, ζ, η, µ) via Wick’s theorem. Before
we need certain preliminary relations.
4 Biorthogonal polynomials and dressed fermions
In this section we introduce transformations Ω and Q and dressed fermions d(α), d̄(α) and
bα, b̄α. We re-write the v.e.v. (3.9) in a form suitable for further calculation by means of
Wick’s theorem.
(a) Biorthogonal polynomials.
Consider the sequence of biorthogonal polynomials associated to the measure dµ(x, y):
Pn(x)Sm(y)dµ(x, y) = δn,m, n,m ≥ 0 (4.1)
It is convenient to write down the biorthogonal polynomials in the following form
Pn(x) =
m, Sn(y) =
ym(K̄−1)mn, n ≥ 0 (4.2)
where Knm and (K̄
−1)mn are respectively viewed as entries of semi-infinite matrices K
and (K̄−1). K is a lower triangular matrix and K̄ is an upper triangular one, both have
units on the diagonal: Knn = K̄nn = 1, n = 0, 1, 2, . . . .
As is well-known, the orthogonality relation (4.1) implies
HK̄ = KB, (4.3)
where H = diag(hn) and B is the bi-moment matrix,
Bnm =
xnymdµ(x, y), m, n ≥ 0, (4.4)
In time (see [11]), (4.3) was made good use to relate matrix models to integrable
systems via the factorization method widely used in soliton theory starting from the basic
paper [23]. Then K and K̄ may be identified with the Mikhailov-Ueno-Takasaki dressing
matrices (see [20], [22]), whose rows give rise to a pair the so-called Baker functions, while
columns of K−1and of K̄−1 give rise to a pair of adjoint Baker functions. In case the
dressing matrices are semi-infinite, the first of the Baker functions (related to rows of K)
and the second of the adjoint Baker function (related to columns K̄−1) take the form of
quasi-polynomials.
Here we shall use (4.3) differently.
(b) Canonical transformation Ω.
First, we introduce
S̃n(x) :=
x−m−1(K−1)mn
hn, P̃n(y) :=
K̄nmy
hn, n ≥ 0 (4.5)
Properties of S̃n(x) and P̃n(y) are described in the Appendix C. The main is that they
coincide with Hilbert transforms of biorthogonal polynomials (1.8).
Now, let us define matrices ω(1) and ω(2) via
= K, eω
= K̄ (4.6)
(Respectively, strictly upper and strictly lower) triangular matrices ω(1),(2) may be defined
in a unique way by a recursion procedure.
We introduce
Ω = exp
n>m≥0
ω(1)n,mf
m + ω
−m−1f̄
(4.7)
Note that, for any N1 and N2,
〈N1,−N2|Ω−1 = 〈N1,−N2| , Ω|0, 0〉 = |0, 0〉 (4.8)
The first equality in (4.8) follows from (2.39),(2.41) and from the restriction n > m in the
summation in (4.7). The second equality in (4.8) follows from the restriction m ≥ 0 and
from (2.39),(2.41).
Consider
d(α)(x) = Ωf (α)(x)Ω−1, d̄(α)(x) = Ωf̄ (α)(x)Ω−1, α = 1, 2 (4.9)
which we refer as dressed fermions.
We have (see Appendix B)
d(1)(x) =
f (1)n Pn(x)
hn (4.10)
d̄(1)(x) =
f̄ (1)n
S̃n(x)√
(4.11)
d(2)(y) =
P̃n(y)√
(4.12)
(2)(y) =
−n−1Sn(y)
hn (4.13)
where for n < 0 we use the following notational convention
Pn(x) = Sn(x) = x
n, S̃n(x) = P̃n(x) = x
−n−1, hn = 1, n < 0 (4.14)
A kind of dressed fermionic operators (where powers of x were replaced by Baker
functions similar to (4.10)-(4.13)), in a different context, were also introduced in [30]
where they were called Krichever-Novikov fermions (see Appendix to [30]).
If we write
d(α)(x) =
d(α)n x
n, d̄(α)(x) =
d̄(α)n x
−n−1, α = 1, 2
where
d(α)n = Ωf
−1, d̄(α)n = Ωf̄
n≥i≥0
f (1)n Kni, i ≥ 0, d
i = f
i , i < 0 (4.15)
i≥n≥0
(K−1)inf̄
n , i ≥ 0, d̄
i = f̄
i , i < 0 (4.16)
−i−1 =
i≥n≥0
−n−1K̄ni, i ≥ 0, d
−i−1 = f
−i−1, i < 0 (4.17)
−i−1 =
n≥i≥0
(K̄−1)inf̄
−n−1, i ≥ 0, d̄
−i−1 = f̄
−i−1, i < 0 (4.18)
(c) Useful formulae for charged vacuum vectors.
First, we introduce
gn := e
−n−1 = 1 + f (1)n f̄
−n−1, (4.19)
en := e
−n−1 = 1 + f̄ (1)n f
−n−1 (4.20)
We see that gn, en ∈ ĜL∞.
We shall also consider powers of these operators
p = epf
−n−1 = 1 + pf (1)n f
−n−1, (4.21)
p = epf̄
−n−1 = 1 + pf̄ (1)n f
−n−1 (4.22)
where n ∈ Z, p ∈ C.
Now, due to
〈0, 0|engn = 〈0, 0|(1+f̄ (1)n f
−n−1)(1+f
−n−1) = 〈0, 0|(1−1+f̄ (1)n f
−n−1) = 〈0, 0|f̄ (1)n f
−n−1,
which is true for n > 0, and due to (A.1), (A.2), we conclude that for N > 0
〈N,−N | = (−1)
N(N−1)〈0, 0|
where 〈N,−N | was defined in (2.30)-(2.32).
In the similar way we obtain a representation we shall need later
N2 > N1 ≥ 0 : 〈N1,−N2| = (−1)
N1(N1−1)〈0, 0|f (2)−N1−1 · · · f
gn (4.23)
(d) Evaluation of Ωg|0, 0〉.
Taking into account that f
−n−1|0, 0〉 = f̄
n |0, 0〉 = 0 for n > 0, one obtains
Ωg|0, 0〉 = e
i,j,n,m≥0 KinBnmK̄mjf
−j−1 |0, 0〉 = Q−1|0, 0〉, Q−1 :=
hn (4.24)
where we used (4.15),(4.18) and (4.3).
(e) Fermionic operators bα, b̄α.
We shall also need the following fermionic operators
bα(x) := Qd
(α)(x)Q−1 = QΩ f (α)(QΩ)−1, b̄α(x) := Qd̄
(α)(x)Q−1 = QΩ f̄ (α)(QΩ)−1
(4.25)
Using (A.3)-(A.6) we write down
b1(ξ) =
f (1)n Pn(ξ)
hn = d
(1)(ξ) , (4.26)
b̄1(η) =
f̄ (1)n
S̃n(η)√
−n−1S̃n(η)
hn , (4.27)
b2(µ) =
P̃n(µ)√
f (1)n P̃n(µ)
hn , (4.28)
b̄2(ζ) =
−n−1Sn(ζ)
hn = d̄
(2)(ζ) (4.29)
Notice that each of b̄1 and b2 contains both component fermions.
The fermionic operators bα, b̄α may be also called dressed fermions. Similarly to (3.4)-
(3.7), for their products, we shall use large letters, namely, we define
B1(ξ) := QΩ F
(1)(ξ) (QΩ)−1, B2(µ) := QΩ F
(2)(µ) (QΩ)−1,
B̄1(η) := QΩ F̄
(1)(η) (QΩ)−1, B̄2(ζ) := QΩ F̄
(2)(ζ) (QΩ)−1
(f) IN as a re-written vacuum expectation value
Now we restate (3.12) as
IN(ξ, ζ, η, µ)R
N = 〈N1,−N2|Q
−1 B2(µ)B1(ξ)B̄1(η)B̄2(ζ)|0, 0〉 (4.30)
where RN was defined by (3.10).
Now we have to consider the cases (1),(2),(3) separately.
5 Determinantal expression for integrals of rational
symmetric functions
Below we apply Wick’s theorem to evaluate v.e.v. in the right hand side of (4.30), getting
answers for I
N (ξ, ζ, η, µ) for all the cases listed in the Introduction.
5.1 Determinantal representation in the case (1): N2 ≥ N1 ≥ 0
In the case (1) we have the following formulae for 〈N1,−N2|Q−1:
N2 ≥ N1 ≥ 0 : 〈N1,−N2|Q−1 = (−1)
N1(N1+1)〈0, 0|f (2)−N1−1 · · · f
hn , (5.1)
where
n (5.2)
Formula (5.1) follows from relations
〈0, 0|eng1+hnn = 〈0, 0|(1 + f̄ (1)n f
−n−1)(1 + (1 + hn)f
−n−1) = 〈0, 0|(−hn + f̄ (1)n f
−n−1)
= −hn〈0, 0|e
n , n ≥ 0
and from (4.23).
It is important that
E|0, 0〉 = 0 , (5.3)
which is true as each epn|0, 0〉 = 0, n ≥ 0.
Using (A.3)-(A.6), we may evaluate
a1(ξ) := Eb1(ξ)E
f (1)n Pn(ξ)
Pn(ξ)√
, (5.4)
ā1(η) := Eb̄1(η)E
f̄ (1)n
S̃n(η)√
f̄ (1)n
S̃n(η)√
−n−1S̃n(η)
hn , (5.5)
a2(µ) := Eb
(2)(µ)E−1 =
P̃n(µ)√
P̃n(µ)√
f (1)n P̃n(µ)
hn , (5.6)
ā2(ζ) := Eb̄
(2)(ζ)E−1 =
−n−1Sn(ζ)
f̄ (1)n
Sn(ζ)√
(5.7)
Using notations
A1(ξ) := EQΩ F
(1)(ξ) (EQΩ)−1, A2(µ) := EQΩ F
(2)(µ) (EQΩ)−1,
Ā1(η) := EQΩ F̄
(1)(η) (EQΩ)−1, Ā2(ζ) := EQΩ F̄
(2)(ζ) (EQΩ)−1
for the products, by analogy with (3.4)-(3.7), by (5.1) and (5.3), we arrive at
IN(ξ, ζ, η, µ)R
N = (−1)
N1(N1+1)
hn〈0, 0|f (2)−N1−1 · · · f
A2(µ)A1(ξ)Ā1(η)Ā2(ζ)|0, 0〉
(5.8)
Finely, this form is applicable to apply the Wick’s formula (2.8). Indeed, each ai is
a linear combination of fermions f (1) and f (2), while each āi is a linear combination of
fermions f̄ (1) and f̄ (2).
Then, by (2.8), the vacuum expectation value in formula (5.8) is equal to the deter-
minant of a L1 +M2 by L1 +M2 matrix, which consists of six blocks formed by pair-wise
v.e.v.:
〈a1(ξα)ā1(ηj)〉 〈a2(µk)ā1(ηj)〉 〈f (2)i−1−N−L2+M2 ā1(ηj)〉
〈a1(ξα)ā2(ζβ)〉 〈a2(µk)ā2(ζβ)〉 〈f (2)i−1−N−L2+M2ā2(ζβ)〉
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2
i = 1, . . . ,M1 + L2 − L1 −M2
Now, taking into account (2.29)-(2.29), from the explicit formulae (5.4)-(5.7), and also
from (C.3) and (C.4), we compute all relevant pair-wise vacuum expectation values
〈a1(ξi)ā1(ηj)〉 =
ξi − ηj
Pn(ξi)S̃n(ηj),
〈a2(µi)ā2(ζj)〉 =
Sn(ζj)P̃n(µi) = −
ζj − µi
Sn(ζj)P̃n(µi),
〈a1(ξi)ā2(ζj)〉 =
Pn(ξi)Sn(ζj),
〈a2(µi)ā1(ηj)〉 =
S̃n(ηj)P̃n(µi) =
dµ(x, y)
(ηj − x)(µi − y)
S̃n(ηj)P̃n(µi),
〈f (2)i−1−N−L2+M2 ā1(ηj)〉 =
hN+L2−M2−iS̃N+L2−M2−i(ηj),
〈f (2)i−1−N−L2+M2 ā2(ζj)〉 =
hN+L2−M2−iSN+L2−M2−i(ζj)
After trivial manipulations with rows and columns of the matrix of pair-wise v.e.v.
we obtain the answer:
IN(ξ, ζ, η, µ) = (−1)
(M1+M2)(M1+M2−1)(−1)L2M1
N+L1−M1−1
N+L2−M2−1
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG,
(5.9)
where the matrix G is (L2 +M1)× (L2 +M1) matrix which consists of six blocks:
K11(µk, ζβ)
K21(ξα, ζβ) SN+L1−M1(ζβ) . . . SN+L2−M2−1(ζβ)
K12(µk, ηj)
K22(ξα, ηj) S̃N+L1−M1(ηj) . . . S̃N+L2−M2−1(ηj)
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2
and where N1 = N + L1 −M1 and
K11(µ, ζ) =
P̃n(µ)Sn(ζ) +
ζ − µ
(5.10)
K22(ξ, η) =
Pn(ξ)S̃n(η) +
ξ − η
(5.11)
K21(ξ, ζ) =
Pn(ξ)Sn(ζ) (5.12)
K12(µ, η) =
P̃n(µ)S̃n(η)−
dµ(x, y)
(η − x)(µ− y)
(5.13)
SN+L1−M1+i−1(ζj), i = 1, . . . ,M1 − L1 −M2 + L2, j = 1, . . . , L2 (5.14)
S̃N+L1−M1+i−1(ηj), i = 1, . . . ,M1 − L1 −M2 + L2, j = 1, . . . ,M1, (5.15)
Now we shall consider six examples, related to the six-block structure, where, in each
case, only one entry contributes.
Example 1. M1 = 1 and L1 = L2 = M2 = 0, thus N2 > N1 ≥ 0. We put η1 = η. In
this case the matrix has only one non-vanishing element, giving the well-known formula
IN(η) =
S̃N−1(η) (5.16)
Example 2. L2 = 1 and L1 = M1 = M2 = 0, thus N2 > N1 ≥ 0. We put ζ1 = ζ . In
this case the matrix has only one non-vanishing element, giving the well-known formula
IN(ζ) =
hNSN(ζ) (5.17)
Examples below are related to the equality N2 = N1 ≥ 0.
Example 3. M1 = L1 = 1 and L2 = M2 = 0. We put ξ1 = ξ and η1 = η. In this case
the matrix has only one non-vanishing element and we obtain
IN(ξ, η) = 1 + (ξ − η)
Pn(ξ)S̃n(η) (5.18)
Similarly, we have
Example 4. L2 = M2 = 1 and L1 = M1 = 0. In this case the matrix has only one
non-vanishing element, and we obtain
IN(ζ, µ) = −(ζ − µ)
Sn(ζ)P̃n(µ) = 1 + (ζ − µ)
Sn(ζ)P̃n(µ) (5.19)
Example 5. M1 = M2 = 1 and L1 = L2 = 0. In this case the matrix has only one
non-vanishing element. We obtain
IN(η, µ) =
n=N−1
S̃n(η)P̃n(µ) =
dµ(x, y)
(η − x)(µ − y)
S̃n(η)P̃n(µ)
(5.20)
Example 6. L1 = L2 = 1 and M1 = M2 = 0. In this case the matrix has only one
non-vanishing element and we obtain
IN(ξ, ζ) = hN
Pn(ξ)Sn(ζ) (5.21)
Evaluation for the case N1 ≥ N2 ≥ 0
This case may be obtained from the previous one, by interchanging subscripts L1 ↔ L2,
M1 ↔ M2, N1 ↔ N2, and ξ ↔ ζ , µ↔ η and also Pn ↔ Sn, P̃n ↔ S̃n.
We obtain the answer which coincides with the answer given by formulae (1.8)-(1.16)
of [1]:
IN(ξ, ζ, η, µ) = (−1)
(M1+M2)(M1+M2−1)(−1)L1M2
N+L2−M2−1
N+L1−M1−1
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG,
(5.22)
where (L1 +M2)× (L1 +M2) matrix G consists of six blocks:
K11(ξα, ηj)
K12(ξα, ζβ) PN+L2−M2(ξα) . . . PN+L1−M1−1(ξα)
K21(µk, ηj)
K22(µk, ζβ) P̃N+L2−M2(µk) . . . P̃N+L1−M1−1(µk)
where
α = 1, . . . , L1; β = 1, . . . , L2; ; j = 1, . . . ,M1; k = 1, . . . ,M2
and where N2 = N + L2 −M2 and
K11(ξ, η) =
Pn(ξ)S̃n(η) +
ξ − η
(5.23)
K22(µ, ζ) =
P̃n(µ)Sn(ζ) +
ζ − µ
(5.24)
K12(ξ, ζ) =
Pn(ξ)Sn(ζ) (5.25)
K21(µ, η) =
P̃n(µ)S̃n(η)−
dµ(x, y)
(η − x)(µ− y)
(5.26)
Example 7. M2 = 1 and L2 = L1 =M1 = 0, thus N1 > N2 ≥ 0. We put µ1 = µ. Then
IN(µ) =
P̃N−1(µ) (5.27)
Example 8. L1 = 1 and L2 = M2 = M1 = 0, thus N1 > N2 ≥ 0. We put ξ1 = ξ. We
obtain the well-known formula
IN(ξ) =
hNPN(ξ) (5.28)
5.2 When N1 < 0
In cases listed in the Introduction as (2) and (3) we have N1 < 0. Let us explicitly write
N1 ≤ 0 ≤ N2 : 〈N1,−N2| = 〈0, 0|f (1)−1 · · · f
−1 · · · f
, (5.29)
N1 ≤ N2 ≤ 0 : 〈N1,−N2| = 〈0, 0|f (1)−1 · · · f
0 · · · f̄
−N2−1
(5.30)
Then, as it follows from (A.3)-(A.4), in either case
〈N1,−N2|Q−1 = 〈N1,−N2| (5.31)
Thus, in the both cases, we re-write (4.30) as
IN(ξ, ζ, η, µ)R
N = 〈N1,−N2|B2(µ)B1(ξ)B̄1(η)B̄2(ζ)|0, 0〉 (5.32)
5.3 Evaluation for the case (2): N1 ≤ 0 ≤ N2.
Thus, by (5.32) and (5.29) we have
IN(ξ, ζ, η, µ)R
N = 〈0, 0|f
−1 · · · f
−1 · · · f
B2(µ)B1(ξ)B̄1(η)B̄2(ζ)|0, 0〉 (5.33)
By Wick’s formula (2.8) the right hand side is the determinant of a L2+M1 by L2+M1
matrix which consists of eight blocks:
〈b1(ξα)b̄1(ηj)〉 〈b2(µk)b̄1(ηj)〉 〈f (2)b−1−N−L2+M2 b̄1(ηj)〉 〈f
m−1+N+L1−M1
b̄1(ηj)〉
〈b1(ξα)b̄2(ζβ)〉 〈b2(µk)b̄2(ζβ)〉 〈f (2)b−1−N−L2+M2 b̄2(ζβ)〉 〈f
m−1+N+L1−M1
b̄2(ζβ)〉
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2;
b = 1, . . . , N + L2 −M2; m = 1, . . . ,−N − L1 +M1
Then using (4.26)-(4.29), we evaluate all pair-wise vacuum expectation values:
〈b1(ξα)b̄1(ηj)〉 =
1− ηj
ξα − ηj
〈b2(µk)b̄2(ζβ)〉 =
P̃n(µk)Sn(ζβ) =
µk − ζβ
〈b2(µk)b̄1(ηj)〉 =
P̃n(µk)S̃n(ηj) = H(µk, ηj)
〈b1(ξα)b̄2(ζβ)〉 = 0,
〈f (2)
b−1−N−L2+M2
b̄1(ηj)〉 =
hN+L2−M2−bS̃N+L2−M2−b(ηj), b = 1, . . . , N + L2 −M2,
〈f (2)
b−1−N−L2+M2
b̄2(ζβ)〉 =
hN+L2−M2−bSN+L2−M2−b(ζβ), b = 1, . . . , N + L2 −M2,
〈f (1)m−1+N+L1−M1 b̄1(ηj)〉 = η
−N−L1+M1−m
j , m = 1, . . . ,−N − L1 +M1,
〈f (1)m−1+N+L1−M1 b̄2(ζβ)〉 = 0, m = 1, . . . ,−N − L1 +M1
After trivial manipulations with rows and columns of the matrix of pair-wise v.e.v.
we obtain the answer
IN(ξ, ζ, η, µ) =
N+L1−M1−1
N+L2−M2−1
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG, (5.34)
ǫ = (−1)
N(N−1)+N(L1−M1)+
M1(M1−1)+
L2(L2−1)+L1(L2−M2)+L2M2−M2
where the (L2 +M1)× (L2 +M1) matrix G consists of eight blocks
µk−ζβ
0 Sb(ζβ) 0
H(µk, ηj)
ξα−ηj
S̃b(ηj) Sm(ηj)
(5.35)
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2;
b = 1, . . . , N + L2 −M2; m = 1, . . . ,−N − L1 +M1
Evaluation for the case N2 ≤ 0 ≤ N1.
The answer for this case may be obtained by the answer for the previous case, if we
inter-change subscripts L1 ↔ L2, M1 ↔ M2, N1 ↔ N2, and the variables ξ ↔ ζ , µ ↔ η
and also Pn ↔ Sn, P̃n ↔ S̃n, and (due to the definition (C.4)) H(µ, η) → H(µ, η) .
The answer coincides with the answer of [1] (see (3.21)-(3.22) there):
IN(ξ, ζ, η, µ) =
N+L2−M2−1
N+L1−M1−1
(ξα − ηj)
(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG, (5.36)
ǫ = (−1)
N(N−1)+N(L2−M2)+
M2(M2−1)+
L1(L1−1)+L2(L1−M1)+L1M1−M1
where the (L1 +M2)× (L1 +M2) matrix G consists of eight blocks
ξα−ηi
0 Pb(ξα) 0
H(µk, ηi)
µk−ζβ
P̃b(µk) Sm(µk)
(5.37)
where
α = 1, . . . , L1; β = 1, . . . , L2; i = 1, . . . ,M1; k = 1, . . . ,M2;
b = 1, . . . , N + L1 −M1; m = 1, . . . ,M2 − L2 −N
5.4 Evaluation for the case (3): N1 ≤ N2 ≤ 0
In this case we have (5.32).
Using transformation (2.42)-(2.43) which leaves vacuum expectation values invariant,
in form:
〈∗, ∗| → 〈∗, ∗+N2|, |∗, ∗〉 → |∗, ∗+N2〉,
i → f
i → f̄
, i ∈ Z,
we write
IN(ξ, ζ, η, µ)R
N = 〈N1,−N2|B2(µ)B1(ξ)B̄1(η)B̄2(ζ)|0, 0〉
= 〈N1, 0|C2(µ)C1(ξ)C̄1(η)C̄2(ζ)|0, N2〉 (5.38)
where we shift the charges of the second component of the vacuum vector by N2 and
replaced fermionic operators b1(ξ), b̄1(η), b2(µ) and b̄2(ζ) given by (4.26)-(4.29), by
c1(ξ) =
n Pn(ξ)
hn , (5.39)
c1(η) =
S̃n(η)√
−n−1+N2
S̃n(η)
hn , (5.40)
c2(µ) =
−n−1+N2
P̃n(µ)√
f (1)n P̃n(µ)
hn , (5.41)
c̄2(ζ) =
−n−1+N2
Sn(ζ)
hn (5.42)
We produced this shift of the vacuum charge and the fermion numbering in order to
come to the form of v.e.v., where all fermions with bar are situated to the right:
〈0, 0|f (1)−1 · · · f
c2(µn)
c1(ξn)
c̄1(ηn)
c̄2(ζn)f̄
· · · f̄ (2)−1 |0, 0〉, (5.43)
where formula (2.8) is applicable. We have nine-block matrix:
〈c1(ξα)c̄1(ηj)〉 〈c1(ξα)c̄2(ζβ)〉 〈c1(ξα)f̄ (2)m−1+N+L2−M2〉
〈c2(µk)c̄1(ηj)〉 〈c2(µk)c̄2(ζβ)〉 〈c2(µk)f̄ (2)m−1+N+L2−M2〉
〈f (1)
ℓ−1+N+L1−M1
c̄1(ηj)〉 〈f (1)ℓ−1+N+L1−M1 c̄2(ζβ)〉 0
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2;
ℓ = 1, . . . ,−N − L1 +M1; m = 1, . . . , N + L2 −M2
Pair-wise expectation values are
〈c1(ξ)c̄1(η)〉 =
ξ − η
〈c2(µ)c̄2(ζ)〉 =
P̃n(µ)Sn(ζ) =
〈c2(µ)c̄1(η)〉 =
P̃n(µ)S̃n(η) = H(µ, η),
〈c1(ξ)c̄2(ζ)〉 = 0,
〈f (1)
ℓ−1+N+L1−M1
c̄1(η)〉 = S̃ℓ−1+N+L1−M1(η)
hℓ−1+N+L1−M1 = η
−N−L1+M1−ℓ, ℓ = 1, . . . ,−N−L1+M1,
〈f (1)
ℓ−1+N+L1−M1
c̄2(ζ)〉 = 0, ℓ = 1, . . . ,−N − L1 +M1,
〈c1(ξ)f̄ (2)m−1+N+L2−M2〉 = 0, m = 1, . . . , N + L2 −M2,
〈c2(µ)f̄ (2)m−1+N+L2−M2〉 = P̃−m(µ)
h−m = µ
ℓ−1, m = 1, . . . , N + L2 −M2
After trivial manipulations with rows and columns of the matrix of pair-wise v.e.v. we
obtain the answer which coincides with the answer of [1] (given by (3.35)-(3.36) there):
IN(ξ, ζ, η, µ) =
N+L2−M2−1
N+L1−M1−1
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG, (5.44)
ǫ = (−1)
L2(L2−1)+
M1(M1−1)+
N(N+1)+L2(L1+M1)+N(L1+M1)+M2L2
where G is (M1 +M2 −N)× (M1 +M2 −N) matrix which consists of nine blocks:
ξα−ηj
H(µk, ηj)
µk−ζβ
Sm(µk)
Pℓ(ηj) 0 0
(5.45)
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2;
ℓ = 1, . . . ,M1 − L1 −N ; m = 1, . . . , N + L2 −M2
Evaluation for the case N2 ≤ N1 ≤ 0
After replacements we obtain
IN(ξ, ζ, η, µ) =
N+L2−M2−1
N+L1−M1−1
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG, (5.46)
ǫ = (−1)
L1(L1−1)+
M2(M2−1)+
N(N+1)+L1(L2+M2)+N(L2+M2)+M1L1
where G is (M1 +M2 −N)× (M1 +M2 −N) matrix which consists of nine blocks
ζβ−µk
H(µk, ηj)
ηj−ξα
Pm(ηj)
Sℓ(µk) 0 0
(5.47)
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2;
ℓ = 1, . . . ,M2 − L2 −N ; m = 1, . . . ,M1 − L1 −N
Acknowledgements
The authors would like to thank T. Shiota and J. van de Leur for helpful discussions,
and (A.O.) thanks A. Odzijevicz for kind hospitality during his stay in Bialystok in June
2005.
A Appendix A
By (2.24) we have for p, q ∈ C:
[(gn)
, (gm)
q] = 0, [(en)
, (em)
q] = 0, n,m ∈ Z (A.1)
[(gn)
p, (em)
q] = 0, n 6= m (A.2)
where [, ] denotes the commutator.
Also:
[(gn)
p, f (1)m ] = [(gn)
p, f̄ (2)m ] = [(en)
p, f̄ (1)m ] = [(en)
p, f (2)m ] = 0, n,m ∈ Z, (A.3)
[(gn)
p, f̄ (1)m ] = [(gn)
−m−1] = [(en)
p, f (1)m ] = [(en)
p, f̄
−m−1] = 0, n 6= m, (A.4)
−pf̄ (1)n (gn)
p = f̄ (1)n + pf̄
−n−1, (gn)
−n−1(gn)
p = f
−n−1 − pf (1)n , n ∈ Z, (A.5)
pf (1)n (en)
−p = f (1)n − pf
−n−1, (en)
−n−1(en)
−p = f̄
−n−1 + pf̄
n n ∈ Z (A.6)
B Appendix B
For ω̂ :=
ωn,mfnf̄m we have
adω̂ fi =
fnωn,i , adω̂ f̄i = −
ωi,nf̄n
Then it follows
Adω̂ fi =
fn (e
, Adω̂ f̄i =
This yields (4.15) and (4.16) which are equivalent respectively to (4.10) and (4.11).
In the same way we prove (4.18) and (4.17) which are equivalent respectively to (4.13)
and (4.12).
C Appendix C
Now, let us show that
S̃n(η) =
Sn(y)
η − x
dµ(x, y), P̃n(µ) =
Pn(x)
dµ(x, y), n ≥ 0 (C.1)
Proof I.
−n−1]+ =
Pn(x)S̃n(η) (C.2)
where the first equality follows from the definitions (4.15) and (4.16), while the equality
of the anti-commutator to the last member follows from (4.10)-(4.11). Multiplying both
sides of the second equality by Sn(y)dµ(x, y) and integrating we come to the first equality
(C.1).
We obtain the second equality (C.1) by the similar integration of
−n−1µ
−n−1y
n]+ =
Sn(y)P̃n(µ) (C.3)
(These equalities results from definitions (4.17) and (4.18), and from (4.12),(4.13)).
We shall also use (in (5.20) below) the corollary of these equalities:
dµ(x, y)
(η − x)(µ− y)
S̃n(η)P̃n(µ) =: H(µ, η) (C.4)
which is obtained by multiplying of the right hand sides of the relations (C.2) and (C.3),
integrating and using the orthogonality of Pn and Sn.
Proof II. Considering the sum of entries weighted with powers of x−m−1 (where m
is the row-number of the entry) of the n-th column of relation K−1H = BK̄−1 (which
follows from the known factorization relation HK̄ = KB), we obtain the first equality of
(C.1) from the first of (4.5) and from the definition of bi-moments. (The second relation
in (C.1) is proved similarly).
D Appendix: Links with soliton theory
In this appendix, we discuss the links between the evaluation of such symmetric rational
integrals and soliton theory. There are two problems:
(A) To find which measure dµ is related to a given matrix integral? We do not know
the general answer to this problem. A partial answer can be found in the papers [24], [25]
and the Appendices to [26] and to [19].
(B) To find which measure dµ is related to soliton theory. This problem is addressed
in [19] and [28].
We restrict ourselves to the relation to the usual (one-component) TL hierarchy.
Indeed, one can easily show that the expression
(1), t(2)) = 〈N,−N |g(t(1), t(2))|0, 0〉, (D-1)
where g(t(1), t(2)) = eA(t
(1),t(2)), and A is of form
A(t(1), t(2)) =
i,j=1,2
eV (x,t
(1))−V (y,t(2))f (1)(x)f̄ (2)(y)dµ0(x, y) (D-2)
where
V (x, t(1)) =
t(1)n x
n, V (y, t(2)) =
t(2)n y
n (D-3)
and where dµ0(x, y) is a rather arbitrary measure, fits into a general expression for tau
functions of the two-component KP hierarchy and also the TL hierarchy developed in [14]
(see also [22], [21]).
Remark: the notations t
m and t
m are related respectively to the notations −umm and
of [19]. Our V (x, t(j)) here is Vj(x) of [19].
In [19] it is shown that the expression (D-1) on the one hand gives rise to multiple
integrals, which, in turn, for certain choices of the measure dµ0(x, y) (see Problem (A)
above) can be identified with a partition function of some model of random matrices. On
the other hand it is also an example of TL tau function.
This means that if we choose
dµ(x, y) = dµ(x, y, t(1), t(2)) = eV (x,t
(1))−V (y,t(2))dµ0(x, y) (D-4)
then the expectation value (D-1) is a TL (and also two-component KP) tau function.
Remark: if dµ0 solves this or that differential equation, than, one can write a corre-
spondent ”string equation”.
Notice that the first non-trivial member of the TL hierarchy was introduced and in-
tegrated in [20] and called the relativistic two-dimensional Toda lattice. In this paper a
factorization problem similar to (4.3) (but with matrices that are infinite in both direc-
tions) was considered (see also [22]).
Notational Remark. In formulas below we shall denote the pair of sets of times
(t(1), t(2)) by a single letter t. The only quantity which depends on a single such set
(either t(1) or t(2)) is V , where we shall point out which set it depends on.
Then from general consideration in [14] (see also [22], [21]) one finds that the inte-
grals of rational functions IN (η, ξ, µ, ζ), see (3.9), may be obtained via the bosonization
procedure as follows
IN(ξ, ζ, η, µ; t) =
τN(t)
ξNi D1(ξi)
η−Ni D1(ηi)
µ−Ni D2(µi)
ζNi D2(ζi)
−1τN (t), (D-5)
where
Dj(z) = exp
(D-6)
is a vertex operator.
Notational Remark. In [1] we use the notations
Pn(x) =
pn(x), (D-7)
S̃n(x) =
n(x) (D-8)
P̃ (y) =
n(y, t), (D-9)
Sn(y) =
sn(y) (D-10)
The functions
n (x, t) := e
V (x,t(1))
Pn(x, t)
hn = e
V (x,t(1))
pn(x, t), (D-11)
ψ∗(1)n (x, t) := e
−V (x,t(1)) S̃n−1(x, t)
= e−V (x,t
(1))p∗n−1(x, t) (D-12)
are to be interpreted as first component Baker functions (respectively, adjoint first com-
ponent Baker functions) which depend on a spectral parameter x, while
ψ(2)n (y, t) := e
V (y,t(2)) P̃n−1(y, t)
= eV (y,t
(2))s∗n−1(y, t), (D-13)
ψ∗(2)n (y, t) := e
−V (y,t(2))Sn(y, t)
hn = e
−V (y,t(2))sn(y, t) (D-14)
are to be interpreted as second component Baker functions (respectively, adjoint second
component Baker functions) which depend on a spectral parameter y.
This fact is due to the formulae (notice the factor (−1)N which result from re-ordering
the fermions of two different types to achieve true sign according to (2.30))
ψ(1)n (x, t) = (−1)NeV (x,t
(1)) 〈N + 1,−N |f (1)(x)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
= xNeV (x,t
(1))D1(x)τN (t)
τN(t)
(D-15)
ψ∗(1)n (x, t) = (−1)Ne−V (x,t
(1)) 〈N − 1,−N |f̄ (1)(x)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
= x−Ne−V (x,t
(1))D1(x)
−1τN (t)
τN (t)
(D-16)
ψ(2)n (y, t) = e
V (y,t(2)) 〈N, 1−N |f (2)(y)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
= y−NeV (y,t
(2))D2(y)τN(t)
τN(t)
(D-17)
ψ∗(2)n (y, t) = e
−V (y,t(2)) 〈N,−1−N |f̄ (2)(y)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
= yNe−V (y,t
(2))D2(y)
−1τN (t)
τN (t)
(D-18)
The first equality in each of the relations (D-15),(D-16),(D-17) and (D-18) is an example
of the evaluation of IN . Formulae (D-16) and (D-18) fit into the case N2 ≥ N1 ≥ 0 and
are just particular cases of formula (5.9, see the Examples following (5.9)). The second
equalities in each of the relations (D-15)-(D-18) are examples of bosonization formula
(D-5).
Notice that different examples of IN ,
K11(ξ, η, t) =
〈N,−N |f (1)(ξ)f̄ (1)(η)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
ξNη−ND1(ξ)D
1 (η)τN(t)
τN (t)
, (D-19)
K22(µ, ζ, t) =
〈N,−N |f (2)(µ)f̄ (2)(ζ)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
µ−NζND2(µ)D
2 (ζ)τN(t)
τN (t)
, (D-20)
K12(ξ, ζ, t) =
〈N + 1,−N − 1|f (1)(ξ)f̄ (2)(ζ)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
ξNζND1(ξ)D
2 (ζ)τN(t)
τN (t)
(D-21)
K21(µ, η, t) =
〈N − 1,−N + 1|f (2)(µ)f̄ (1)(η)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
µ−Nη−ND2(µ)D
1 (η)τN(t)
τN (t)
(D-22)
(where for the sake of brevity we omit t(1), t(2)-dependence in the l.h. sides), may be con-
sidered as 2-component analogue of a modified Cauchy-Baker-Akhiezer kernel, introduced
in [30], [31] to present an explicit version of Segal-Wilson construction to study Virasoro
deformations of tau functions for the quasi-periodical solutions of the KP (and actually
for the TL) hierarchies.
As we have obtained (see Examples following (5.9)):
K11(µ, ζ) =
P̃n(µ)Sn(ζ) +
ζ − µ
(D-23)
K22(ξ, η) =
Pn(ξ)S̃n(η) +
ξ − η
(D-24)
K21(ξ, ζ) =
Pn(ξ)Sn(ζ) (D-25)
K12(µ, η) =
P̃n(µ)S̃n(η)−
dµ(x, y)
(η − x)(µ− y)
(D-26)
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http://arxiv.org/abs/math-ph/0209019
	Introduction
	Summary of free fermions
	One component fermions
	Two-component fermions
	The integral of rational symmetric functions as a certain expectation value
	Biorthogonal polynomials and dressed fermions
	Determinantal expression for integrals of rational symmetric functions
	Determinantal representation in the case (1): N2N10
	When N1<0
	Evaluation for the case (2): N10N2.
	Evaluation for the case (3): N1N20 
	Appendix A
	Appendix B
	Appendix C
	Appendix: Links with soliton theory

  We use the fermionic construction of two-matrix model partition functions to
evaluate integrals over rational symmetric functions. This approach is
complementary to the one used in the paper ``Integrals of Rational Symmetric
Functions, Two-Matrix Models and Biorthogonal Polynomials'' \cite{paper2},
where these integrals were evaluated by a direct method.

Introduction
In this work, we consider the following integral
N (ξ, ζ, η, µ) :=
dµ(x1, y1) . . .
dµ(xN , yN)∆N(x)∆N (y)
1Work of (J.H.) supported in part by the Natural Sciences and Engineering Research Council of
Canada (NSERC) and the Fonds FCAR du Québec; that of (A.O.) by the Russian Academy of Science
program “Mathematical Methods in Nonlinear Dynamics” and RFBR grant No 05-01-00498.
2harnad@crm.umontreal.ca
3orlovs@wave.sio.rssi.ru
http://arxiv.org/abs/0704.1150v1
α=1(ξα − xa)
β=1(ζα − ya)
j=1(ηj − xa)
k=1(µk − ya)
(1.1)
where
∆N(x) :=
(xi − xj), ∆N (y) :=
(yi − yj) (1.2)
are Vandermonde determinants, and
dµ(x1, y1) . . .
dµ(xN , yN)∆N (x)∆N(y), (1.3)
is a normalization constant, which in the context of two-matrix models is interpreted as
the partition function. Here dµ(x, y) is a measure (in general, complex), supported on a
finite set of products of curves in the complex x and y planes.
It is known (see for instance [5]) that this integral can be presented in the form of a
determinant of an N ×N matrix; namely,
N (ξ, ζ, η, µ) =
α=1(ξα − x)
β=1(ζβ − y)
j=1(ηj − x)
k=1(µk − y)
dµ(x, y)(x, y)xjyk
0≤j,k≤N−1
(1.4)
However for some purposes, it is more useful to express it as the determinant of a matrix
whose size does not depend on N .
The problem of evaluation of integrals of such symmetric functions of N variables is
of importance in the context of matrix models [4], where N is the number of eigenvalues.
Expressing integrals that determine correlation functions and expectation values as de-
terminants of matrices whose size is independent of N is of importance e.g., in the study
of N → ∞ limits. For one-matrix models such integrals were studied in [3], [6], [7], [8]
and [9]. In the case of complex, normal or two-matrix models [10] the problem was
considered in [8], [2], [1].
Remark 1.1. In order to compare with the results in [8] and [2] on complex integrals the
variables {xa, ya}a=1,...N must be replaced by complex conjugate pairs {za, z̄a}a=1,...N , and the
integration domains taken as N copies of the complex plane.
The present paper is complementary to ref. [1], where such integrals were evaluated
by a “direct” method based on partial fraction expansions and the Cauchy-Binet identity.
Ref. [1] and the present work are intended to provide concise presentations and new
derivations of previously known as well as new results, using two distinct methods.
The approach used here is based on constructing a fermionic representation of integrals
of such rational symmetric functions, introduced in [19], using two–component fermions.
This representation is different from the ones used previously in the context of matrix
models in [16], [17] and [18], which were based on one-component fermions.
Remark 1.2. Although there is a close connection between matrix integrals, orthogonal poly-
nomials and the spectral transform approach to the theory of integrable systems, in the present
work we do not develop this latter aspect, which concerns the deformation theory of the mea-
sures involved. Our starting point however is similar to the previous paper [19], where the
deformation theory is addressed, and the fermionic approach to integrable systems developed
in [14] is utilized. The relation of these results to the objects appearing in the spectral transform
approach is, however, briefly explained in the Appendices, where the biorthogonal polynomials
and their Hilbert transforms are interpreted as Baker functions and adjoint Baker functions.
The language of free two-component fermions used here is borrowed from [14]. The
integrals that we are interested in are expressed as vacuum state expectation values of
operator products formed from free fermion generators. Besides successive application
of Wick’s theorem, the main computational device used consists of applying canonical
(“dressing”) transformations to pass from free fermions fields, whose Laurent coefficients
are the free Fermi creation and annihilation operators, to “dressed’ ones, where the mono-
mials are replaced by biorthogonal polynomials. Wick’s theorem then serves to express the
same operator product vacuum state expectation value (VEV) both as a multiple ratio-
nal integral and as the determinant of a matrix formed from elementary factors involving
orthogonal polynomials and their Hilbert transforms.
Let dµ(x, y) be a measure (in general, complex), supported on a finite set of products
of curves in the complex x and y planes, for which the semi-infinite matrix of bimoments
is finite:
Bjk :=
dµ(x, y)xjyk <∞, 0, ∀j, k ∈ N. (1.5)
The integrals are understood to be evaluated on a specified linear combination of products
of the support curves. Assuming that, for all N ≥ 1, the N×N submatrix (Bjk)0≤j,k,≤N−1
is nonsingular, the Gram-Schmidt process may be used to construct an infinite sequence
of pairs of biorthogonal polynomials {Pj(x), Sj(y)}j=0...∞, unique up to signs, satisfying
dµ(x, y)Pj(x)Sk(y) = δjk, (1.6)
and normalized to have leading coefficients that are equal:
Pj(x) =
+O(xj−1), Sj(x) =
+O(yj−1). (1.7)
We will also assume that the Hilbert transforms of these biorthogonal polynomials,
P̃n(µ) :=
dµ(x, y)
Pm(x)
, S̃n(η) :=
dµ(x, y)
Sm(x)
η − x
, (1.8)
exist for all n ∈ N.
The following expression for Z
N in terms of the leading term normalization factors
hn is then easily shown to hold (see, e.g., [4]).
N = N !det (Bjk) |j,k=1,...,N = N !
hn (1.9)
Defining
N1 := N + L1 −M1, N2 := N + L2 −M2, (1.10)
we consider different cases. In each case the answer is written in form of the determinant
of a matrix G which is different for different cases. Results are given
(1) N2 ≥ N1 ≥ 0: by formulae (5.9)-(5.15). G is a (L2 +M1)× (L2 +M1) matrix
(2) N1 ≤ 0 ≤ N2: by formulae (5.34)-(5.35). G is a (L2 +M1)× (L2 +M1) matrix
(3) N1 ≤ N2 ≤ 0: by formulae (5.44)-(5.45). G is a (M1 +M2 −N)× (M1 +M2 −N)
matrix
The cases N1 ≥ N2 ≥ 0, N2 ≤ 0 ≤ N1 and N2 ≤ N1 ≤ 0 are related to previous ones
by interchanging quantities L1 ↔ L2, M1 ↔ M2, N1 ↔ N2, ξ ↔ ζ , µ ↔ η, Pn ↔ Sn and
P̃n ↔ S̃n in the final formulae. (See respectively (5.9)-(5.15) vs. (5.22)-(5.26), (5.34)-
(5.35) vs. (5.36)-(5.37) and (5.44)-(5.45) vs. (5.46)-(5.47).)
For instance, for the case N1 ≥ N2 ≥ 0 the answer is
N (ξ, ζ, η, µ) = (−1)
(M1+M2)(M1+M2−1)(−1)L1M2
N+L2−M2−1
N+L1−M1−1
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG,
(1.11)
where G is a (L1+M2)×(L1+M2) matrix whose entries are expressed in terms of the values
of Pn, Sn, P̃n and S̃n for n = 0, . . . ,max(N1, N2) evaluated at the points {ξα, ζβ, ηj , µk}.
Remark 1.3. It is assumed that the support curves for the integrals involved do not pass
through the points {ηj , µk}j=1,...M1,k=1,...M2 . The definition of the integrals may be extended
throughout the complex plane of these parameters by analytic continuation, but might then take
on multiple values. However, if the measures dµ(x, y) vanish with sufficient rapidity as these
points approach the support curves, the integrals considered in the present paper become single
valued. If the replacements {xa, ya}a=1,...N→{za, z̄}a=1,...N , are made, in order to compare with
the results of [8], [2], it must be assumed that the corresponding measures dµ(z, z̄ in the complex
plane vanish sufficiently rapidly at the points {z = ηj , z̄ = µk} for the integrals to converge.
In the following sections, we present the integral (1.1) as a vacuum state expectation
value of a certain fermionic expression, using two-component fermions (see (3.9) below).
To evaluate this expectation value we use Wick’s theorem, which expresses the vacuum
state expectation value of products of linear combinations of fermions as a determinant
of the matrix formed by evaluating the pair-wise VEV’s. Since the expression (3.9) is not
as yet of the form of a vacuum expectation value of a product of fermions, we use a set of
tricks (namely, canonical transformations of fermions and re-writing of charged vacuum
vectors in a suitable form) which serve also to reduce the final number of fermions inside
the vacuum expectation value, and therefore to reduce as much as possible the size of the
matrix G.
The structure of the paper is the following. First we recall some facts about free
fermions, including Wick’s theorem. Then we equal I
N (ξ, ζ, η, µ) to a certain vacuum
expectation value, see (3.12). After certain preliminaries, via (4.25), we introduce dressed
fermions bα, b̄α to adapt formula (3.12) for a usage of Wick’s theorem in its determinantal
form. Then we consider each of the numbered cases separately. For the case (1) (which
is the most involved case) we introduce fermionic operators aα, āα, obtained from bα, b̄α
via a sort of conjugation (see (5.4)-(5.7)), and whose vacuum expectation evaluated via
Wick’s theorem yields the final result given by formulae (5.9)-(5.15) and are illustrated by
six examples. For the case (2), instead of aα, āα, we similarly use operators bα, b̄α. At last,
for the case (3), instead of aα, āα, we exploit operators cα, c̄α introduced in (5.39)-(5.42).
Remark 1.4. One can obtain the corresponding results for N -fold integrals arising in one-
matrix models by specifying that the measure be proportional to Dirac delta function, say
δ(x − y). As this specialization will not be consider detailed here we shall write IN instead of
2 Summary of free fermions
The following is a summary regarding the one and two-component free fermion algebra
based on the introductory section of [19]. The reader may refer to [19], or [15], [14] for
further details.
2.1 One component fermions
In the following, A denotes the complex Clifford algebra over C generated by charged free
fermions {fi, f̄i}i∈Z, satisfying the anticommutation relations
[fi, fj]+ = [f̄i, f̄j]+ = 0, [fi, f̄j ]+ = δij . (2.1)
where [, ]+ denotes the anticommutator.
Elements of the linear part
W := (⊕m∈ZCfm)⊕
⊕m∈ZCf̄m
(2.2)
will be referred to as a free fermions. The fermionic free fields
f(x) :=
k, f̄(y) :=
−k−1, (2.3)
may be viewed as generating functions for the fj , f̄j’s.
This Clifford algebra has a standard Fock space representation F and dual space
F̄ (see [19]) which contain unique vacuum states |0〉 and 〈0| respectively satisfying the
properties
fm|0〉 = 0 (m < 0), f̄m|0〉 = 0 (m ≥ 0),
〈0|fm = 0 (m ≥ 0), 〈0|f̄m = 0 (m < 0). (2.4)
The Fock spaces F and F̄ are mutually dual, with the hermitian pairing defined via the
linear form 〈0||0〉 on A called the vacuum expectation value. This satisfies
〈0|1|0〉 = 1; 〈0|fmf̄m|0〉 = 1, m < 0; 〈0|f̄mfm|0〉 = 1, m ≥ 0, (2.5)
〈0|fn|0〉 = 〈0|f̄n|0〉 = 〈0|fmfn|0〉 = 〈0|f̄mf̄n|0〉 = 0; 〈0|fmf̄n|0〉 = 0, m 6= n, . (2.6)
Wick’s theorem implies that for any finite set of elements {wk ∈ W}, we have
〈0|w1 · · ·w2n+1|0〉 = 0,
〈0|w1 · · ·w2n|0〉 =
σ∈S2n
sgnσ〈0|wσ(1)wσ(2)|0〉 · · · 〈0|wσ(2n−1)wσ(2n)|0〉. (2.7)
Here σ runs over permutations for which σ(1) < σ(2), . . . , σ(2n− 1) < σ(2n) and σ(1) <
σ(3) < · · · < σ(2n− 1).
If {wi}i=1,...,N , are linear combinations of the fj ’s only, j ∈ Z, and {w̄i}i=1,...,N linear
combinations of the f̄j ’s, j ∈ Z, then(2.7) implies
〈0|w1 · · ·wN w̄N · · · w̄1|0〉 = det (〈0|wiw̄j|0〉) |i,j=1,...,N (2.8)
Following [14], [15], for all N ∈ Z, we also introduce the states
〈N | := 〈0|CN (2.9)
where
CN := f̄0 · · · f̄N−1 if N > 0 (2.10)
CN := f−1 · · · fN if N < 0 (2.11)
CN := 1 if N = 0 (2.12)
|N〉 := C̄N |0〉 (2.13)
where
C̄N := fN−1 · · · f0 if N > 0 (2.14)
C̄N := f̄N · · · f̄−1 if N < 0 (2.15)
C̄N := 1 if N = 0 (2.16)
The states (2.9) and (2.13) are referred to as the left and right charged vacuum vectors,
respectively, with charge N .
From the relations
〈0|f̄N−kf(xi)|0〉 = xN−ki , 〈0|f−N+k−1f̄(yi)|0〉 = yN−ki , k = 1, 2, . . .N, (2.17)
and (2.8), it follows that
〈N |f(xn) · · ·f(x1)|0〉 = δn,N∆N(x), N ∈ Z, (2.18)
〈−N |f̄(yn) · · · f̄(y1)|0〉 = δn,N∆N(y), N ∈ Z. (2.19)
For free fermion generators with |x| 6= |y|,
〈0|f(x)f̄(y)|0〉 = 1
(2.20)
Note that the expression on the right hand side is actually defined, by (2.5), as the
infinite series
n=0 y
nx−n−1 which converges only inside |x| < |y|. However one can
consider expression (2.20) for the whole region of x and y (when |x| 6= |y|) in the sense of
analytical continuation.
From Wick’s theorem it follows that
〈n−m|f(xn) · · ·f(x1)f̄(ym) · · · f̄(y1)|0〉 =
∆n(x)∆m(y)
i=1,...,n
j=1,...,m
(xi − yj)
(2.21)
2.2 Two-component fermions
The two-component fermion formalism is obtained by relabelling the above as follows.
f (α)n := f2n+α−1 , f̄
n := f̄2n+α−1 , (2.22)
(α)(z) :=
k , f̄
(α)(z) :=
k , (2.23)
where α = 1, 2. Then (2.1) is equivalent to
[f (α)n , f
m ]+ = [f̄
n , f̄
m ]+ = 0, [f
n , f̄
m ]+ = δα,βδnm. (2.24)
We denote the right and left vacuum vectors respectively as
|0, 0〉 := |0〉, 〈0, 0| := 〈0|. (2.25)
Relations (2.4) then become, for α = 1, 2,
f (α)m |0, 0〉 = 0 (m < 0), f̄ (α)m |0, 0〉 = 0 (m ≥ 0), (2.26)
〈0, 0|f (α)m = 0 (m ≥ 0), 〈0, 0|f̄ (α)m = 0 (m < 0). (2.27)
For n,m ∈ Z, i, j = 1, 2 it follows from (2.5)-(2.6) that
〈0, 0|f (i)n f (j)m |0, 0〉 = 〈0, 0|f̄ (i)n f̄ (j)m |0, 0〉 = 0, (2.28)
〈0, 0|f (i)n f̄ (j)m |0, 0〉 = δijδnm, n < 0 (2.29)
Following [14], [15], we also introduce the states
〈n(1), n(2)| := 〈0, 0|Cn(1)Cn(2), (2.30)
where
Cn(α) := f̄
0 · · · f̄
n(α)−1
if n(α) > 0 (2.31)
Cn(α) := f
−1 · · · f
if n(α) < 0 (2.32)
Cn(α) := 1 if n
(α) = 0 (2.33)
|n(1), n(2)〉 := C̄n(2)C̄n(1)|0, 0〉 (2.34)
where
C̄n(α) := f
n(α)−1
· · ·f (α)0 if n(α) > 0 (2.35)
C̄n(α) := f̄
· · · f̄ (α)−1 if n(α) < 0 (2.36)
C̄n(α) := 1 if n
(α) = 0 (2.37)
The states (2.30) and (2.34) will be referred to, respectively, as left and right charged
vacuum vectors with charges (n(1), n(2)).
It follows that
f (1)m |n, ∗〉 = 0 (m < n), f̄ (1)m |n, ∗〉 = 0 (m ≥ n), (2.38)
〈n, ∗|f (1)m = 0 (m ≥ n), 〈n, ∗|f̄ (1)m = 0 (m < n), (2.39)
f (2)m |∗, n〉 = 0 (m < n), f̄ (2)m |∗, n〉 = 0 (m ≥ n), (2.40)
〈∗, n|f (2)m = 0 (m ≥ n), 〈∗, n|f̄ (2)m = 0 (m < n). (2.41)
Remark 2.1. Note that if we shift the charges of the vacuum vectors
〈∗, ∗| → 〈∗, ∗+ n|, |∗, ∗〉 → |∗, ∗ + n〉 (2.42)
and, at the same time, re-label
i → f
i+n, f̄
i → f̄
i+n, i, n ∈ Z, (2.43)
then the vacuum expectation values remain invariant.
Remark 2.2. Wick’s theorem will be used in the form (2.8) in the following. In the two
component notation, there are two possible ways to do this; either
(1) Use formula (2.8), remembering that 2-component fermions consist just of the usual even
and odd ones (see (2.22)), or
(2) Use formula (2.8) separately for each component. To calculate the vacuum state expec-
tation value of an operator O, we first present it in the form
i . (2.44)
〈0, 0|O|0, 0〉 =
〈0, 0|O(1)i O
i |0, 0〉 =
〈0, 0|O(1)i |0, 0〉〈0, 0|O
i |0, 0〉 (2.45)
where Wick’s theorem in the form (2.8) may be applied to each factor 〈0, 0|O(α)i |0, 0〉
The version (2) is used in the Section 3, and the version (1) is used in the Section 5.
Now define
F (j)(x(j)) := f (j)(x(j)nj ) · · ·f
(j)(x
F̄ (j)(y(j)) := f̄ (j)(y(j)mj ) · · · f̄
(j)(y
1 ), j = 1, 2 (2.46)
Combining (2.21) and (2.45) we obtain
〈n1 −m1, n2 −m2|F (2)(x(2))F (1)(x(1))F̄ (1)(y(1))F̄ (2)(y(2))|0, 0〉
= (−1)m2(m1+n1) ∆n1(x
(1))∆m1(y
(1))∆n2(x
(2))∆m2(y
i=1,...,n1
j=1,...,m1
i − y
i=1,...,n2
j=1,...,m2
i − y
. (2.47)
expectation value
3 The integral of rational symmetric functions as a
certain expectation value
We shall use the following notations
η = (η1, . . . , ηM1), ξ = (ξ1, . . . , ξL1), µ = (µ1, . . . , µM2), ζ = (ζ1, . . . , ζL2) (3.1)
Nj = N + Lj −Mj, j = 1, 2, (3.2)
dµ(x,y)(·) =
dµ(x1, y1) . . .
dµ(xN , yN)(·) (3.3)
F (1)(ξ) := f (1)(ξL1) · · ·f (1)(ξ1), (3.4)
F (2)(µ) := f (2)(µM2) · · ·f (2)(µ1), (3.5)
F̄ (1)(η) := f̄ (1)(ηM1) · · · f̄ (1)(η1), (3.6)
F̄ (2)(ζ) := f̄ (2)(ζL2) · · · f̄ (2)(ζ1), (3.7)
g := eA, A :=
f (1)(x)f̄ (2)(y)dµ(x, y), (3.8)
and consider the expression
JN(ξ, ζ, η, µ) := RN 〈N1,−N2|F (2)(µ)F (1)(ξ) g F̄ (1)(η)F̄ (2)(ζ)|0, 0〉 (3.9)
where
RN = RN (ξ, ζ, η, µ) :=
s(L1, L2,M1,M2)
n=0 hn
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
(3.10)
and s(L1, L2,M1,M2) is the sign factor
s(L1, L2,M1,M2)) = (−1)
N(N+1)+L2(L1+M1)+N(L1+M1)+M2L2 (3.11)
Remark 3.1. Because of the form (3.8), g commutes with both F (1)(ξ) and F̄ (2)(µ). Moreover
he conditions given in Remark (1.3) also imply that g commutes with F̄ (1)(η) and F (2)(ζ).
The main result of this subsection is the equality
IN(ξ, ζ, η, µ) = JN(ξ, ζ, η, µ) (3.12)
Proof. Inserting
into (3.9), we note that only the N -th power contributes, giving
JN(ξ, ζ, η, µ) =
〈N1,−N2|F (2)(µ)F (1)(ξ)AN F̄ (1)(η)F̄ (2)(ζ)|0, 0〉 (3.13)
Collecting the fermion terms of the same types (which gives rise to a sign factor), the
right hand side of (3.13) may be expressed as
N(N−1)+NM1
dµ(x,y)〈N1,−N2|F (2)(µ)F (1)(ξ, x)F̄ (1)(η)F̄ (2)(y, ζ)|0, 0〉
where (ξ, x) = (ξ1, . . . , ξL1, x1, . . . , xN ) and (y, ζ) = (y1, . . . , yN , ζ1, . . . , ζL2). Using (2.47)
to evaluate the expectation value in the integrand we get
JN(ξ, ζ, η, µ) =
dµ(x,y)∆N(x)∆N (y)
α=1(ξα − xa)
β=1(ζα − ya)
j=1(ηj − xa)
k=1(µk − ya)
= IN(ξ, ζ, η, µ).
Remark 3.2. From (3.9) and (3.12) in the absence of the F (1), F (2), F̄ (1), F̄ (2) terms, we obtain
the representation which we used in [19]
= (−1)
N(N+1)N !〈N,−N |g|0, 0〉.
Having the representation (3.12) we evaluate IN(ξ, ζ, η, µ) via Wick’s theorem. Before
we need certain preliminary relations.
4 Biorthogonal polynomials and dressed fermions
In this section we introduce transformations Ω and Q and dressed fermions d(α), d̄(α) and
bα, b̄α. We re-write the v.e.v. (3.9) in a form suitable for further calculation by means of
Wick’s theorem.
(a) Biorthogonal polynomials.
Consider the sequence of biorthogonal polynomials associated to the measure dµ(x, y):
Pn(x)Sm(y)dµ(x, y) = δn,m, n,m ≥ 0 (4.1)
It is convenient to write down the biorthogonal polynomials in the following form
Pn(x) =
m, Sn(y) =
ym(K̄−1)mn, n ≥ 0 (4.2)
where Knm and (K̄
−1)mn are respectively viewed as entries of semi-infinite matrices K
and (K̄−1). K is a lower triangular matrix and K̄ is an upper triangular one, both have
units on the diagonal: Knn = K̄nn = 1, n = 0, 1, 2, . . . .
As is well-known, the orthogonality relation (4.1) implies
HK̄ = KB, (4.3)
where H = diag(hn) and B is the bi-moment matrix,
Bnm =
xnymdµ(x, y), m, n ≥ 0, (4.4)
In time (see [11]), (4.3) was made good use to relate matrix models to integrable
systems via the factorization method widely used in soliton theory starting from the basic
paper [23]. Then K and K̄ may be identified with the Mikhailov-Ueno-Takasaki dressing
matrices (see [20], [22]), whose rows give rise to a pair the so-called Baker functions, while
columns of K−1and of K̄−1 give rise to a pair of adjoint Baker functions. In case the
dressing matrices are semi-infinite, the first of the Baker functions (related to rows of K)
and the second of the adjoint Baker function (related to columns K̄−1) take the form of
quasi-polynomials.
Here we shall use (4.3) differently.
(b) Canonical transformation Ω.
First, we introduce
S̃n(x) :=
x−m−1(K−1)mn
hn, P̃n(y) :=
K̄nmy
hn, n ≥ 0 (4.5)
Properties of S̃n(x) and P̃n(y) are described in the Appendix C. The main is that they
coincide with Hilbert transforms of biorthogonal polynomials (1.8).
Now, let us define matrices ω(1) and ω(2) via
= K, eω
= K̄ (4.6)
(Respectively, strictly upper and strictly lower) triangular matrices ω(1),(2) may be defined
in a unique way by a recursion procedure.
We introduce
Ω = exp
n>m≥0
ω(1)n,mf
m + ω
−m−1f̄
(4.7)
Note that, for any N1 and N2,
〈N1,−N2|Ω−1 = 〈N1,−N2| , Ω|0, 0〉 = |0, 0〉 (4.8)
The first equality in (4.8) follows from (2.39),(2.41) and from the restriction n > m in the
summation in (4.7). The second equality in (4.8) follows from the restriction m ≥ 0 and
from (2.39),(2.41).
Consider
d(α)(x) = Ωf (α)(x)Ω−1, d̄(α)(x) = Ωf̄ (α)(x)Ω−1, α = 1, 2 (4.9)
which we refer as dressed fermions.
We have (see Appendix B)
d(1)(x) =
f (1)n Pn(x)
hn (4.10)
d̄(1)(x) =
f̄ (1)n
S̃n(x)√
(4.11)
d(2)(y) =
P̃n(y)√
(4.12)
(2)(y) =
−n−1Sn(y)
hn (4.13)
where for n < 0 we use the following notational convention
Pn(x) = Sn(x) = x
n, S̃n(x) = P̃n(x) = x
−n−1, hn = 1, n < 0 (4.14)
A kind of dressed fermionic operators (where powers of x were replaced by Baker
functions similar to (4.10)-(4.13)), in a different context, were also introduced in [30]
where they were called Krichever-Novikov fermions (see Appendix to [30]).
If we write
d(α)(x) =
d(α)n x
n, d̄(α)(x) =
d̄(α)n x
−n−1, α = 1, 2
where
d(α)n = Ωf
−1, d̄(α)n = Ωf̄
n≥i≥0
f (1)n Kni, i ≥ 0, d
i = f
i , i < 0 (4.15)
i≥n≥0
(K−1)inf̄
n , i ≥ 0, d̄
i = f̄
i , i < 0 (4.16)
−i−1 =
i≥n≥0
−n−1K̄ni, i ≥ 0, d
−i−1 = f
−i−1, i < 0 (4.17)
−i−1 =
n≥i≥0
(K̄−1)inf̄
−n−1, i ≥ 0, d̄
−i−1 = f̄
−i−1, i < 0 (4.18)
(c) Useful formulae for charged vacuum vectors.
First, we introduce
gn := e
−n−1 = 1 + f (1)n f̄
−n−1, (4.19)
en := e
−n−1 = 1 + f̄ (1)n f
−n−1 (4.20)
We see that gn, en ∈ ĜL∞.
We shall also consider powers of these operators
p = epf
−n−1 = 1 + pf (1)n f
−n−1, (4.21)
p = epf̄
−n−1 = 1 + pf̄ (1)n f
−n−1 (4.22)
where n ∈ Z, p ∈ C.
Now, due to
〈0, 0|engn = 〈0, 0|(1+f̄ (1)n f
−n−1)(1+f
−n−1) = 〈0, 0|(1−1+f̄ (1)n f
−n−1) = 〈0, 0|f̄ (1)n f
−n−1,
which is true for n > 0, and due to (A.1), (A.2), we conclude that for N > 0
〈N,−N | = (−1)
N(N−1)〈0, 0|
where 〈N,−N | was defined in (2.30)-(2.32).
In the similar way we obtain a representation we shall need later
N2 > N1 ≥ 0 : 〈N1,−N2| = (−1)
N1(N1−1)〈0, 0|f (2)−N1−1 · · · f
gn (4.23)
(d) Evaluation of Ωg|0, 0〉.
Taking into account that f
−n−1|0, 0〉 = f̄
n |0, 0〉 = 0 for n > 0, one obtains
Ωg|0, 0〉 = e
i,j,n,m≥0 KinBnmK̄mjf
−j−1 |0, 0〉 = Q−1|0, 0〉, Q−1 :=
hn (4.24)
where we used (4.15),(4.18) and (4.3).
(e) Fermionic operators bα, b̄α.
We shall also need the following fermionic operators
bα(x) := Qd
(α)(x)Q−1 = QΩ f (α)(QΩ)−1, b̄α(x) := Qd̄
(α)(x)Q−1 = QΩ f̄ (α)(QΩ)−1
(4.25)
Using (A.3)-(A.6) we write down
b1(ξ) =
f (1)n Pn(ξ)
hn = d
(1)(ξ) , (4.26)
b̄1(η) =
f̄ (1)n
S̃n(η)√
−n−1S̃n(η)
hn , (4.27)
b2(µ) =
P̃n(µ)√
f (1)n P̃n(µ)
hn , (4.28)
b̄2(ζ) =
−n−1Sn(ζ)
hn = d̄
(2)(ζ) (4.29)
Notice that each of b̄1 and b2 contains both component fermions.
The fermionic operators bα, b̄α may be also called dressed fermions. Similarly to (3.4)-
(3.7), for their products, we shall use large letters, namely, we define
B1(ξ) := QΩ F
(1)(ξ) (QΩ)−1, B2(µ) := QΩ F
(2)(µ) (QΩ)−1,
B̄1(η) := QΩ F̄
(1)(η) (QΩ)−1, B̄2(ζ) := QΩ F̄
(2)(ζ) (QΩ)−1
(f) IN as a re-written vacuum expectation value
Now we restate (3.12) as
IN(ξ, ζ, η, µ)R
N = 〈N1,−N2|Q
−1 B2(µ)B1(ξ)B̄1(η)B̄2(ζ)|0, 0〉 (4.30)
where RN was defined by (3.10).
Now we have to consider the cases (1),(2),(3) separately.
5 Determinantal expression for integrals of rational
symmetric functions
Below we apply Wick’s theorem to evaluate v.e.v. in the right hand side of (4.30), getting
answers for I
N (ξ, ζ, η, µ) for all the cases listed in the Introduction.
5.1 Determinantal representation in the case (1): N2 ≥ N1 ≥ 0
In the case (1) we have the following formulae for 〈N1,−N2|Q−1:
N2 ≥ N1 ≥ 0 : 〈N1,−N2|Q−1 = (−1)
N1(N1+1)〈0, 0|f (2)−N1−1 · · · f
hn , (5.1)
where
n (5.2)
Formula (5.1) follows from relations
〈0, 0|eng1+hnn = 〈0, 0|(1 + f̄ (1)n f
−n−1)(1 + (1 + hn)f
−n−1) = 〈0, 0|(−hn + f̄ (1)n f
−n−1)
= −hn〈0, 0|e
n , n ≥ 0
and from (4.23).
It is important that
E|0, 0〉 = 0 , (5.3)
which is true as each epn|0, 0〉 = 0, n ≥ 0.
Using (A.3)-(A.6), we may evaluate
a1(ξ) := Eb1(ξ)E
f (1)n Pn(ξ)
Pn(ξ)√
, (5.4)
ā1(η) := Eb̄1(η)E
f̄ (1)n
S̃n(η)√
f̄ (1)n
S̃n(η)√
−n−1S̃n(η)
hn , (5.5)
a2(µ) := Eb
(2)(µ)E−1 =
P̃n(µ)√
P̃n(µ)√
f (1)n P̃n(µ)
hn , (5.6)
ā2(ζ) := Eb̄
(2)(ζ)E−1 =
−n−1Sn(ζ)
f̄ (1)n
Sn(ζ)√
(5.7)
Using notations
A1(ξ) := EQΩ F
(1)(ξ) (EQΩ)−1, A2(µ) := EQΩ F
(2)(µ) (EQΩ)−1,
Ā1(η) := EQΩ F̄
(1)(η) (EQΩ)−1, Ā2(ζ) := EQΩ F̄
(2)(ζ) (EQΩ)−1
for the products, by analogy with (3.4)-(3.7), by (5.1) and (5.3), we arrive at
IN(ξ, ζ, η, µ)R
N = (−1)
N1(N1+1)
hn〈0, 0|f (2)−N1−1 · · · f
A2(µ)A1(ξ)Ā1(η)Ā2(ζ)|0, 0〉
(5.8)
Finely, this form is applicable to apply the Wick’s formula (2.8). Indeed, each ai is
a linear combination of fermions f (1) and f (2), while each āi is a linear combination of
fermions f̄ (1) and f̄ (2).
Then, by (2.8), the vacuum expectation value in formula (5.8) is equal to the deter-
minant of a L1 +M2 by L1 +M2 matrix, which consists of six blocks formed by pair-wise
v.e.v.:
〈a1(ξα)ā1(ηj)〉 〈a2(µk)ā1(ηj)〉 〈f (2)i−1−N−L2+M2 ā1(ηj)〉
〈a1(ξα)ā2(ζβ)〉 〈a2(µk)ā2(ζβ)〉 〈f (2)i−1−N−L2+M2ā2(ζβ)〉
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2
i = 1, . . . ,M1 + L2 − L1 −M2
Now, taking into account (2.29)-(2.29), from the explicit formulae (5.4)-(5.7), and also
from (C.3) and (C.4), we compute all relevant pair-wise vacuum expectation values
〈a1(ξi)ā1(ηj)〉 =
ξi − ηj
Pn(ξi)S̃n(ηj),
〈a2(µi)ā2(ζj)〉 =
Sn(ζj)P̃n(µi) = −
ζj − µi
Sn(ζj)P̃n(µi),
〈a1(ξi)ā2(ζj)〉 =
Pn(ξi)Sn(ζj),
〈a2(µi)ā1(ηj)〉 =
S̃n(ηj)P̃n(µi) =
dµ(x, y)
(ηj − x)(µi − y)
S̃n(ηj)P̃n(µi),
〈f (2)i−1−N−L2+M2 ā1(ηj)〉 =
hN+L2−M2−iS̃N+L2−M2−i(ηj),
〈f (2)i−1−N−L2+M2 ā2(ζj)〉 =
hN+L2−M2−iSN+L2−M2−i(ζj)
After trivial manipulations with rows and columns of the matrix of pair-wise v.e.v.
we obtain the answer:
IN(ξ, ζ, η, µ) = (−1)
(M1+M2)(M1+M2−1)(−1)L2M1
N+L1−M1−1
N+L2−M2−1
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG,
(5.9)
where the matrix G is (L2 +M1)× (L2 +M1) matrix which consists of six blocks:
K11(µk, ζβ)
K21(ξα, ζβ) SN+L1−M1(ζβ) . . . SN+L2−M2−1(ζβ)
K12(µk, ηj)
K22(ξα, ηj) S̃N+L1−M1(ηj) . . . S̃N+L2−M2−1(ηj)
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2
and where N1 = N + L1 −M1 and
K11(µ, ζ) =
P̃n(µ)Sn(ζ) +
ζ − µ
(5.10)
K22(ξ, η) =
Pn(ξ)S̃n(η) +
ξ − η
(5.11)
K21(ξ, ζ) =
Pn(ξ)Sn(ζ) (5.12)
K12(µ, η) =
P̃n(µ)S̃n(η)−
dµ(x, y)
(η − x)(µ− y)
(5.13)
SN+L1−M1+i−1(ζj), i = 1, . . . ,M1 − L1 −M2 + L2, j = 1, . . . , L2 (5.14)
S̃N+L1−M1+i−1(ηj), i = 1, . . . ,M1 − L1 −M2 + L2, j = 1, . . . ,M1, (5.15)
Now we shall consider six examples, related to the six-block structure, where, in each
case, only one entry contributes.
Example 1. M1 = 1 and L1 = L2 = M2 = 0, thus N2 > N1 ≥ 0. We put η1 = η. In
this case the matrix has only one non-vanishing element, giving the well-known formula
IN(η) =
S̃N−1(η) (5.16)
Example 2. L2 = 1 and L1 = M1 = M2 = 0, thus N2 > N1 ≥ 0. We put ζ1 = ζ . In
this case the matrix has only one non-vanishing element, giving the well-known formula
IN(ζ) =
hNSN(ζ) (5.17)
Examples below are related to the equality N2 = N1 ≥ 0.
Example 3. M1 = L1 = 1 and L2 = M2 = 0. We put ξ1 = ξ and η1 = η. In this case
the matrix has only one non-vanishing element and we obtain
IN(ξ, η) = 1 + (ξ − η)
Pn(ξ)S̃n(η) (5.18)
Similarly, we have
Example 4. L2 = M2 = 1 and L1 = M1 = 0. In this case the matrix has only one
non-vanishing element, and we obtain
IN(ζ, µ) = −(ζ − µ)
Sn(ζ)P̃n(µ) = 1 + (ζ − µ)
Sn(ζ)P̃n(µ) (5.19)
Example 5. M1 = M2 = 1 and L1 = L2 = 0. In this case the matrix has only one
non-vanishing element. We obtain
IN(η, µ) =
n=N−1
S̃n(η)P̃n(µ) =
dµ(x, y)
(η − x)(µ − y)
S̃n(η)P̃n(µ)
(5.20)
Example 6. L1 = L2 = 1 and M1 = M2 = 0. In this case the matrix has only one
non-vanishing element and we obtain
IN(ξ, ζ) = hN
Pn(ξ)Sn(ζ) (5.21)
Evaluation for the case N1 ≥ N2 ≥ 0
This case may be obtained from the previous one, by interchanging subscripts L1 ↔ L2,
M1 ↔ M2, N1 ↔ N2, and ξ ↔ ζ , µ↔ η and also Pn ↔ Sn, P̃n ↔ S̃n.
We obtain the answer which coincides with the answer given by formulae (1.8)-(1.16)
of [1]:
IN(ξ, ζ, η, µ) = (−1)
(M1+M2)(M1+M2−1)(−1)L1M2
N+L2−M2−1
N+L1−M1−1
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG,
(5.22)
where (L1 +M2)× (L1 +M2) matrix G consists of six blocks:
K11(ξα, ηj)
K12(ξα, ζβ) PN+L2−M2(ξα) . . . PN+L1−M1−1(ξα)
K21(µk, ηj)
K22(µk, ζβ) P̃N+L2−M2(µk) . . . P̃N+L1−M1−1(µk)
where
α = 1, . . . , L1; β = 1, . . . , L2; ; j = 1, . . . ,M1; k = 1, . . . ,M2
and where N2 = N + L2 −M2 and
K11(ξ, η) =
Pn(ξ)S̃n(η) +
ξ − η
(5.23)
K22(µ, ζ) =
P̃n(µ)Sn(ζ) +
ζ − µ
(5.24)
K12(ξ, ζ) =
Pn(ξ)Sn(ζ) (5.25)
K21(µ, η) =
P̃n(µ)S̃n(η)−
dµ(x, y)
(η − x)(µ− y)
(5.26)
Example 7. M2 = 1 and L2 = L1 =M1 = 0, thus N1 > N2 ≥ 0. We put µ1 = µ. Then
IN(µ) =
P̃N−1(µ) (5.27)
Example 8. L1 = 1 and L2 = M2 = M1 = 0, thus N1 > N2 ≥ 0. We put ξ1 = ξ. We
obtain the well-known formula
IN(ξ) =
hNPN(ξ) (5.28)
5.2 When N1 < 0
In cases listed in the Introduction as (2) and (3) we have N1 < 0. Let us explicitly write
N1 ≤ 0 ≤ N2 : 〈N1,−N2| = 〈0, 0|f (1)−1 · · · f
−1 · · · f
, (5.29)
N1 ≤ N2 ≤ 0 : 〈N1,−N2| = 〈0, 0|f (1)−1 · · · f
0 · · · f̄
−N2−1
(5.30)
Then, as it follows from (A.3)-(A.4), in either case
〈N1,−N2|Q−1 = 〈N1,−N2| (5.31)
Thus, in the both cases, we re-write (4.30) as
IN(ξ, ζ, η, µ)R
N = 〈N1,−N2|B2(µ)B1(ξ)B̄1(η)B̄2(ζ)|0, 0〉 (5.32)
5.3 Evaluation for the case (2): N1 ≤ 0 ≤ N2.
Thus, by (5.32) and (5.29) we have
IN(ξ, ζ, η, µ)R
N = 〈0, 0|f
−1 · · · f
−1 · · · f
B2(µ)B1(ξ)B̄1(η)B̄2(ζ)|0, 0〉 (5.33)
By Wick’s formula (2.8) the right hand side is the determinant of a L2+M1 by L2+M1
matrix which consists of eight blocks:
〈b1(ξα)b̄1(ηj)〉 〈b2(µk)b̄1(ηj)〉 〈f (2)b−1−N−L2+M2 b̄1(ηj)〉 〈f
m−1+N+L1−M1
b̄1(ηj)〉
〈b1(ξα)b̄2(ζβ)〉 〈b2(µk)b̄2(ζβ)〉 〈f (2)b−1−N−L2+M2 b̄2(ζβ)〉 〈f
m−1+N+L1−M1
b̄2(ζβ)〉
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2;
b = 1, . . . , N + L2 −M2; m = 1, . . . ,−N − L1 +M1
Then using (4.26)-(4.29), we evaluate all pair-wise vacuum expectation values:
〈b1(ξα)b̄1(ηj)〉 =
1− ηj
ξα − ηj
〈b2(µk)b̄2(ζβ)〉 =
P̃n(µk)Sn(ζβ) =
µk − ζβ
〈b2(µk)b̄1(ηj)〉 =
P̃n(µk)S̃n(ηj) = H(µk, ηj)
〈b1(ξα)b̄2(ζβ)〉 = 0,
〈f (2)
b−1−N−L2+M2
b̄1(ηj)〉 =
hN+L2−M2−bS̃N+L2−M2−b(ηj), b = 1, . . . , N + L2 −M2,
〈f (2)
b−1−N−L2+M2
b̄2(ζβ)〉 =
hN+L2−M2−bSN+L2−M2−b(ζβ), b = 1, . . . , N + L2 −M2,
〈f (1)m−1+N+L1−M1 b̄1(ηj)〉 = η
−N−L1+M1−m
j , m = 1, . . . ,−N − L1 +M1,
〈f (1)m−1+N+L1−M1 b̄2(ζβ)〉 = 0, m = 1, . . . ,−N − L1 +M1
After trivial manipulations with rows and columns of the matrix of pair-wise v.e.v.
we obtain the answer
IN(ξ, ζ, η, µ) =
N+L1−M1−1
N+L2−M2−1
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG, (5.34)
ǫ = (−1)
N(N−1)+N(L1−M1)+
M1(M1−1)+
L2(L2−1)+L1(L2−M2)+L2M2−M2
where the (L2 +M1)× (L2 +M1) matrix G consists of eight blocks
µk−ζβ
0 Sb(ζβ) 0
H(µk, ηj)
ξα−ηj
S̃b(ηj) Sm(ηj)
(5.35)
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2;
b = 1, . . . , N + L2 −M2; m = 1, . . . ,−N − L1 +M1
Evaluation for the case N2 ≤ 0 ≤ N1.
The answer for this case may be obtained by the answer for the previous case, if we
inter-change subscripts L1 ↔ L2, M1 ↔ M2, N1 ↔ N2, and the variables ξ ↔ ζ , µ ↔ η
and also Pn ↔ Sn, P̃n ↔ S̃n, and (due to the definition (C.4)) H(µ, η) → H(µ, η) .
The answer coincides with the answer of [1] (see (3.21)-(3.22) there):
IN(ξ, ζ, η, µ) =
N+L2−M2−1
N+L1−M1−1
(ξα − ηj)
(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG, (5.36)
ǫ = (−1)
N(N−1)+N(L2−M2)+
M2(M2−1)+
L1(L1−1)+L2(L1−M1)+L1M1−M1
where the (L1 +M2)× (L1 +M2) matrix G consists of eight blocks
ξα−ηi
0 Pb(ξα) 0
H(µk, ηi)
µk−ζβ
P̃b(µk) Sm(µk)
(5.37)
where
α = 1, . . . , L1; β = 1, . . . , L2; i = 1, . . . ,M1; k = 1, . . . ,M2;
b = 1, . . . , N + L1 −M1; m = 1, . . . ,M2 − L2 −N
5.4 Evaluation for the case (3): N1 ≤ N2 ≤ 0
In this case we have (5.32).
Using transformation (2.42)-(2.43) which leaves vacuum expectation values invariant,
in form:
〈∗, ∗| → 〈∗, ∗+N2|, |∗, ∗〉 → |∗, ∗+N2〉,
i → f
i → f̄
, i ∈ Z,
we write
IN(ξ, ζ, η, µ)R
N = 〈N1,−N2|B2(µ)B1(ξ)B̄1(η)B̄2(ζ)|0, 0〉
= 〈N1, 0|C2(µ)C1(ξ)C̄1(η)C̄2(ζ)|0, N2〉 (5.38)
where we shift the charges of the second component of the vacuum vector by N2 and
replaced fermionic operators b1(ξ), b̄1(η), b2(µ) and b̄2(ζ) given by (4.26)-(4.29), by
c1(ξ) =
n Pn(ξ)
hn , (5.39)
c1(η) =
S̃n(η)√
−n−1+N2
S̃n(η)
hn , (5.40)
c2(µ) =
−n−1+N2
P̃n(µ)√
f (1)n P̃n(µ)
hn , (5.41)
c̄2(ζ) =
−n−1+N2
Sn(ζ)
hn (5.42)
We produced this shift of the vacuum charge and the fermion numbering in order to
come to the form of v.e.v., where all fermions with bar are situated to the right:
〈0, 0|f (1)−1 · · · f
c2(µn)
c1(ξn)
c̄1(ηn)
c̄2(ζn)f̄
· · · f̄ (2)−1 |0, 0〉, (5.43)
where formula (2.8) is applicable. We have nine-block matrix:
〈c1(ξα)c̄1(ηj)〉 〈c1(ξα)c̄2(ζβ)〉 〈c1(ξα)f̄ (2)m−1+N+L2−M2〉
〈c2(µk)c̄1(ηj)〉 〈c2(µk)c̄2(ζβ)〉 〈c2(µk)f̄ (2)m−1+N+L2−M2〉
〈f (1)
ℓ−1+N+L1−M1
c̄1(ηj)〉 〈f (1)ℓ−1+N+L1−M1 c̄2(ζβ)〉 0
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2;
ℓ = 1, . . . ,−N − L1 +M1; m = 1, . . . , N + L2 −M2
Pair-wise expectation values are
〈c1(ξ)c̄1(η)〉 =
ξ − η
〈c2(µ)c̄2(ζ)〉 =
P̃n(µ)Sn(ζ) =
〈c2(µ)c̄1(η)〉 =
P̃n(µ)S̃n(η) = H(µ, η),
〈c1(ξ)c̄2(ζ)〉 = 0,
〈f (1)
ℓ−1+N+L1−M1
c̄1(η)〉 = S̃ℓ−1+N+L1−M1(η)
hℓ−1+N+L1−M1 = η
−N−L1+M1−ℓ, ℓ = 1, . . . ,−N−L1+M1,
〈f (1)
ℓ−1+N+L1−M1
c̄2(ζ)〉 = 0, ℓ = 1, . . . ,−N − L1 +M1,
〈c1(ξ)f̄ (2)m−1+N+L2−M2〉 = 0, m = 1, . . . , N + L2 −M2,
〈c2(µ)f̄ (2)m−1+N+L2−M2〉 = P̃−m(µ)
h−m = µ
ℓ−1, m = 1, . . . , N + L2 −M2
After trivial manipulations with rows and columns of the matrix of pair-wise v.e.v. we
obtain the answer which coincides with the answer of [1] (given by (3.35)-(3.36) there):
IN(ξ, ζ, η, µ) =
N+L2−M2−1
N+L1−M1−1
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG, (5.44)
ǫ = (−1)
L2(L2−1)+
M1(M1−1)+
N(N+1)+L2(L1+M1)+N(L1+M1)+M2L2
where G is (M1 +M2 −N)× (M1 +M2 −N) matrix which consists of nine blocks:
ξα−ηj
H(µk, ηj)
µk−ζβ
Sm(µk)
Pℓ(ηj) 0 0
(5.45)
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2;
ℓ = 1, . . . ,M1 − L1 −N ; m = 1, . . . , N + L2 −M2
Evaluation for the case N2 ≤ N1 ≤ 0
After replacements we obtain
IN(ξ, ζ, η, µ) =
N+L2−M2−1
N+L1−M1−1
j=1(ξα − ηj)
k=1(ζβ − µk)
∆L1(ξ)∆L2(ζ)∆M1(η)∆M2(µ)
detG, (5.46)
ǫ = (−1)
L1(L1−1)+
M2(M2−1)+
N(N+1)+L1(L2+M2)+N(L2+M2)+M1L1
where G is (M1 +M2 −N)× (M1 +M2 −N) matrix which consists of nine blocks
ζβ−µk
H(µk, ηj)
ηj−ξα
Pm(ηj)
Sℓ(µk) 0 0
(5.47)
where
α = 1, . . . , L1; β = 1, . . . , L2; j = 1, . . . ,M1; k = 1, . . . ,M2;
ℓ = 1, . . . ,M2 − L2 −N ; m = 1, . . . ,M1 − L1 −N
Acknowledgements
The authors would like to thank T. Shiota and J. van de Leur for helpful discussions,
and (A.O.) thanks A. Odzijevicz for kind hospitality during his stay in Bialystok in June
2005.
A Appendix A
By (2.24) we have for p, q ∈ C:
[(gn)
, (gm)
q] = 0, [(en)
, (em)
q] = 0, n,m ∈ Z (A.1)
[(gn)
p, (em)
q] = 0, n 6= m (A.2)
where [, ] denotes the commutator.
Also:
[(gn)
p, f (1)m ] = [(gn)
p, f̄ (2)m ] = [(en)
p, f̄ (1)m ] = [(en)
p, f (2)m ] = 0, n,m ∈ Z, (A.3)
[(gn)
p, f̄ (1)m ] = [(gn)
−m−1] = [(en)
p, f (1)m ] = [(en)
p, f̄
−m−1] = 0, n 6= m, (A.4)
−pf̄ (1)n (gn)
p = f̄ (1)n + pf̄
−n−1, (gn)
−n−1(gn)
p = f
−n−1 − pf (1)n , n ∈ Z, (A.5)
pf (1)n (en)
−p = f (1)n − pf
−n−1, (en)
−n−1(en)
−p = f̄
−n−1 + pf̄
n n ∈ Z (A.6)
B Appendix B
For ω̂ :=
ωn,mfnf̄m we have
adω̂ fi =
fnωn,i , adω̂ f̄i = −
ωi,nf̄n
Then it follows
Adω̂ fi =
fn (e
, Adω̂ f̄i =
This yields (4.15) and (4.16) which are equivalent respectively to (4.10) and (4.11).
In the same way we prove (4.18) and (4.17) which are equivalent respectively to (4.13)
and (4.12).
C Appendix C
Now, let us show that
S̃n(η) =
Sn(y)
η − x
dµ(x, y), P̃n(µ) =
Pn(x)
dµ(x, y), n ≥ 0 (C.1)
Proof I.
−n−1]+ =
Pn(x)S̃n(η) (C.2)
where the first equality follows from the definitions (4.15) and (4.16), while the equality
of the anti-commutator to the last member follows from (4.10)-(4.11). Multiplying both
sides of the second equality by Sn(y)dµ(x, y) and integrating we come to the first equality
(C.1).
We obtain the second equality (C.1) by the similar integration of
−n−1µ
−n−1y
n]+ =
Sn(y)P̃n(µ) (C.3)
(These equalities results from definitions (4.17) and (4.18), and from (4.12),(4.13)).
We shall also use (in (5.20) below) the corollary of these equalities:
dµ(x, y)
(η − x)(µ− y)
S̃n(η)P̃n(µ) =: H(µ, η) (C.4)
which is obtained by multiplying of the right hand sides of the relations (C.2) and (C.3),
integrating and using the orthogonality of Pn and Sn.
Proof II. Considering the sum of entries weighted with powers of x−m−1 (where m
is the row-number of the entry) of the n-th column of relation K−1H = BK̄−1 (which
follows from the known factorization relation HK̄ = KB), we obtain the first equality of
(C.1) from the first of (4.5) and from the definition of bi-moments. (The second relation
in (C.1) is proved similarly).
D Appendix: Links with soliton theory
In this appendix, we discuss the links between the evaluation of such symmetric rational
integrals and soliton theory. There are two problems:
(A) To find which measure dµ is related to a given matrix integral? We do not know
the general answer to this problem. A partial answer can be found in the papers [24], [25]
and the Appendices to [26] and to [19].
(B) To find which measure dµ is related to soliton theory. This problem is addressed
in [19] and [28].
We restrict ourselves to the relation to the usual (one-component) TL hierarchy.
Indeed, one can easily show that the expression
(1), t(2)) = 〈N,−N |g(t(1), t(2))|0, 0〉, (D-1)
where g(t(1), t(2)) = eA(t
(1),t(2)), and A is of form
A(t(1), t(2)) =
i,j=1,2
eV (x,t
(1))−V (y,t(2))f (1)(x)f̄ (2)(y)dµ0(x, y) (D-2)
where
V (x, t(1)) =
t(1)n x
n, V (y, t(2)) =
t(2)n y
n (D-3)
and where dµ0(x, y) is a rather arbitrary measure, fits into a general expression for tau
functions of the two-component KP hierarchy and also the TL hierarchy developed in [14]
(see also [22], [21]).
Remark: the notations t
m and t
m are related respectively to the notations −umm and
of [19]. Our V (x, t(j)) here is Vj(x) of [19].
In [19] it is shown that the expression (D-1) on the one hand gives rise to multiple
integrals, which, in turn, for certain choices of the measure dµ0(x, y) (see Problem (A)
above) can be identified with a partition function of some model of random matrices. On
the other hand it is also an example of TL tau function.
This means that if we choose
dµ(x, y) = dµ(x, y, t(1), t(2)) = eV (x,t
(1))−V (y,t(2))dµ0(x, y) (D-4)
then the expectation value (D-1) is a TL (and also two-component KP) tau function.
Remark: if dµ0 solves this or that differential equation, than, one can write a corre-
spondent ”string equation”.
Notice that the first non-trivial member of the TL hierarchy was introduced and in-
tegrated in [20] and called the relativistic two-dimensional Toda lattice. In this paper a
factorization problem similar to (4.3) (but with matrices that are infinite in both direc-
tions) was considered (see also [22]).
Notational Remark. In formulas below we shall denote the pair of sets of times
(t(1), t(2)) by a single letter t. The only quantity which depends on a single such set
(either t(1) or t(2)) is V , where we shall point out which set it depends on.
Then from general consideration in [14] (see also [22], [21]) one finds that the inte-
grals of rational functions IN (η, ξ, µ, ζ), see (3.9), may be obtained via the bosonization
procedure as follows
IN(ξ, ζ, η, µ; t) =
τN(t)
ξNi D1(ξi)
η−Ni D1(ηi)
µ−Ni D2(µi)
ζNi D2(ζi)
−1τN (t), (D-5)
where
Dj(z) = exp
(D-6)
is a vertex operator.
Notational Remark. In [1] we use the notations
Pn(x) =
pn(x), (D-7)
S̃n(x) =
n(x) (D-8)
P̃ (y) =
n(y, t), (D-9)
Sn(y) =
sn(y) (D-10)
The functions
n (x, t) := e
V (x,t(1))
Pn(x, t)
hn = e
V (x,t(1))
pn(x, t), (D-11)
ψ∗(1)n (x, t) := e
−V (x,t(1)) S̃n−1(x, t)
= e−V (x,t
(1))p∗n−1(x, t) (D-12)
are to be interpreted as first component Baker functions (respectively, adjoint first com-
ponent Baker functions) which depend on a spectral parameter x, while
ψ(2)n (y, t) := e
V (y,t(2)) P̃n−1(y, t)
= eV (y,t
(2))s∗n−1(y, t), (D-13)
ψ∗(2)n (y, t) := e
−V (y,t(2))Sn(y, t)
hn = e
−V (y,t(2))sn(y, t) (D-14)
are to be interpreted as second component Baker functions (respectively, adjoint second
component Baker functions) which depend on a spectral parameter y.
This fact is due to the formulae (notice the factor (−1)N which result from re-ordering
the fermions of two different types to achieve true sign according to (2.30))
ψ(1)n (x, t) = (−1)NeV (x,t
(1)) 〈N + 1,−N |f (1)(x)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
= xNeV (x,t
(1))D1(x)τN (t)
τN(t)
(D-15)
ψ∗(1)n (x, t) = (−1)Ne−V (x,t
(1)) 〈N − 1,−N |f̄ (1)(x)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
= x−Ne−V (x,t
(1))D1(x)
−1τN (t)
τN (t)
(D-16)
ψ(2)n (y, t) = e
V (y,t(2)) 〈N, 1−N |f (2)(y)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
= y−NeV (y,t
(2))D2(y)τN(t)
τN(t)
(D-17)
ψ∗(2)n (y, t) = e
−V (y,t(2)) 〈N,−1−N |f̄ (2)(y)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
= yNe−V (y,t
(2))D2(y)
−1τN (t)
τN (t)
(D-18)
The first equality in each of the relations (D-15),(D-16),(D-17) and (D-18) is an example
of the evaluation of IN . Formulae (D-16) and (D-18) fit into the case N2 ≥ N1 ≥ 0 and
are just particular cases of formula (5.9, see the Examples following (5.9)). The second
equalities in each of the relations (D-15)-(D-18) are examples of bosonization formula
(D-5).
Notice that different examples of IN ,
K11(ξ, η, t) =
〈N,−N |f (1)(ξ)f̄ (1)(η)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
ξNη−ND1(ξ)D
1 (η)τN(t)
τN (t)
, (D-19)
K22(µ, ζ, t) =
〈N,−N |f (2)(µ)f̄ (2)(ζ)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
µ−NζND2(µ)D
2 (ζ)τN(t)
τN (t)
, (D-20)
K12(ξ, ζ, t) =
〈N + 1,−N − 1|f (1)(ξ)f̄ (2)(ζ)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
ξNζND1(ξ)D
2 (ζ)τN(t)
τN (t)
(D-21)
K21(µ, η, t) =
〈N − 1,−N + 1|f (2)(µ)f̄ (1)(η)g(t)|0, 0〉
〈N,−N |g(t)|0, 0〉
µ−Nη−ND2(µ)D
1 (η)τN(t)
τN (t)
(D-22)
(where for the sake of brevity we omit t(1), t(2)-dependence in the l.h. sides), may be con-
sidered as 2-component analogue of a modified Cauchy-Baker-Akhiezer kernel, introduced
in [30], [31] to present an explicit version of Segal-Wilson construction to study Virasoro
deformations of tau functions for the quasi-periodical solutions of the KP (and actually
for the TL) hierarchies.
As we have obtained (see Examples following (5.9)):
K11(µ, ζ) =
P̃n(µ)Sn(ζ) +
ζ − µ
(D-23)
K22(ξ, η) =
Pn(ξ)S̃n(η) +
ξ − η
(D-24)
K21(ξ, ζ) =
Pn(ξ)Sn(ζ) (D-25)
K12(µ, η) =
P̃n(µ)S̃n(η)−
dµ(x, y)
(η − x)(µ− y)
(D-26)
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	Introduction
	Summary of free fermions
	One component fermions
	Two-component fermions
	The integral of rational symmetric functions as a certain expectation value
	Biorthogonal polynomials and dressed fermions
	Determinantal expression for integrals of rational symmetric functions
	Determinantal representation in the case (1): N2N10
	When N1<0
	Evaluation for the case (2): N10N2.
	Evaluation for the case (3): N1N20 
	Appendix A
	Appendix B
	Appendix C
	Appendix: Links with soliton theory
