CRM-xxxx (2005)
nlin.SI/05xxxxxx
Fermionic construction of partition function for
multi-matrix models and multi-component TL hierarchy1
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 p-component fermions (p = 2, 3, . . . ) to present (2p−2)N -fold integrals as a fermionic
expectation value. This yields fermionic representation for various (2p−2)-matrix models. Links
with the p-component KP hierarchy and also with the p-component TL hierarchy are discussed.
We show that the set of all (but two) flows of p-component TL changes standard matrix models
to new ones.
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 “Fundamental Methods in Nonlinear Dynamics” and RFBR grant No 05-01-00498.
2harnad@crm.umontreal.ca
3 orlovs@wave.sio.rssi.ru
http://arxiv.org/abs/0704.1145v1
1 Introduction
Let dµα(x, y) be a set of measures (in general, complex), supported on a finite set of
products of curves in the complex x and y planes.
Let ρα, α = 2, . . . , p− 1, be a set of functions in two variables.
Let x(α) = (x
1 , . . . , x
N ) and y
(α) = (y
1 , . . . , y
N ), α = 1, . . . , p are two sets of
variables, where x(p) and y(1) are fixed by
i = y
i = N − i, i = 1, . . . , N (1.1)
We shall use the following notation
dµα(x
(α), y(α+1)) :=
dµα(x
i , y
(α+1)
i ) (1.2)
We consider the following integral over (2p− 2)N variables x(α) = (x
1 , . . . , x
N ) and
y(α+1) = (y
(α+1)
1 , . . . , y
(α+1)
N ), α = 1, . . . , p− 1:
(α), x(α))
dµα(x
(α), y(α+1)) (1.3)
where
(1), x(1)) = det
i,j=1,...,N
i − x
j ) =: ∆N (x
(1)),
(p), x(p)) = det
i,j=1,...,N
i − y
j ) =: ∆N(y
are Vandermonde determinants, and where
(α), x(α)) = det
i , x
i,j=1,...,N
, α = 2, . . . , p− 1 (1.4)
Developing each detρα into N ! monomial terms (each is labeled by an element of the
permutation group SN), and, for given choice of the element of the permutation group,
say σ, using the change of variables inside of each N -fold integral to the left (namely,
x(β) → σ(x(β)), y(β) → σ(y(β)) for all β ≤ α), then, using the anti-symmetry of ∆N (x
(which is the integrand of the very left N -fold integral), one finds that each term of the
mentioned development yields the same contribution. This is a standard way to re-write
(1.3) as
(N !)2p−2
. . .
∆N (x
dµ1(x
i , y
i )ρ2(y
i , x
. . .
i , x
i )dµ2(x
i , y
· · ·
. . .
ρp−1(y
(p−1)
i , x
(p−1)
i )dµp−2(x
(p−2)
i , y
(p−1)
. . .
dµp−1(x
(p−1)
i , y
i )∆N(y
Integrals (1.3) may be related to the so-called determinantal ensembles [6].
For special choice of measures dµα and functions ρα, integrals (1.3) arose in the
study of multi-matrix models, where matrices M1,M2,M3, . . . ,M2p−2 with eigenvalues
respectively equal to the sets {x
i , i = 1, . . . , N},{y
i , i = 1, . . . , N},{x
i , i =
1, . . . , N},. . . ,{y
i , i = 1, . . . , N}, are coupled in an open chain. It occurs in case when
one can reduce the integration over matrix entries to the integrals over eigenvalues of
each matrix (for these topic see [16], [17] and Appendices to [18], [1]). Depending on
dµα and functions ρα, these are models of normal matrices, and certain models of ran-
dom Hermitian (anti-Hermitian) matrices and certain models of random unitary matrices,
see [4], [5], [11], [3], [8], [16], [17], together with discrete versions of these matrix mod-
els [18].
For instance, to obtain the partition function for the model of random N by N Her-
mitian matrices, M1, . . . ,M2p−2, coupled in a chain,
P2p−2
Vk(Mk)+Tr(c1M1M2+···+c2p−3M2p−3M2p−2)
one takes
dµα(x, y) = e
c2α−1xy+V2α−1(x)+V2α(y), α = 1, . . . , p− 1 (1.5)
ρα(x, y) = e
c2αxy, α = 2, . . . , p− 1 (1.6)
Then x
i , i = 1, . . . , N , are eigenvalues of Hermitian matrices with odd numbers, say
M2α−1, while y
i , i = 1, . . . , N , are eigenvalues of M2α, α = 1, . . . , p − 1. For future
purpose, let us use the obvious freedom to re-write dµα and ρα in form
dµ1(x, y) → e
c1xy+V1(x), dµp−1(x, y) → e
c2p−3xy+V2p−2(x), (1.7)
dµα(x, y) → e
c2α−1xy, α = 2, . . . , p− 2, ρα(x, y) → e
c2αxy+V(x)+V(y), α = 2, . . . , p− 1
(1.8)
In the present paper we have two tasks.
First, we equate the integral (1.3) to the fermionic vacuum expectation value. Here
we use the so-called p-component fermions. This may be considered as a continuation of
of the work [11].
Second, as a continuation of [7], we relate ZN to the coupled p-component KP hier-
archies, or, the same to the p component TL hierarchy. For this purpose we consider the
following deformation of the first and the last measures
dµ1(x, y) → dµ1(x, y|t
(1), n, t̄(1)) := xn1eV (x,t
(α))+V (x−1,t̄(1))dµ1(x, y), (1.9)
dµp−1(x, y) → dµp−1(x, y|t
(p), n, t̄(p)) := ynpeV (y,t
(p))+V (y−1,t̄(p))dµp−1(x, y) (1.10)
V (x, t(α)) =
xmt(α)m , V (x
−1, t̄(α)) =
x−mt̄(α)m , α = 1, p, (1.11)
and also the following deformations of functions ρα, α = 2, . . . , p− 1,
ρα(x, y) → τn(α)(t
(α) + [x], t̄(α) + [y]), α = 2, . . . , p− 1, (1.12)
where in the right hand side we have tau functions (labeled by α = 2, . . . , p − 1) of the
one-component TL hierarchy, and where +[x] and +[y] denote the so-called Miwa shift of
a TL (a one-component TL) higher times, details are written down below.
The deformation (1.9)-(1.12) relates integrals (1.3) to the coupled p-component KP
hierarchies. If in (1.3) we take the deformed measures and the deformed functions ρα as
described above, then, ZN turns out to be a certain tau function of coupled p-component
KP, or the same, p-component TL hierarchy, where the sets of complex numbers t(α) =
1 , t
2 , . . . ), t̄
(α) = (t̄
1 , t̄
2 , . . . ), and the set of integers n
(α), α = 1, . . . , p, play the role
of higher p-component TL times. For the sake of brevity we shall also use the notations
t = (t(1), . . . , t(p)) and t̄ = (t̄(1), . . . , t̄(p)).
Important to mark, that the deformation (1.9), (1.10) and (1.12) seems do not keep the
form (1.7)-(1.8). In our case the interaction ec2αM2αM2α+1+V2α(M2α)+V2α+1(M2α+1) is replaced
by arbitrary chosen one-component TL tau function (1.12) where x = x(α) is the collection
of eigenvalues of the matrixM2α while y = y
(α) is the collection of eigenvalues of the matrix
M2α+1.
Let us note that one can consider (p− 1)N -fold integrals if he specifies the measures
dµα(x, y) to be proportional to Dirac delta function which equate x to a function of y (it
may be δ(x− y)).
The present paper is a part of series of papers devoted to fermionic approaches to
multi-fold integrals, see [12], [1], [2]. Let us mark that our fermionic constructions of
papers [1], [2] and of the present paper are different from what was considered in [11] and
also different of [12].
1.1 Free fermions
LetA be 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 . (1.13)
Any element of the linear part
W := (⊕m∈ZCfm)⊕
⊕m∈ZCf̄m
(1.14)
will be referred to as a free fermion. We also introduce the fermionic free fields
f(x) :=
k, f̄(y) :=
−k−1, (1.15)
which may be viewed as generating functions for the fj , f̄j’s.
This Clifford algebra has a standard Fock space representation defined as follows.
Define the complementary, totally null (with respect to the underlying quadratic form)
and mutually dual subspaces
Wan := (⊕m<0Cfm)⊕
⊕m≥0Cf̄m
, Wcr := (⊕m≥0Cfm)⊕
⊕m<0Cf̄m
, (1.16)
and consider the left and right A-modules
F := A/AWan, F̄ := WcrA\A. (1.17)
These are cyclic A-modules generated by the vectors
|0〉 = 1 mod AWan, 〈0| = 1 mod WcrA, (1.18)
respectively, with the properties
fm|0〉 = 0 (m < 0), f̄m|0〉 = 0 (m ≥ 0),
〈0|fm = 0 (m ≥ 0), 〈0|f̄m = 0 (m < 0). (1.19)
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 is determined by
〈0|1|0〉 = 1; 〈0|fmf̄m|0〉 = 1, m < 0; 〈0|f̄mfm|0〉 = 1, m ≥ 0, (1.20)
〈0|fn|0〉 = 〈0|f̄n|0〉 = 〈0|fmfn|0〉 = 〈0|f̄mf̄n|0〉 = 0; 〈0|fmf̄n|0〉 = 0, m 6= n,
(1.21)
together with the Wick theorem which implies, for any finite set of elements {wk ∈ W},
〈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〉. (1.22)
Here σ runs over permutations for which σ(1) < σ(2), . . . , σ(2n− 1) < σ(2n) and σ(1) <
σ(3) < · · · < σ(2n− 1).
Now let {wi}i=1,...,N , be linear combinations of the fj’s only, j ∈ Z, and {w̄i}i=1,...,N
linear combinations of the f̄j ’s, j ∈ Z. Then(1.22) implies
〈0|w1 · · ·wN w̄N · · · w̄1|0〉 = det (〈0|wiw̄j|0〉) |i,j=1,...,N (1.23)
Following refs. [9], [10], for all N ∈ Z, we also introduce the states
〈N | := 〈0|CN (1.24)
where
CN := f̄0 · · · f̄
N−1 if N > 0 (1.25)
CN := f−1 · · · fN if N < 0 (1.26)
CN := 1 if N = 0 (1.27)
|N〉 := C̄N |0〉 (1.28)
where
C̄N := fN−1 · · · f0 if N > 0 (1.29)
C̄N := f̄N · · · f̄−1 if N < 0 (1.30)
C̄N := 1 if N = 0 (1.31)
The states (1.24) and (1.28) are referred to as the left and right charged vacuum vectors,
respectively, with charge N .
In what follows we use the notational convention
∆N(x) = det (x
i )|i,k=1,...,N (N > 0), ∆0(x) = 1, ∆N(x) = 0 (N < 0). (1.32)
From the relations
〈0|f̄N−kf(xi)|0〉 = x
i , 〈0|f−N+k−1f̄(yi)|0〉 = y
i , k = 1, 2, . . . , (1.33)
and (1.23), it follows that
〈N |f(x1) · · · f(xn)|0〉 = δn,N∆N(x), N ∈ Z, (1.34)
〈−N |f̄(y1) · · · f̄(yn)|0〉 = δn,N∆N(y), N ∈ Z. (1.35)
Following [9], [10] we consider ĜL∞ element
g = eh, h =
hi,jfif̄j , hi,j ∈ C (1.36)
Via the conjugation, (·) → g(·)g−1 , each g ∈ ĜL∞ acts on the spaces (⊕m∈ZCfm) and
⊕m∈ZCf̄m
as linear transformations [9], [10].
We suppose that the following factorization condition is valid:
g = g+g−, 〈0|g
+ = 〈0|, g−|0〉 = |0〉, (1.37)
where g+, g− ∈ ĜL∞.
Remark. Though, the property (1.37) is valid for a rather wide class of (1.36) (which
includes all cases when the sum in (1.36) is finite) , however, we do not know the general
theorem providing sufficient and necessary conditions to have this property in case the
sum in (1.36) is infinite.
Consider
〈0|vN · · · v1gv̄1 · · · v̄N |0〉,
where each vi ∈ (⊕m∈ZCfm) and each v̄i ∈
⊕m∈ZCf̄m
, i = 1, . . . , N . Denoting wi =
−1vig+ ∈ (⊕m∈ZCfm) and w̄i = (g−)v̄i(g−)
⊕m∈ZCf̄m
we have
〈0|vN · · · v1gv̄1 · · · v̄N |0〉 = 〈0|wN · · ·w1w̄1 · · · w̄N |0〉 = det〈0|wiw̄j|0〉|i,j=1,...,N
where the second equality is due to the Wick theorem (1.23). Thus
〈0|vN · · · v1gv̄1 · · · v̄N |0〉 = det〈0|vigv̄j · · · |0〉|i,j=1,...,N (1.38)
1.2 Multi-component fermions
One obtains the so-called p-component fermion formalism by re-numerating the above
free fermions (1.13) as follows
f (α)n := fpn+α−1 , f̄
n := f̄pn+α−1 , (1.39)
f (α)(z) :=
k , f̄
(α)(z) :=
z−k−1f̄
k , (1.40)
where α = 1, . . . , p. From (1.13) we obviously have
[f (α)n , f
m ]+ = [f̄
n , f̄
m ]+ = 0, [f
n , f̄
m ]+ = δα,βδn,m. (1.41)
Right and left vacuum vectors are respectively defined
|0, . . . , 0
︸ ︷︷ ︸
〉 := |0〉, 〈0, . . . , 0
︸ ︷︷ ︸
| := 〈0| (1.42)
where |0〉 and 〈0| were introduced in (1.19).
As it follows from (1.19)
f (α)m |0, . . . , 0〉 = 0 (m < 0), f̄
m |0, . . . , 0〉 = 0 (m ≥ 0), (1.43)
〈0, . . . , 0|f (α)m = 0 (m ≥ 0), 〈0, . . . , 0|f̄
m = 0 (m < 0). (1.44)
We also introduce the states
〈n(1), . . . , n(p)| := 〈0, 0|Cn(1) · · ·Cn(p) (1.45)
where
Cn(α) := f̄
0 · · · f̄
n(α)−1
if n(α) > 0 (1.46)
Cn(α) := f
−1 · · · f
if n(α) < 0 (1.47)
Cn(α) := 1 if n
(α) = 0 (1.48)
|n(1), . . . , n(p)〉 := C̄n(p) · · · C̄n(1)|0, 0〉 (1.49)
where
C̄n(α) := f
n(α)−1
· · · f
0 if n
(α) > 0 (1.50)
C̄n(α) := f̄
· · · f̄
−1 if n
(α) < 0 (1.51)
C̄n(α) := 1 if n
(α) = 0 (1.52)
Let us call (1.45) and (1.49) respectively left and right charged vacuum vectors with the
charge (n(1), . . . , n(p)).
We easily verify that
f (α)m |∗, n
(α), ∗〉 = 0 (m < n(α)), f̄ (1)m |∗, n
(α), ∗〉 = 0 (m ≥ n(α)), (1.53)
〈∗, n(α), ∗|f (α)m = 0 (m ≥ n
(α)), 〈∗, n(α), ∗|f̄ (α)m = 0 (m < n
(α)), (1.54)
where ∗ serve for irrelevant components in vacuum vectors.
Remark 1.1. For calculations we use the Wick theorem in form (1.23). There are two
ways to do it:
(1) The first one is to use (1.23) just remembering that p-component fermions are
composed of usual ones, see (1.39).
(2) The second way is to use formula (1.23) separately for each component. Namely,
to calculate the vacuum expectation value of an operator O, first, we present it in form
i · · ·O
i (1.55)
〈0|O|0〉 =
i · · ·O
i |0〉 =
i |0〉 · · · 〈0|O
i |0〉 (1.56)
where the Wick theorem in form (1.23) is applied to each of 〈0, . . . , 0|O
i |0, . . . , 0〉.
2 Fermionic representation for ZN
Consider the element of the Clifford algebra of the following form
g = eA1g2e
A2g3 · · · e
Ap−2gp−1e
Ap−1 , (2.1)
where
f (α)(x)f̄ (α+1)(y)dµα(x, y), α = 1, . . . , p− 1, (2.2)
with measure dµα(x, y), which we do not specify.
In (2.1)
gα = e
hα , hα =
i,j f
j , h
i,j ∈ C, (2.3)
so that we have
i gα = gαf
i , α 6= β, i ∈ Z (2.4)
We also suppose that each gα = e
h(α), α = 2, . . . , p − 1, may be factorized into ĜL
elements g+α and g
α as follows (see (1.37))
gα = g
α , 〈∗,
0̂, ∗|g+α = 〈∗,
0̂, ∗|, g−α |∗,
0̂, ∗〉 = |∗,
0̂, ∗〉 (2.5)
where by ∗ we denote irrelevant components of a vacuum vector (different from the com-
ponent α marked by hats).
Now, let us notice that by (1.38) we have
〈0|f̄ (α)(y1) · · · f̄
(α)(yN)gαf
(α)(xN ) · · ·f
(α)(x1)|0〉 = det
〈0|f̄ (α)(yi)gαf
(α)(xj)|0〉
i,j=1,...,N
(2.6)
Now let us prove, that for special choice of functions ρα, α = 2, . . . , p− 1, namely, for
〈0|f̄ (α)(y)gαf
(α)(x)|0〉 = ρα(y, x) (2.7)
we have
(N !)p−1〈N, 0, . . . , 0,−N |g|0, 0, . . . , 0, 0〉 = ZN (2.8)
Indeed, to get a non-vanishing expectation value in the left hand side, we have to pick
up only N -th term,
, in the Taylor series for eA1 (this is because eA1 is the only factor of
g which contains the first component fermions, and the matrix element 〈N, ∗|An1 |0, ∗〉 ≡ 0
until n = N). Using the known formula (1.34), we obtain, that the left hand side of (2.8)
is equal to the integral
dµ1(x
1 , y
1 ) . . .
dµ1(x
N , y
N )∆N (x
(1))R1,
R1 = 〈∗, 0, . . . , 0,−N |f̄
(2)(y
1 ) · · · f̄
(2)(y
N )g2 · · · |∗, 0, . . . , 0〉
where we put ∗ on the first place of the left and right vacuum vectors to show that we
forget about the first component fermions. This is the first step.
Then, we have to pick up only N -th term,
, when developing the next factor eA2.
Otherwise, the vacuum expectation values of the second component fermions vanishes.
This is because the second component fermions are in presence only in eA1 , g2 and in e
factors of g, and the g2 is a sum of monomials, each of which contains equal number of
f (2) and f̄ (2) fermions, while eA1 contains only f (2) , and eA2 contains only f̄ (2) fermions.
Thus, second component fermions yields the expression
f̄ (2)(y
1 ) · · · f̄
(2)(y
N )g2f
(2)(x
1 ) · · · f
(2)(x
which should be integrated with measures
i=1 dµ(∗, y
i )dµ(x
i , ∗), and then substituted
inside 〈N, 0, . . . , 0,−N | and |0, 0, . . . , 0, 0〉. Denoting
〈0|f̄ (2)(y
1 ) · · · f̄
(2)(y
N )g2f
(2)(x
1 ) · · · f
(2)(x
N )|0〉 = ̺2(y
(2), x(2)) (2.9)
(which, by (2.6), is equal to detρ2(y
i , x
j )) we obtain that the l.h.s of (2.8) is equal to
the integral ∫
dµ1(x
1 , y
1 ) . . .
dµ1(x
N , y
N )∆N (x
dµ2(x
1 , y
1 ) . . .
dµ2(x
N , y
N )̺3(y
(3), x(3))R2,
R2 = 〈∗, ∗, 0, . . . , 0,−N |f̄
(2)(y
1 ) · · · f̄
(2)(y
N )g2 · · · |∗, ∗, 0, . . . , 0〉
where we put ∗ on the first and second places of the left and right vacuum vectors to show
that we forget about the first and the second component fermions. This is the second
step.
Then, it is easy to see that each exponential eAα should be replaced by their N -th
Taylor term, otherwise the l.h.s. of (2.8) vanishes, it means we have
〈N, 0, . . . , 0,−N |g|0, 0, . . . , 0, 0〉 =
(N !)p−1
〈N, 0, . . . , 0,−N |AN1 g2A
2 · · · gp−1A
p−1|0, 0, . . . , 0, 0〉 (2.10)
Continuing excluding step by step third- forth- and so on component fermions, and,
on the last step, using the known formula (1.35), we obtain that (2.10) is equal to (1.3).
At last we want to make the following remark
Remark 2.1. Insert additional factors to (2.1) as follows
g = eA1g2e
A2g3 · · · e
Ap−2gp−1e
Ap−1 → g	 := e
A1g2e
A2g3 · · · e
Ap−2gp−1e
Ap−1gpg1e
Ap (2.11)
where
gα = e
i,j h
j , Aα =
f (α)(x)f̄ (α+1)(y)dµα(x, y), α = 1, . . . , p (2.12)
where f̄ (p+1)(y) ≡ f̄ (1)(y). (Thus we add g1, gp and dµp(x, y) to our collection of data, gα, α =
2, . . . , p− 1 and dµα(x, y), α = 1, . . . , p− 1). Then
〈0, 0, . . . , 0, 0|g	|0, 0, . . . , 0, 0〉 =
(2.13)
where cN are certain numbers and each Z
is the following integral over 2pN variables x(α) =
1 , . . . , x
) and y(α) = (y
1 , . . . , y
), α = 1, . . . , p:
(α), x(α))
dµα(x
(α), y(α+1)), y(p+1) ≡ y(1) (2.14)
(notice that variables y(1) and x(p) are not fixed by (1.1))
In (2.14) dµα(x
(α), y(α+1)) (α = 1, . . . , p − 1) are defined by (1.2) and
dµp(x
(p), y(1)) :=
dµp(x
i , y
i ), (2.15)
and each ̺α(y
(α), x(α)) is defined by (2.6)-(2.7), where now α = 1, . . . , p.
Sums (2.13) and their relation to the grand partition function of closed chains of coupled
random matrices and to integrable equations will be considered in a forthcoming paper.
3 Deformation of measure and relations to integrable
hierarchies
The described deformation
dµ1(x, y) → dµ1(x, y|t
(1), n, t̄(1)) := xn1eV (x,t
(α))+V (x−1,t̄(1)))dµ1(x, y), (3.1)
dµp−1(x, y) → dµp−1(x, y|t
(p), n, t̄(p)) := ynpeV (y,t
(p))+V (y−1,t̄(p)))dµp−1(x, y) (3.2)
V (x, t(α)) =
xmt(α)m , V (x
−1, t̄(α)) =
x−mt̄(α)m , α = 1, p, (3.3)
Then, it is quite known fact that in this case ZN = τN (t
(1), t̄(p)), where τN (t
(1), t̄(p)) is
a tau function of the (one-component) TL hierarchy. Indeed, one just re-writes (1.3) as
2N -fold integral with a modified measure dµmod (the latter depends on the choice of ρα) :
dµmod(x
i , y
,t̄(1))+V (y
,t̄(p))eV (x
,t(1))+V (y
,t(p))∆N(x
(1))∆N(y
Moreover, as a function of t(1), t(p), t̄(1), t̄(p), the integral ZN(t
(1), t(p), n1, np, t̄
(1), t̄(p)) is
a tau function of the coupled two-component KP, or, the same, a tau function of the
two-component TL hierarchy, see [1].
Now, in addition, we consider the following deformations of functions ρα, α = 2, . . . , p−
ρα(x, y) → ρα(x, y|t
(α), n(α), t̄(α)) := (3.4)
〈n(α)|eH
(α)(t(α))f̄ (α)(y
1 ) · · · f̄
(α)(y
N )gαf
(α)(x
1 ) · · ·f
(α)(x
H̄(α)(t̄(α))|n(α)〉 (3.5)
where t(α) = (t
1 , t
1 , . . . ) and t̄
(α) = (t̄
1 , t̄
2 , . . . ) are the deformation parameters, and
where the “Hamiltonians” H
, k = ±1,±2, . . . , are defined by
H(α)(t(α)) =
, H̄(α)(t̄(α)) =
f (α)n f̄
n+k (3.6)
(for future purpose we define them for α = 1, . . . , p range).
Let us note that the expectation value (3.5), by definition [9], [10], [15], is a tau
function of one component TL and in our case may be denoted by
τn(α)(t
(α) + [x(α)], t̄(α) + [y(α)])
where
[x] :=
, . . .
Now let us prove that the combination of deformations (3.1)-(3.2) and (3.4) is equiv-
alent to the replacement
〈N, 0, . . . , 0,−N |g|0, 0, . . . , 0, 0〉 →
τN(t,n, t̄) := 〈N + n
(1), n(2), . . . , n(p−1),−N − n(p)|eH(t)geH̄(t̄)|n(1), n(2), . . . , n(p−1),−n(p)〉,
(3.7)
where
H(t) =
, H̄(t̄) =
the “Hamiltonians” H
k were defined earlier by (3.6).
Proof. Indeed, we have that each terms of type (2.9), namely, each
〈0|f̄ (α)(y
1 ) · · · f̄
(α)(y
N )gαf
(α)(x
1 ) · · ·f
(α)(x
N )|0〉 = ̺(y
(α), x(α)), α = 2, . . . , p− 1,
(3.8)
is now replaced by
〈n(α)|eH
(α)(t(α))f̄ (α)(y
1 ) · · · f̄
(α)(y
N )gαf
(α)(x
1 ) · · ·f
(α)(x
H̄(α)(t̄(α))|n(α)〉 (3.9)
which is, by definition [9], [10], [15], a tau function of one component TL. Due to (1.38)
it is equal to
detρα(xi, yj|t,n, t̄)
where
ρα(xi, yj|t,n, t̄) := 〈n
(α)|eH
(α)(t(α))f̄ (α)(y
i )gαf
(α)(x
H̄(α)(t̄(α))|n(α)〉
As for α = 1, p we have
〈N, ∗|eH
(1)(t(1))f(x1) · · ·f(xN)e
H̄(1)(t̄(1))|0, ∗〉 = a1∆N (x)e
i=1 V (xi,t
(1))+V (x
,t̄(1)), (3.10)
〈∗, 0|eH̄
(p)(t̄(p))f̄(yN) · · · f̄(y1)e
H̄(p)(t̄(p))|∗,−N〉 = ap∆N (y)e
i=1 V (yi,t
(p))+V (y
,t̄(p)), (3.11)
where aα = e
k=1 kt
k , which contribute to the deformation respectively of dµ1 and of
dµp−1. The end of proof.
Thus, we obtain that the deformation of functions ρα, α = 2, . . . , p − 1, and also of
dµ1, dµp−1 reduce to the fact that ZN is equal to τN(t,n, t̄). It is known [9], [10], [14] that
thus constructed τN (t,n, t̄) is a tau function of the coupled p-component KP hierarchy,
or, the same, p-component TL hierarchy.
4 Conclusion
We equate the multi-integral (1.3) to the fermionic expectation value (2.8). On the one
hand we hope that the fermionic representation allows to evaluate different magnitudes re-
lated to the matrix models, like spectral determinants (compare with [2]), or perturbative
series generalizing [12], [13]. On the other hand it allows to incorporate the study of these
integrals and related multi-matrix models to the study of multi-component integrable
hierarchies
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, which helped stimulate ideas leading to this work.
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http://arxiv.org/abs/math-ph/0501017
	Introduction
	Free fermions 
	Multi-component fermions 
	Fermionic representation for ZN
	Deformation of measure and relations to integrable hierarchies
	Conclusion

  We use $p$-component fermions $(p=2,3,...)$ to present $(2p-2)N$-fold
integrals as a fermionic expectation value. This yields fermionic
representation for various $(2p-2)$-matrix models. Links with the $p$-component
KP hierarchy and also with the $p$-component TL hierarchy are discussed. We
show that the set of all (but two) flows of $p$-component TL changes standard
matrix models to new ones.

Introduction
Let dµα(x, y) be a set of measures (in general, complex), supported on a finite set of
products of curves in the complex x and y planes.
Let ρα, α = 2, . . . , p− 1, be a set of functions in two variables.
Let x(α) = (x
1 , . . . , x
N ) and y
(α) = (y
1 , . . . , y
N ), α = 1, . . . , p are two sets of
variables, where x(p) and y(1) are fixed by
i = y
i = N − i, i = 1, . . . , N (1.1)
We shall use the following notation
dµα(x
(α), y(α+1)) :=
dµα(x
i , y
(α+1)
i ) (1.2)
We consider the following integral over (2p− 2)N variables x(α) = (x
1 , . . . , x
N ) and
y(α+1) = (y
(α+1)
1 , . . . , y
(α+1)
N ), α = 1, . . . , p− 1:
(α), x(α))
dµα(x
(α), y(α+1)) (1.3)
where
(1), x(1)) = det
i,j=1,...,N
i − x
j ) =: ∆N (x
(1)),
(p), x(p)) = det
i,j=1,...,N
i − y
j ) =: ∆N(y
are Vandermonde determinants, and where
(α), x(α)) = det
i , x
i,j=1,...,N
, α = 2, . . . , p− 1 (1.4)
Developing each detρα into N ! monomial terms (each is labeled by an element of the
permutation group SN), and, for given choice of the element of the permutation group,
say σ, using the change of variables inside of each N -fold integral to the left (namely,
x(β) → σ(x(β)), y(β) → σ(y(β)) for all β ≤ α), then, using the anti-symmetry of ∆N (x
(which is the integrand of the very left N -fold integral), one finds that each term of the
mentioned development yields the same contribution. This is a standard way to re-write
(1.3) as
(N !)2p−2
. . .
∆N (x
dµ1(x
i , y
i )ρ2(y
i , x
. . .
i , x
i )dµ2(x
i , y
· · ·
. . .
ρp−1(y
(p−1)
i , x
(p−1)
i )dµp−2(x
(p−2)
i , y
(p−1)
. . .
dµp−1(x
(p−1)
i , y
i )∆N(y
Integrals (1.3) may be related to the so-called determinantal ensembles [6].
For special choice of measures dµα and functions ρα, integrals (1.3) arose in the
study of multi-matrix models, where matrices M1,M2,M3, . . . ,M2p−2 with eigenvalues
respectively equal to the sets {x
i , i = 1, . . . , N},{y
i , i = 1, . . . , N},{x
i , i =
1, . . . , N},. . . ,{y
i , i = 1, . . . , N}, are coupled in an open chain. It occurs in case when
one can reduce the integration over matrix entries to the integrals over eigenvalues of
each matrix (for these topic see [16], [17] and Appendices to [18], [1]). Depending on
dµα and functions ρα, these are models of normal matrices, and certain models of ran-
dom Hermitian (anti-Hermitian) matrices and certain models of random unitary matrices,
see [4], [5], [11], [3], [8], [16], [17], together with discrete versions of these matrix mod-
els [18].
For instance, to obtain the partition function for the model of random N by N Her-
mitian matrices, M1, . . . ,M2p−2, coupled in a chain,
P2p−2
Vk(Mk)+Tr(c1M1M2+···+c2p−3M2p−3M2p−2)
one takes
dµα(x, y) = e
c2α−1xy+V2α−1(x)+V2α(y), α = 1, . . . , p− 1 (1.5)
ρα(x, y) = e
c2αxy, α = 2, . . . , p− 1 (1.6)
Then x
i , i = 1, . . . , N , are eigenvalues of Hermitian matrices with odd numbers, say
M2α−1, while y
i , i = 1, . . . , N , are eigenvalues of M2α, α = 1, . . . , p − 1. For future
purpose, let us use the obvious freedom to re-write dµα and ρα in form
dµ1(x, y) → e
c1xy+V1(x), dµp−1(x, y) → e
c2p−3xy+V2p−2(x), (1.7)
dµα(x, y) → e
c2α−1xy, α = 2, . . . , p− 2, ρα(x, y) → e
c2αxy+V(x)+V(y), α = 2, . . . , p− 1
(1.8)
In the present paper we have two tasks.
First, we equate the integral (1.3) to the fermionic vacuum expectation value. Here
we use the so-called p-component fermions. This may be considered as a continuation of
of the work [11].
Second, as a continuation of [7], we relate ZN to the coupled p-component KP hier-
archies, or, the same to the p component TL hierarchy. For this purpose we consider the
following deformation of the first and the last measures
dµ1(x, y) → dµ1(x, y|t
(1), n, t̄(1)) := xn1eV (x,t
(α))+V (x−1,t̄(1))dµ1(x, y), (1.9)
dµp−1(x, y) → dµp−1(x, y|t
(p), n, t̄(p)) := ynpeV (y,t
(p))+V (y−1,t̄(p))dµp−1(x, y) (1.10)
V (x, t(α)) =
xmt(α)m , V (x
−1, t̄(α)) =
x−mt̄(α)m , α = 1, p, (1.11)
and also the following deformations of functions ρα, α = 2, . . . , p− 1,
ρα(x, y) → τn(α)(t
(α) + [x], t̄(α) + [y]), α = 2, . . . , p− 1, (1.12)
where in the right hand side we have tau functions (labeled by α = 2, . . . , p − 1) of the
one-component TL hierarchy, and where +[x] and +[y] denote the so-called Miwa shift of
a TL (a one-component TL) higher times, details are written down below.
The deformation (1.9)-(1.12) relates integrals (1.3) to the coupled p-component KP
hierarchies. If in (1.3) we take the deformed measures and the deformed functions ρα as
described above, then, ZN turns out to be a certain tau function of coupled p-component
KP, or the same, p-component TL hierarchy, where the sets of complex numbers t(α) =
1 , t
2 , . . . ), t̄
(α) = (t̄
1 , t̄
2 , . . . ), and the set of integers n
(α), α = 1, . . . , p, play the role
of higher p-component TL times. For the sake of brevity we shall also use the notations
t = (t(1), . . . , t(p)) and t̄ = (t̄(1), . . . , t̄(p)).
Important to mark, that the deformation (1.9), (1.10) and (1.12) seems do not keep the
form (1.7)-(1.8). In our case the interaction ec2αM2αM2α+1+V2α(M2α)+V2α+1(M2α+1) is replaced
by arbitrary chosen one-component TL tau function (1.12) where x = x(α) is the collection
of eigenvalues of the matrixM2α while y = y
(α) is the collection of eigenvalues of the matrix
M2α+1.
Let us note that one can consider (p− 1)N -fold integrals if he specifies the measures
dµα(x, y) to be proportional to Dirac delta function which equate x to a function of y (it
may be δ(x− y)).
The present paper is a part of series of papers devoted to fermionic approaches to
multi-fold integrals, see [12], [1], [2]. Let us mark that our fermionic constructions of
papers [1], [2] and of the present paper are different from what was considered in [11] and
also different of [12].
1.1 Free fermions
LetA be 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 . (1.13)
Any element of the linear part
W := (⊕m∈ZCfm)⊕
⊕m∈ZCf̄m
(1.14)
will be referred to as a free fermion. We also introduce the fermionic free fields
f(x) :=
k, f̄(y) :=
−k−1, (1.15)
which may be viewed as generating functions for the fj , f̄j’s.
This Clifford algebra has a standard Fock space representation defined as follows.
Define the complementary, totally null (with respect to the underlying quadratic form)
and mutually dual subspaces
Wan := (⊕m<0Cfm)⊕
⊕m≥0Cf̄m
, Wcr := (⊕m≥0Cfm)⊕
⊕m<0Cf̄m
, (1.16)
and consider the left and right A-modules
F := A/AWan, F̄ := WcrA\A. (1.17)
These are cyclic A-modules generated by the vectors
|0〉 = 1 mod AWan, 〈0| = 1 mod WcrA, (1.18)
respectively, with the properties
fm|0〉 = 0 (m < 0), f̄m|0〉 = 0 (m ≥ 0),
〈0|fm = 0 (m ≥ 0), 〈0|f̄m = 0 (m < 0). (1.19)
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 is determined by
〈0|1|0〉 = 1; 〈0|fmf̄m|0〉 = 1, m < 0; 〈0|f̄mfm|0〉 = 1, m ≥ 0, (1.20)
〈0|fn|0〉 = 〈0|f̄n|0〉 = 〈0|fmfn|0〉 = 〈0|f̄mf̄n|0〉 = 0; 〈0|fmf̄n|0〉 = 0, m 6= n,
(1.21)
together with the Wick theorem which implies, for any finite set of elements {wk ∈ W},
〈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〉. (1.22)
Here σ runs over permutations for which σ(1) < σ(2), . . . , σ(2n− 1) < σ(2n) and σ(1) <
σ(3) < · · · < σ(2n− 1).
Now let {wi}i=1,...,N , be linear combinations of the fj’s only, j ∈ Z, and {w̄i}i=1,...,N
linear combinations of the f̄j ’s, j ∈ Z. Then(1.22) implies
〈0|w1 · · ·wN w̄N · · · w̄1|0〉 = det (〈0|wiw̄j|0〉) |i,j=1,...,N (1.23)
Following refs. [9], [10], for all N ∈ Z, we also introduce the states
〈N | := 〈0|CN (1.24)
where
CN := f̄0 · · · f̄
N−1 if N > 0 (1.25)
CN := f−1 · · · fN if N < 0 (1.26)
CN := 1 if N = 0 (1.27)
|N〉 := C̄N |0〉 (1.28)
where
C̄N := fN−1 · · · f0 if N > 0 (1.29)
C̄N := f̄N · · · f̄−1 if N < 0 (1.30)
C̄N := 1 if N = 0 (1.31)
The states (1.24) and (1.28) are referred to as the left and right charged vacuum vectors,
respectively, with charge N .
In what follows we use the notational convention
∆N(x) = det (x
i )|i,k=1,...,N (N > 0), ∆0(x) = 1, ∆N(x) = 0 (N < 0). (1.32)
From the relations
〈0|f̄N−kf(xi)|0〉 = x
i , 〈0|f−N+k−1f̄(yi)|0〉 = y
i , k = 1, 2, . . . , (1.33)
and (1.23), it follows that
〈N |f(x1) · · · f(xn)|0〉 = δn,N∆N(x), N ∈ Z, (1.34)
〈−N |f̄(y1) · · · f̄(yn)|0〉 = δn,N∆N(y), N ∈ Z. (1.35)
Following [9], [10] we consider ĜL∞ element
g = eh, h =
hi,jfif̄j , hi,j ∈ C (1.36)
Via the conjugation, (·) → g(·)g−1 , each g ∈ ĜL∞ acts on the spaces (⊕m∈ZCfm) and
⊕m∈ZCf̄m
as linear transformations [9], [10].
We suppose that the following factorization condition is valid:
g = g+g−, 〈0|g
+ = 〈0|, g−|0〉 = |0〉, (1.37)
where g+, g− ∈ ĜL∞.
Remark. Though, the property (1.37) is valid for a rather wide class of (1.36) (which
includes all cases when the sum in (1.36) is finite) , however, we do not know the general
theorem providing sufficient and necessary conditions to have this property in case the
sum in (1.36) is infinite.
Consider
〈0|vN · · · v1gv̄1 · · · v̄N |0〉,
where each vi ∈ (⊕m∈ZCfm) and each v̄i ∈
⊕m∈ZCf̄m
, i = 1, . . . , N . Denoting wi =
−1vig+ ∈ (⊕m∈ZCfm) and w̄i = (g−)v̄i(g−)
⊕m∈ZCf̄m
we have
〈0|vN · · · v1gv̄1 · · · v̄N |0〉 = 〈0|wN · · ·w1w̄1 · · · w̄N |0〉 = det〈0|wiw̄j|0〉|i,j=1,...,N
where the second equality is due to the Wick theorem (1.23). Thus
〈0|vN · · · v1gv̄1 · · · v̄N |0〉 = det〈0|vigv̄j · · · |0〉|i,j=1,...,N (1.38)
1.2 Multi-component fermions
One obtains the so-called p-component fermion formalism by re-numerating the above
free fermions (1.13) as follows
f (α)n := fpn+α−1 , f̄
n := f̄pn+α−1 , (1.39)
f (α)(z) :=
k , f̄
(α)(z) :=
z−k−1f̄
k , (1.40)
where α = 1, . . . , p. From (1.13) we obviously have
[f (α)n , f
m ]+ = [f̄
n , f̄
m ]+ = 0, [f
n , f̄
m ]+ = δα,βδn,m. (1.41)
Right and left vacuum vectors are respectively defined
|0, . . . , 0
︸ ︷︷ ︸
〉 := |0〉, 〈0, . . . , 0
︸ ︷︷ ︸
| := 〈0| (1.42)
where |0〉 and 〈0| were introduced in (1.19).
As it follows from (1.19)
f (α)m |0, . . . , 0〉 = 0 (m < 0), f̄
m |0, . . . , 0〉 = 0 (m ≥ 0), (1.43)
〈0, . . . , 0|f (α)m = 0 (m ≥ 0), 〈0, . . . , 0|f̄
m = 0 (m < 0). (1.44)
We also introduce the states
〈n(1), . . . , n(p)| := 〈0, 0|Cn(1) · · ·Cn(p) (1.45)
where
Cn(α) := f̄
0 · · · f̄
n(α)−1
if n(α) > 0 (1.46)
Cn(α) := f
−1 · · · f
if n(α) < 0 (1.47)
Cn(α) := 1 if n
(α) = 0 (1.48)
|n(1), . . . , n(p)〉 := C̄n(p) · · · C̄n(1)|0, 0〉 (1.49)
where
C̄n(α) := f
n(α)−1
· · · f
0 if n
(α) > 0 (1.50)
C̄n(α) := f̄
· · · f̄
−1 if n
(α) < 0 (1.51)
C̄n(α) := 1 if n
(α) = 0 (1.52)
Let us call (1.45) and (1.49) respectively left and right charged vacuum vectors with the
charge (n(1), . . . , n(p)).
We easily verify that
f (α)m |∗, n
(α), ∗〉 = 0 (m < n(α)), f̄ (1)m |∗, n
(α), ∗〉 = 0 (m ≥ n(α)), (1.53)
〈∗, n(α), ∗|f (α)m = 0 (m ≥ n
(α)), 〈∗, n(α), ∗|f̄ (α)m = 0 (m < n
(α)), (1.54)
where ∗ serve for irrelevant components in vacuum vectors.
Remark 1.1. For calculations we use the Wick theorem in form (1.23). There are two
ways to do it:
(1) The first one is to use (1.23) just remembering that p-component fermions are
composed of usual ones, see (1.39).
(2) The second way is to use formula (1.23) separately for each component. Namely,
to calculate the vacuum expectation value of an operator O, first, we present it in form
i · · ·O
i (1.55)
〈0|O|0〉 =
i · · ·O
i |0〉 =
i |0〉 · · · 〈0|O
i |0〉 (1.56)
where the Wick theorem in form (1.23) is applied to each of 〈0, . . . , 0|O
i |0, . . . , 0〉.
2 Fermionic representation for ZN
Consider the element of the Clifford algebra of the following form
g = eA1g2e
A2g3 · · · e
Ap−2gp−1e
Ap−1 , (2.1)
where
f (α)(x)f̄ (α+1)(y)dµα(x, y), α = 1, . . . , p− 1, (2.2)
with measure dµα(x, y), which we do not specify.
In (2.1)
gα = e
hα , hα =
i,j f
j , h
i,j ∈ C, (2.3)
so that we have
i gα = gαf
i , α 6= β, i ∈ Z (2.4)
We also suppose that each gα = e
h(α), α = 2, . . . , p − 1, may be factorized into ĜL
elements g+α and g
α as follows (see (1.37))
gα = g
α , 〈∗,
0̂, ∗|g+α = 〈∗,
0̂, ∗|, g−α |∗,
0̂, ∗〉 = |∗,
0̂, ∗〉 (2.5)
where by ∗ we denote irrelevant components of a vacuum vector (different from the com-
ponent α marked by hats).
Now, let us notice that by (1.38) we have
〈0|f̄ (α)(y1) · · · f̄
(α)(yN)gαf
(α)(xN ) · · ·f
(α)(x1)|0〉 = det
〈0|f̄ (α)(yi)gαf
(α)(xj)|0〉
i,j=1,...,N
(2.6)
Now let us prove, that for special choice of functions ρα, α = 2, . . . , p− 1, namely, for
〈0|f̄ (α)(y)gαf
(α)(x)|0〉 = ρα(y, x) (2.7)
we have
(N !)p−1〈N, 0, . . . , 0,−N |g|0, 0, . . . , 0, 0〉 = ZN (2.8)
Indeed, to get a non-vanishing expectation value in the left hand side, we have to pick
up only N -th term,
, in the Taylor series for eA1 (this is because eA1 is the only factor of
g which contains the first component fermions, and the matrix element 〈N, ∗|An1 |0, ∗〉 ≡ 0
until n = N). Using the known formula (1.34), we obtain, that the left hand side of (2.8)
is equal to the integral
dµ1(x
1 , y
1 ) . . .
dµ1(x
N , y
N )∆N (x
(1))R1,
R1 = 〈∗, 0, . . . , 0,−N |f̄
(2)(y
1 ) · · · f̄
(2)(y
N )g2 · · · |∗, 0, . . . , 0〉
where we put ∗ on the first place of the left and right vacuum vectors to show that we
forget about the first component fermions. This is the first step.
Then, we have to pick up only N -th term,
, when developing the next factor eA2.
Otherwise, the vacuum expectation values of the second component fermions vanishes.
This is because the second component fermions are in presence only in eA1 , g2 and in e
factors of g, and the g2 is a sum of monomials, each of which contains equal number of
f (2) and f̄ (2) fermions, while eA1 contains only f (2) , and eA2 contains only f̄ (2) fermions.
Thus, second component fermions yields the expression
f̄ (2)(y
1 ) · · · f̄
(2)(y
N )g2f
(2)(x
1 ) · · · f
(2)(x
which should be integrated with measures
i=1 dµ(∗, y
i )dµ(x
i , ∗), and then substituted
inside 〈N, 0, . . . , 0,−N | and |0, 0, . . . , 0, 0〉. Denoting
〈0|f̄ (2)(y
1 ) · · · f̄
(2)(y
N )g2f
(2)(x
1 ) · · · f
(2)(x
N )|0〉 = ̺2(y
(2), x(2)) (2.9)
(which, by (2.6), is equal to detρ2(y
i , x
j )) we obtain that the l.h.s of (2.8) is equal to
the integral ∫
dµ1(x
1 , y
1 ) . . .
dµ1(x
N , y
N )∆N (x
dµ2(x
1 , y
1 ) . . .
dµ2(x
N , y
N )̺3(y
(3), x(3))R2,
R2 = 〈∗, ∗, 0, . . . , 0,−N |f̄
(2)(y
1 ) · · · f̄
(2)(y
N )g2 · · · |∗, ∗, 0, . . . , 0〉
where we put ∗ on the first and second places of the left and right vacuum vectors to show
that we forget about the first and the second component fermions. This is the second
step.
Then, it is easy to see that each exponential eAα should be replaced by their N -th
Taylor term, otherwise the l.h.s. of (2.8) vanishes, it means we have
〈N, 0, . . . , 0,−N |g|0, 0, . . . , 0, 0〉 =
(N !)p−1
〈N, 0, . . . , 0,−N |AN1 g2A
2 · · · gp−1A
p−1|0, 0, . . . , 0, 0〉 (2.10)
Continuing excluding step by step third- forth- and so on component fermions, and,
on the last step, using the known formula (1.35), we obtain that (2.10) is equal to (1.3).
At last we want to make the following remark
Remark 2.1. Insert additional factors to (2.1) as follows
g = eA1g2e
A2g3 · · · e
Ap−2gp−1e
Ap−1 → g	 := e
A1g2e
A2g3 · · · e
Ap−2gp−1e
Ap−1gpg1e
Ap (2.11)
where
gα = e
i,j h
j , Aα =
f (α)(x)f̄ (α+1)(y)dµα(x, y), α = 1, . . . , p (2.12)
where f̄ (p+1)(y) ≡ f̄ (1)(y). (Thus we add g1, gp and dµp(x, y) to our collection of data, gα, α =
2, . . . , p− 1 and dµα(x, y), α = 1, . . . , p− 1). Then
〈0, 0, . . . , 0, 0|g	|0, 0, . . . , 0, 0〉 =
(2.13)
where cN are certain numbers and each Z
is the following integral over 2pN variables x(α) =
1 , . . . , x
) and y(α) = (y
1 , . . . , y
), α = 1, . . . , p:
(α), x(α))
dµα(x
(α), y(α+1)), y(p+1) ≡ y(1) (2.14)
(notice that variables y(1) and x(p) are not fixed by (1.1))
In (2.14) dµα(x
(α), y(α+1)) (α = 1, . . . , p − 1) are defined by (1.2) and
dµp(x
(p), y(1)) :=
dµp(x
i , y
i ), (2.15)
and each ̺α(y
(α), x(α)) is defined by (2.6)-(2.7), where now α = 1, . . . , p.
Sums (2.13) and their relation to the grand partition function of closed chains of coupled
random matrices and to integrable equations will be considered in a forthcoming paper.
3 Deformation of measure and relations to integrable
hierarchies
The described deformation
dµ1(x, y) → dµ1(x, y|t
(1), n, t̄(1)) := xn1eV (x,t
(α))+V (x−1,t̄(1)))dµ1(x, y), (3.1)
dµp−1(x, y) → dµp−1(x, y|t
(p), n, t̄(p)) := ynpeV (y,t
(p))+V (y−1,t̄(p)))dµp−1(x, y) (3.2)
V (x, t(α)) =
xmt(α)m , V (x
−1, t̄(α)) =
x−mt̄(α)m , α = 1, p, (3.3)
Then, it is quite known fact that in this case ZN = τN (t
(1), t̄(p)), where τN (t
(1), t̄(p)) is
a tau function of the (one-component) TL hierarchy. Indeed, one just re-writes (1.3) as
2N -fold integral with a modified measure dµmod (the latter depends on the choice of ρα) :
dµmod(x
i , y
,t̄(1))+V (y
,t̄(p))eV (x
,t(1))+V (y
,t(p))∆N(x
(1))∆N(y
Moreover, as a function of t(1), t(p), t̄(1), t̄(p), the integral ZN(t
(1), t(p), n1, np, t̄
(1), t̄(p)) is
a tau function of the coupled two-component KP, or, the same, a tau function of the
two-component TL hierarchy, see [1].
Now, in addition, we consider the following deformations of functions ρα, α = 2, . . . , p−
ρα(x, y) → ρα(x, y|t
(α), n(α), t̄(α)) := (3.4)
〈n(α)|eH
(α)(t(α))f̄ (α)(y
1 ) · · · f̄
(α)(y
N )gαf
(α)(x
1 ) · · ·f
(α)(x
H̄(α)(t̄(α))|n(α)〉 (3.5)
where t(α) = (t
1 , t
1 , . . . ) and t̄
(α) = (t̄
1 , t̄
2 , . . . ) are the deformation parameters, and
where the “Hamiltonians” H
, k = ±1,±2, . . . , are defined by
H(α)(t(α)) =
, H̄(α)(t̄(α)) =
f (α)n f̄
n+k (3.6)
(for future purpose we define them for α = 1, . . . , p range).
Let us note that the expectation value (3.5), by definition [9], [10], [15], is a tau
function of one component TL and in our case may be denoted by
τn(α)(t
(α) + [x(α)], t̄(α) + [y(α)])
where
[x] :=
, . . .
Now let us prove that the combination of deformations (3.1)-(3.2) and (3.4) is equiv-
alent to the replacement
〈N, 0, . . . , 0,−N |g|0, 0, . . . , 0, 0〉 →
τN(t,n, t̄) := 〈N + n
(1), n(2), . . . , n(p−1),−N − n(p)|eH(t)geH̄(t̄)|n(1), n(2), . . . , n(p−1),−n(p)〉,
(3.7)
where
H(t) =
, H̄(t̄) =
the “Hamiltonians” H
k were defined earlier by (3.6).
Proof. Indeed, we have that each terms of type (2.9), namely, each
〈0|f̄ (α)(y
1 ) · · · f̄
(α)(y
N )gαf
(α)(x
1 ) · · ·f
(α)(x
N )|0〉 = ̺(y
(α), x(α)), α = 2, . . . , p− 1,
(3.8)
is now replaced by
〈n(α)|eH
(α)(t(α))f̄ (α)(y
1 ) · · · f̄
(α)(y
N )gαf
(α)(x
1 ) · · ·f
(α)(x
H̄(α)(t̄(α))|n(α)〉 (3.9)
which is, by definition [9], [10], [15], a tau function of one component TL. Due to (1.38)
it is equal to
detρα(xi, yj|t,n, t̄)
where
ρα(xi, yj|t,n, t̄) := 〈n
(α)|eH
(α)(t(α))f̄ (α)(y
i )gαf
(α)(x
H̄(α)(t̄(α))|n(α)〉
As for α = 1, p we have
〈N, ∗|eH
(1)(t(1))f(x1) · · ·f(xN)e
H̄(1)(t̄(1))|0, ∗〉 = a1∆N (x)e
i=1 V (xi,t
(1))+V (x
,t̄(1)), (3.10)
〈∗, 0|eH̄
(p)(t̄(p))f̄(yN) · · · f̄(y1)e
H̄(p)(t̄(p))|∗,−N〉 = ap∆N (y)e
i=1 V (yi,t
(p))+V (y
,t̄(p)), (3.11)
where aα = e
k=1 kt
k , which contribute to the deformation respectively of dµ1 and of
dµp−1. The end of proof.
Thus, we obtain that the deformation of functions ρα, α = 2, . . . , p − 1, and also of
dµ1, dµp−1 reduce to the fact that ZN is equal to τN(t,n, t̄). It is known [9], [10], [14] that
thus constructed τN (t,n, t̄) is a tau function of the coupled p-component KP hierarchy,
or, the same, p-component TL hierarchy.
4 Conclusion
We equate the multi-integral (1.3) to the fermionic expectation value (2.8). On the one
hand we hope that the fermionic representation allows to evaluate different magnitudes re-
lated to the matrix models, like spectral determinants (compare with [2]), or perturbative
series generalizing [12], [13]. On the other hand it allows to incorporate the study of these
integrals and related multi-matrix models to the study of multi-component integrable
hierarchies
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, which helped stimulate ideas leading to this work.
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	Introduction
	Free fermions 
	Multi-component fermions 
	Fermionic representation for ZN
	Deformation of measure and relations to integrable hierarchies
	Conclusion
