%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Title: On the Bilinear Equations for Fredholm Determinants
%        Appearing in Random Matrices
% Type:    CRM/MSRI preprint       
%=============================================================
% Author:    J. Harnad 
% Compiler:  PlainTeX 
% Date:     April  1998
% ===========================================================
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\hyphenation{Harnad Neumann}
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\def\RightHeadText{Equations for Fredholm Determinants Appearing in
Random Matrices}
\def\LeftHeadText{J. Harnad}
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%\leftline{ \hfill MSRI 1999-031 (1999) \break} \bigskip 
\bigskip \bigskip
\centerline{\titlefont On the Bilinear Equations for Fredholm Determinants}
\centerline{\titlefont Appearing in Random Matrices \footnote{\hskip
-5pt$^\dagger$}{\eightpoint \baselineskip = 9.5pt minus 2pt
\hskip -7pt Research at MSRI is supported in part by NSF grant
DMS-9701755. Research supported in part by the Natural Sciences and 
Engineering Research Council of Canada and by the Fonds FCAR du 
Qu\'ebec.}}
%\footnote{${}^{\dagger}$}{\eightpoint
%Research supported in part by the Natural Sciences and
%Engineering Research Council of Canada and the Fonds FCAR du Qu\'ebec.}}
\bigskip
\centerline{\authorfont J.~Harnad}
\authoraddr
{Department of Mathematics and Statistics, Concordia University\\
7141 Sherbrooke W., Montr\'eal, Qu\'e., Canada H4B 1R6, {\rm \eightpoint
and} \\ 
Centre de recherches math\'ematiques, Universit\'e de Montr\'eal\\
C.~P.~6128, succ. centre ville, Montr\'eal, Qu\'e., Canada H3C 3J7\\
{\rm \eightpoint e-mail}: harnad\@crm.umontreal.ca} 
\bigskip

\abstract{It is shown how the bilinear differential equations satisfied by Fredholm
determinants of integral operators appearing as spectral distribution functions
for random matrices may be deduced from the associated systems of nonautonomous
Hamiltonian equations satisfied by auxiliary canonical phase space variables
introduced by Tracy and Widom. The essential step is to recast the latter as
isomonodromic deformation equations for families of rational covariant derivative
operators on the Riemann sphere and interpret the Fredholm determinants as
isomonodromic $\tau$-functions.} 
\bigskip \baselineskip 14 pt

%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section 1. Introduction %%%%%%%%%%%%%%%%%%%%%%%
\section 1.  Differential equations for Fredholm determinants in random matrices

   In the theory of random matrices, it is known that in suitably defined double
scaling limits the generating functions for spectral distributions are given by
Fredholm determinants of certain integral operators \cite{M,TW1--3}. For example, in 
the universality class of the Gaussian Unitary Ensemble (GUE), in the bulk of the
spectrum, the probability of having exactly $\{m_1, \cdots m_n\}$ scaled
eigenvalues in the sequence of disjoint intervals
$\{([a_1,a_2], \cdots [a_{2n-1},a_{2n}]\}$ is
$$
E(m_1,\ldots,m_n) = {(-1)^{\bar m}
\over m_1!\cdots m_n!}{\partial^{\bar m} \tau^S\quad\over
\partial\lambda_1^{m_1}\cdots\partial\lambda_{n}^{m_n}}
\Bigr\vert_{\lambda_1=\cdots=\lambda_m=1}, \qquad \bar m=\sum_{j} m_j, \eq..
$$
where $\tau^S$ is the Fredholm determinant
$$
\tau^S:=\det(1-\hat{K}^S)  \eq sineFredholmdet..
$$ 
of the integral operator $\hat{K_s}:L^2(\bfR,\bfC)\ra L^2(\bfR,\bfC)$ with the
sine kernel
$$
(\hat{K}^S v)(x) =\sum_{j=1}^n\l_j\int_{a_{2j-1}}^{a_{2j}}
{\sin(\pi(x-y))\over\pi(x-y)} v(y)dy. 
\eq..
$$
Rescaling  at the edge of the spectrum, the corresponding  quantity is given by
the Fredholm determinant 
$$
\tau^A:=\det(1-\hat{K}^A)  \eq AiryFredholmdet..
$$ 
of the operator with the Airy kernel \cite{TW2}
$$
(\hat{K}^A v)(x)
=\sum_{j=1}^n\l_j\int_{a_{2j-1}}^{a_{2j}}{Ai(x))Ai'(y)-Ai(y)Ai'(x)\over x-y}
v(y)dy,  \eq..
$$
where $Ai(x)$ is the Airy function. If the measure is taken to be the
one associated  with either the Laguerre or Bessel orthogonal polynomials,
rescaling at the edge leads to the Fredholm determinant
$$
\tau_\a^B:=\det(1-\hat{K}_\a^B)  \eq BesselFredholmdet..
$$
of the operator with Bessel kernel \cite{TW3}
$$
(\hat{K}_\a^B v)(x) =\sum_{j=1}^n\l_j\int_{a_{2j-1}}^{a_{2j}}
{J_\a(\sqrt{x})\sqrt{y}J_\a'(\sqrt{y})-J_\a(\sqrt{y})\sqrt{x}J_\a'(\sqrt{x})\over
2(x-y)} v(y)dy,  \eq..
$$
where $J_\a(x)$ is the Bessel function with index $\a$.

  It was shown by Tracy and Widom \cite{TW1--3}, extending earlier results of
the Kyoto school \cite{JMMS}, that all these Fredholm determinants can be computed by
quadratures in terms of solutions of certain associated nonautonomous Hamiltonian
systems in which the end points $\{a_j\}$ play the r\^ole of  multi-time
deformation variables. Moreover, these Fredholm  determinants may be interpreted as
isomonodromic $\tau$-functions \cite{HTW, HI} in the sense of \cite{JMU, JM}.

  More recently, Adler, Shiota and van Moerbeke \cite{ASV1, ASV2} have shown that
the Fredholm determinants $\tau^A$, $\tau^B_\a$ satisfy hierarchies of bilinear
differential equations with respect to the endpoint parameters. These follow
from combining Virasoro constraints satisfied by certain associated KP $\tau$-functions
with the bilinear equations they also satisfy with repect to the KP flow parameters
$\{t_1, t_2, \dots\}$, evaluated at the zero values of these parameters. The approach
of \cite{ASV1, ASV2} was based on the application of vertex operators, integrated over
the intervals $\{[a_{2j-1},a_{2j}]\}$,  to suitable ``vacuum'' KP $\tau$-functions,
effecting thereby a continuous version of Darboux transformations, yielding new KP
$\tau$-functions, such that the  Fredholm determinant equals the ratio of the two.

For the Airy kernel, the first equation in this hierarchy may be expressed as
$$
\DD_0^4F^A - 4\DD_1\DD_0 F^A + 2\DD_0F^A + 6(\DD_0^2F^A)^2 = 0, \eq ASVAiry..
$$
where
$$
F^A:= \ln\tau^A \eq..
$$
and 
$$
\DD_m:=\sum_{j=1}^{2n}a_j^m{\di \over \di a_j}, \qquad m\in \bfN,  \eq..
$$ 
while for the Bessel kernel, it is
$$
\DD_1^4F_\a^B -2\DD_1^4 F_\a^B + (1-\a^2)\DD_1^2F_\a^B + \DD_2\DD_1F_\a^B
-{1\over 2}\DD_2F_\a^B - 4(\DD_1F_\a^B)(\DD_1^2F_\a^B) +
6(\DD_1^2F_\a^B)^2 = 0, \eq ASVBessel..
$$
with
$$
F_\a^B:= \ln\tau_\a^B. \eq..
$$
No analogous equations were derived for the sine kernel, although in
the special case where the intervals $[a_{2j-1},a_{2j}]$ are chosen symmetrically about
the origin, the Fredholm determinant $\tau^S$ may be expressed \cite{M, TW3} as a
product $\tau_{1\over2}^B\tau_{-{1\over2}}^B$ of two Bessel kernel determinants. 


In the case of a single interval, it is easy to see that equations \ASVAiry and
\ASVBessel just give the $\tau$-function form of the Painlev\'e equations $P_{II}$
and $P_V$, respectively, to which the Tracy--Widom systems reduce in the case of
the Airy and Bessel kernels. It seems reasonable to expect that analogous results 
hold for the general case, involving an arbitrary number of intervals. The purpose of
this work is to show how the hierarchies of equations derived in \cite{ASV1, ASV2} can
in fact be deduced directly from the Tracy-Widom Hamiltonian systems for both the Airy
and Bessel cases, and to also apply this approach to the sine kernel case. The main
step is to recognize that the Hamiltonian systems imply isomonodromic deformation
equations for associated families of rational covariant derivative operators on the
Riemann sphere. It is known \cite{JMU, JM} that such isomonodromic deformations give
rise to bilinear equations for indexed sets of isomonodromic $\tau$-functions related by
Schlesinger transformations. The fact that for the systems associated with
the Airy and Bessel kernels such equations may be written in terms of a single scalar
$\tau$-function is due to the presence of a pair of conserved quantities,
allowing the elimination of the additional variables  by fixing the level
sets of these invariants. In the sine kernel case this is not possible, and the
associated bilinear equations therefore involve coupled systems for $\tau^S$ together
with a pair of additional variables $(\tau_+^S,\tau_-^S)$. 

  In section 2, equations  \ASVAiry and \ASVBessel are first derived directly from the
Hamiltonian systems of \cite{TW2-3}.  In section 3, it is shown how the isomonodromic
deformation equations following from the associated Hamiltonian systems may be used to
derive the full hierarchy of $\tau$-function equations for all these cases.
In section 4, these results are related to the rational classical $R$-matrix approach to
isomonodromic and isospectral systems developed in \cite{AHP, H}.

\section 2. Deduction of $\tau$-function equations from the Hamiltonian systems

   To establish notation, following \cite{TW1-3}, we define the quantities:
$$
\eqalignno{x_{2j}&:= 2i\sqrt{\l_j}(\bfI-\hat{K})^{-1}\phi (a_{2j}),
\qquad x_{2j+1}:= 2\sqrt{\l_j}(\bfI-\hat{K})^{-1}\phi (a_{2j+1})  \eqn xjdef.a. \cr 
y_{2j} &:=i\sqrt{\l_j}(\bfI-\hat{K})^{-1}\psi (a_{2j}), 
\qquad 
y_{2j+1}:=\sqrt{\l_j}(\bfI-\hat{K})^{-1}\psi (a_{2j+1}),  \eqn yjdef.b.  \cr
x_0 &:=2\sum_{j=1}^n\l_j\int_{a_{2j-1}}^{a_{2j}}\phi(x)(\bfI-\hat{K})^{-1}\psi (x)dx,
\eqn xzerodef.c. \cr
 y_0 &:=\sum_{j=1}^n \l_j\int_{a_{2j-1}}^{a_{2j}}\phi(x)(\bfI-\hat{K})^{-1}\phi (x)dx,
\eqn yzerodef.d. \cr
 }
$$
where, for the case of the sine kernel $\hat{K}=\hat{K}^S$,
$$
\phi(x):= {\sin(\pi x)\over \pi}, \quad \psi(x): =\cos(\pi x).
\eq xySinedef..
$$
while for the  Airy kernel $\hat{K}=\hat{K}^A$,
$$
\phi(x):= Ai(x), \quad \psi(x): ={d Ai(x)\over dx}, \eq xydef..
$$
and for the Bessel kernel $\hat{K}=\hat{K}_\a^B$,
$$
\phi(x):= J_\a(\sqrt{x}), \quad \psi(x): = x{dJ_\a(\sqrt{x}) \over dx}. \eq..
$$
(An odd number of variables may also occur if we set one of the $a_j$'s equal
to some fixed constant, say $0$ or $\infty$, and eliminate the corresponding
pair $(q_j, p_j)$.) As shown in \cite{TW1-3}, the logarithmic derivatives of the
associated Fredholm determinants are given by:
$$
G^S_j:={\di F^S\over\di a_j} =  {\pi^2\over 4} x_j^2 + y_j^2  -{1\over 4}
\sum_{k=1\atop k\neq j}^n {(x_jy_k-y_jx_k)^2\over a_j-a_k}
\eq GSjdef..
$$
for the sine kernel,
$$
G^A_j:={\di F^A\over\di a_j} = 
y_j^2 +{1\over 4} (x_0 -a_j)x_j^2 -y_0x_jy_j 
-{1\over 4} \sum_{k=1\atop k\neq j}^n {(x_jy_k-y_jx_k)^2\over a_j-a_k}
\eq GAjdef..
$$
for the Airy kernel, and
$$
a_jG^B_{\a,j}:= a_j{\di F^B_\a\over\di a_j} = 
y_j^2 -{1\over 16} (\a^2 -a_j + x_0)x_j^2 +{1\over 4}y_0 x_j y_j 
-{1\over 4} \sum_{k=1\atop k\neq j}^n {a_k(x_jy_k-y_jx_k)^2\over a_j-a_k}
\eq GBjdef..
$$
for the Bessel kernel.

For use in what follows, we also define the quantities
$$
R^S_m:=\DD_m F^S = \sum_{j=1}^{2n}a_j^mG^S_j, \quad m\in \bfN \eq RSmdef..
$$
for the sine kernel case,
$$
R^A_m:=\DD_m F^A = \sum_{j=1}^{2n}a_j^mG^A_j, \quad m\in \bfN \eq RAmdef..
$$
for the Airy case and
$$
R^B_{\a,m}:=\DD_m F^B_{\a,j} = \sum_{j=1}^{2n}a_j^m G^B_{\a,j}, \quad m\in \bfN
\eq RBmdef..
$$
for the Bessel case. For all three cases, we define the following sequence of
bilinear forms
$$
P_m :=\sum_{j=1}^{2n}a_j^my_j^2, \qquad
Q_m :=\sum_{j=1}^{2n}a_j^mx_j^2, \qquad
S_m :=\sum_{j=1}^{2n}a_j^mx_j y_j, \qquad m\in \bfN.  \eq PQS..
$$

  As explained below, the $\{G^A_j\}$'s and $\{G^B_{\a,j}\}$'s may
be viewed as sets of Poisson commuting, nonautonomous Hamiltonians on an
auxiliary phase space with canonical coordinates
$\{x_0,y_0,x_j, y_j\}$, such that the quantities defined in \xjdef-\yzerodef
satisfy the corresponding systems of Hamiltonian equations.
These equations will then be shown to imply equations \ASVAiry and \ASVBessel.

\smallskip
\Subtitle {2a. The Airy kernel system}
\smallskip
\nobreak

  The system of dynamical equations for this case is given \cite{TW2} by 
$$
\eqalignno{  
{\di x_j \over \di a_k} &=-{1\over 2} {(x_j y_k-y_jx_k)x_k\over a_j-a_k},
\qquad  j\neq k, \eqn AiryHameqa.a.\cr
{\di y_j \over \di a_k} &=-{1\over 2} {(x_j y_k-y_jx_k)y_k\over a_j-a_k},
 \qquad j\neq k, \eqn AiryHameqb.b.\cr
{\di x_j \over \di a_j} &={1\over 2}\sum_{k=1\atop k\neq j}^n
{(x_j y_k-y_jx_k)x_k\over a_j-a_k} +2y_j -y_0x_j, \eqn AiryHameqc.c.\cr
{\di y_j \over \di a_j} &={1\over 2}\sum_{k=1\atop k\neq j}^n 
{(x_j y_k-y_jx_k)y_k\over a_j-a_k}+{1\over 2}(a_j-x_0)x_j + y_0x_jy_j,
\eqn AiryHameqd.d. \cr
{\di x_0 \over \di a_j} &= -x_j y_j,  \qquad
{\di y_0 \over \di a_j} =-{1\over 4}x_j^2. \eqn AiryHameqe.e.}
$$
Viewing the $a_j$'s as multi-time parameters, this is a compatible
system of nonautonomous Hamiltonian equations generated by the
Poisson commuting Hamiltonians $\{G_j^A\}$ defined in \GAjdef. There is an additional
functionally independent Hamiltonian, defined by
$$
G^A_0 := y_0^2 -x_0 -{1\over 4}Q_0,  \eq GAzero..
$$
which also Poisson commutes with all the $G_j^A$'s. Since $G_0^A$ is not explicitly
dependent on the parameters $\{a_j\}$, it follows that it is a conserved
quantity. Since all the quantities $\{x_0,y_0,x_j,y_j\}$  defined in
\xjdef-\yzerodef vanish in the limit $\{a_j \ra \infty, \ \forall j\}$, the
invariant
$G^A_0$ must vanish on this particular solution. Therefore we may express $x_0$ in terms
of the other variables as
$$
x_0=y_0^2-{1\over 4}Q_0. \eq xzeroAiry..
$$

  The quantity $R_0^A$ defined in \RAmdef will just be denoted
$$
R:=R^A_0= \sum_{j=1}^{2n}G^A_j = P_0 -{1\over 4}Q_1 +{1\over 4}y_0^2 Q_0
-y_0S_0 -{1\over 16}Q_0^2,  \eq RAeval..
$$
where \xzeroAiry has been used. In terms of $R$, equation \ASVAiry becomes
$$
\DD_0^3 R -4\DD_1 R + 2 R + 6 (\DD_0 R)^2 = 0.  \eq ASVAiryR..
$$
It follows from the Poisson commutativity of the Hamiltonians $\{G_j^A\}_{j=1\dots 2n}$
that their Hamiltonian vector fields applied as derivations to $R$ give zero, and hence
along any integral surface of eqs.~\AiryHameqa-\AiryHameqe, the derivatives of
$R$ with respect to the $a_j$'s are just given by its {\it explicit} dependence on
these parameters. This just comes from the $Q_1$ term in expression \RAeval, 
and therefore we have
$$
{\di R\over \di a_j} = -{1\over 4}x_j^2 \eq DRAaj..
$$
Comparing with \AiryHameqe, this implies that
$$
G^A_\infty := y_0 - R  \eq GAinfy..
$$
is a second conserved quantity.   Since in the limit $\{a_j\ra \infty, \ \forall j\}$,
both $y_0$ and $R$ vanish, $G^A_\infty$ must vanish for all values of the parameters,
and therefore the invariant relation
$$
y_0=R  \eq yzeroAiry..
$$
is satisfied by this solution. Applying the operators $\DD_0$, $\DD_1$ to $R$, it 
follows from \DRAaj that
$$
\eqalignno{
\DD_0 R &= -{1\over 4} Q_0,  \eqn DDzeroR.a. \cr
\DD_1 R &= -{1\over 4} Q_1.  \eqn DDoneR.b.}
$$
\nextnumber
Eqs.~\AiryHameqa-\AiryHameqe also imply that application of $\DD_0$ to
$\{Q_0, S_0, x_0, y_0, Q_1\}$ gives
$$
\eqalignno{
\DD_0 Q_0 &=4S_0 - 2y_0 Q_0, \qquad
\DD_0 S_0 ={1\over 2}Q_1 -{1\over 2}x_0Q_0 +2P_0, \eqn PSzeroAderiv.a.\cr
\DD_0 x_0 &= - S_0, \qquad \qquad \quad \  \ 
\DD_0 y_0 =-{1\over 4}Q_0.  \eqn xyAzeroderiv.b.\cr
\DD_0 Q_1 &=Q_0 +4S_1-2y_0 Q_1.  \eqn Qonederiv.c.}
$$
\nextnumber
 Further application of $\DD_0$ and $\DD_1$, using \DDzeroR,
\PSzeroAderiv, \xyAzeroderiv and \xzeroAiry, therefore gives
$$
\eqalignno{
\DD_0^2 R &= {1\over 2}y_0 Q_0 -S_0,  \eqn RAtwo.a.\cr
\DD_0^3 R &= -{1\over 2} Q_1 +2y_0S_0 -{1\over 2}y_0^2 Q_0 
-{1\over 4}Q_0^2-2P_0.
\eqn RAzerothree.b.}
$$
\nextnumber
Substituting \RAeval, \DDzeroR, \DDoneR \RAzerothree,  into \ASVAiryR and
using \xzeroAiry shows that all terms cancel, verifying the equation.
 
\Subtitle {2b. The Bessel kernel system}
\smallskip
\nobreak

 In this case, the system of dynamical equations is given \cite{TW3} by 
$$
\eqalignno{  
{\di x_j \over \di a_k} &=-{1\over 2} {(x_j y_k-y_jx_k)x_k\over a_j-a_k},
\qquad  j\neq k, \eqn BesselHameqa.a.\cr
{\di y_j \over \di a_k} &=-{1\over 2} {(x_j y_k-y_jx_k)y_k\over a_j-a_k},
 \qquad j\neq k, \eqn BesselHameqb.b.\cr
a_j{\di x_j \over \di a_j} &={1\over 2}\sum_{k=1\atop k\neq j}^n
{a_k(x_j y_k-y_jx_k)x_k\over a_j-a_k} +2y_j +{1\over 4}y_0x_j, 
\eqn BesselHameqc.c.\cr 
a_j{\di y_j \over \di a_j} &={1\over 2}\sum_{k=1\atop k\neq j}^n  
{a_k(x_j y_k-y_jx_k)y_k\over a_j-a_k}+{1\over 8}(\a^2-a_j+x_0)x_j
 -{1\over 4} y_0 y_j, \eqn BesselHameqd.d. \cr
{\di x_0 \over \di a_j} &= -x_j y_j,  \qquad
{\di y_0 \over \di a_j} =-{1\over 4}x_j^2. \eqn BesselHameqe.e.}
$$
This is again a compatible system of nonautonomous Hamiltonian equations
generated by the Poisson commuting Hamiltonians $G_{\a,j}^B$ defined in \GBjdef.

  There again exist two additional conserved quantities for this case. The first
is defined by
$$
G_0^B:=x_0 + {1\over 4}y_0^2 + y_0 + {1\over4}Q_1,  \eq GBzero..
$$
as may be seen directly by differentiating with respect to the $a_j$'s, using
\BesselHameqa-\BesselHameqe.  Since all the quantities appearing in \GBzero vanish 
in the limit $\{a_j \ra 0, \ \forall j\}$, this difference must vanish, and therefore
the invariant relation
$$
x_0=-{1\over 4}y_0^2 - y_0 - {1\over4}Q_1  \eq xzeroBessel..
$$
is satisfied for this solution. The second conserved quantity is
$$
\eqalignno{
G^B_\infty &:= y_0 + 4 \sum_{j=1}^{2n} a_j G^B_{\a,j} = y_0 +4R^B_{\a,1} \cr
& = y_0 -{1\over 4}(\a^2+x_0)Q_0+{1\over 4}Q_1 +y_0 S_0 + 4P_0 + Q_0 P_0  -
S_0^2, \eqn GBinfty..}
$$
Again, due to the Poisson commutativity of the Hamiltonians defined in
\GBjdef, the Hamiltonian vector fields generating the $a_j$ deformations when applied
to the term $R^B_{\a,1}$ give zero,  and therefore only the explicit dependence of this
term upon the parameters need be taken into account when verifying that differentiation
of the sum gives zero. Since all the quantities appearing in \GBinfty vanish in the
limit $\{a_j\ra 0, \ \forall j\}$, the invariant $G^B_\infty$ must also vanish on this
particular solution, and we therefore have the relation
$$
y_0 = -4R^B_{\a,1}=-4\DD_1F^B_{\a}={1\over 4}(\a^2 +x_0)Q_0 -{1\over 4}Q_1
-y_0 S_0-4P_0 -Q_0 P_0 +S_0^2.  \eq yzeroBessel..
$$ 


The quantities $R^B_{\a,1}$, $R^B_{\a,2}$ are given by
$$
\eqalignno{
R^B_{\a,1}&= \DD_1 F^B_{\a}= \sum_{j=1}^{2n}a_j G^B_{\a,j}  \cr
 & = -{1\over16}(\a^2 + x_0)Q_0 +{1\over 16}Q_1
 + {1\over 4}y_0S_0 + P_0  + {1\over 4} Q_0 P_0 - {1\over 4}S_0^2,
\eqn DDoneRB.a.  \cr
R^B_{\a,2}&= \DD_2 F^B_{\a}= \sum_{j=1}^{2n}a_j^2 G^B_{\a,j} \cr
&=  -{1\over 16}(\a^2+ x_0)Q_1 +{1\over 16}Q_2
 + {1\over 4}y_0S_1  + P_1.  \eqn DDtwoRB.b. }
$$
\nextnumber
It again follows from the Poisson commutativity of the Hamiltonians $\{G^B_{\a,j}\}$
that the derivatives of $R^B_{\a,1}$ and  $R^B_{\a,2}$ with respect to the parameters 
are given  by their explicit dependence on these parameters, and hence
$$
\eqalignno{
\DD_1^2F^B_\a &= \DD_1 R^B_{\a,1}= {1\over 16}Q_1,   \eqn DDoneRone.a.\cr
\DD_2\DD_1 F^B_\a &= \DD_2 R^B_{\a,1}= {1\over 16}Q_2.  \eqn DDtwoRone.b.
}
$$
\nextnumber
From \BesselHameqa-\BesselHameqe, application of $\DD_1$ to 
$\{Q_1,S_1,x_0, y_0\}$  gives
$$
\eqalignno{
\DD_1 Q_1 &=Q_1 + 4S_1 +{1\over 2}y_0 Q_1, \qquad
\DD_1 S_1 = S_1 + {1\over 8}\left(\a^2 +x_0\right)Q_1 -{1\over 8}Q_2  + 2P_1, 
\eqn QSBonederiv.a.\cr
\DD_1 x_0 &=  -S_1, \qquad \qquad \qquad \qquad \
\DD_1 y_0 =-{1\over 4}Q_1.  \eqn xyBzeroderiv.b.}
$$
\nextnumber
 Further application of $\DD_1$, using \DDzeroR, \QSBonederiv, \xyBzeroderiv and
\xzeroBessel therefore gives
$$
\eqalignno{
\DD_1^3 F^B_\a &= {1\over 16}\left(1 + {y_0\over 2}\right)Q_1 +{1\over 4} S_1,  
\eqn  DDthree.a. \cr
\DD_1^4 F^B_\a &= {1\over 16}\left(1 + {\a^2\over 2} +{y_0\over 2} 
+ {y_0^2\over 8}\right)Q_1
+{1\over 2}P_1 + \left({1\over 2} +{y_0\over 8}\right) S_1  - {1\over 64}Q_1^2
-{1\over 32} Q_2.  
\eqn  DDfour.b.}
$$
\nextnumber
Substitution of \DDtwoRB, \DDoneRone, \DDtwoRone, \DDthree, \DDfour into
\ASVBessel, and use of  \yzeroBessel to replace the term $-4\DD_1F^B_\a$ by
$y_0$,  and \xzeroBessel to eliminate $x_0$, shows that all the terms cancel,
verifying the equation.

\section 3. Deduction of the $\tau$-function equations from isomonodromic deformations

In this section, we show how the full hierarchies of equations derived in
\cite{ASV1, ASV2} may be deduced from the Hamiltonian systems \AiryHameqa-\AiryHameqe, 
\BesselHameqa-\BesselHameqe and also how the corresponding hierarchy is deduced for
the case of the sine kernel. The key step is to recast these systems as isomonodromic
deformation equations for an associated differential operator in an auxiliary spectral
variable
$z\in \bfP^1$, having rational coefficients with poles at the points $\{z=a_j\}$, and
to interpret the Fredholm determinants
$\tau^S$,
$\tau^A$ and $\tau^B_\a$ as isomonodromic $\tau$-functions.
\smallskip

\Subtitle {3a. The Airy kernel isomonodromic system}
\smallskip
\nobreak
 The Hamiltonian system \AiryHameqa-\AiryHameqe implies that the compatibility
conditions
\nextnumber
$$
\eqalignno{
{\di A_j \over \di a_k} &= {[A_j, A_k] \over a_j -a_k} , \quad j\neq k,  \eqn
AjkAiry.a.\cr {\di A_j \over \di a_j} &= [a_j B +C, A_j] 
- \sum_{k=1\atop k\neq j}^{2n}{[A_j, A_k] \over a_j -a_k}, 
\eqn AjjAiry.b.\cr 
{\di C \over \di a_k} &= [B, A_j] \eqn CkAiry.c.}
$$ 
\nextnumber
are satisfied for the following overdetermined system \cite{HTW}
$$
\eqalignno{
{\di \Psi^A \over \di z} &= X^A(z)\Psi,  \eqn PsiAz.a. \cr
{\di \Psi^A \over \di a_j} &= -{A_j \over z- a_j}\Psi^A, \quad j=1, \dots 2n,  
\eqn PsiAaj.b. \cr
 X^A(z) & := zB + C + \sum_{j=1}^{2n} {A_j \over z-a_j}, \eqn XAdef.c.  \cr}
$$
\nextnumber
where $\Psi^A(z, a_1, \dots a_{2n})$ is a $2\times 2$ matrix, invertible
where defined, and
$$
\eqalignno{
A_j  &:= -{1\over 2} \pmatrix{ x_j y_j & y_j^2 \cr -x_j^2 & x_j y_j}, \eqn Ajdef.a.
\cr
B  &:=  \pmatrix{ 0 & -{1\over 2} \cr 0 & 0}, 
\qquad
C  :=  \pmatrix{ y_0 & {x_0 \over 2} \cr -2 & - y_0}. \eqn BCAirydef.b.}
$$
This implies the invariance of the monodromy of the operator ${\di \over \di z} -
X^A(z)$ under changes in the parameters $\{a_j\}$. In view of eq.~\GAjdef, according
to the constructions of \cite{JMU, JM}, the Fredholm determinant $\tau^A$ is just the
isomonodromic $\tau$-function of the system \AjkAiry-\XAdef.

Now define the sequence of $2\times 2$ matrices
$$
B_m := \sum_{j=1}^{2n} a_j^mA_j = -{1\over 2} \pmatrix{ S_m & P_m \cr -Q_m & -S_m},
\qquad m\in \bfN, \eq Bmdef..
$$
where the quantities $P_m$, $Q_m$, $S_m$ were defined in \PQS.
Expanding $X^A(z)$ for large $z$ gives
$$
X^A(z)= zB + C + \sum_{m=0}^\infty {B_m \over z^{m+1}}.  \eq XAexp.. 
$$
Since
$$
G^A_j = {1\over 2}\res_{z=\infty}  \tr((X^A)^2(z)),  \eq..
$$
we have
$$
{1\over 2}\tr((X^A)^2(z))= z + G^A_0 + \sum_{m=0}^{\infty} {R^A_m\over z^{m+1}}, \eq..
$$
where
$$
R^A_m:= \sum_{j=1}^{2n}a_j^mG^A_j =  \tr(B B_{m+1} + CB_m) + {1\over
2}\tr\sum_{k=0}^{m-1} B_k B_{m-k-1} 
\eq RAmexp..
$$
(with the last term absent if $m=0$) are the quantities defined in \RAmdef. 

  Using the fact that the Hamiltonian vector fields generating the $a_j$ deformations
give zero when applied to the $G^A_j$'s, and hence also the $R^A_m$'s, it follows that
the effect of applying the operators $\DD_k$ to  $R^A_m$ gives just
the explicit derivatives,
$$
\DD_k R^A_m = (m+1)\tr (B B_{m+k}) + m\,\tr (CB_{m+k-1}) +
\sum_{l=1}^{m-1} l \,\tr(B_{l+k-1}B_{m-l-1})  \eq DkRAm..
$$
(with the sum in the last term absent if $m=0$ and the second term absent if $m+k=0$).

  Applying the operator $\DD_m$ to $\Psi^A$, using \PsiAaj and \Bmdef gives the
sequence of equations
$$
\DD_m\Psi^A = -\sum_{k=0}^\infty {B_{m+k}\over z^{k+1}}\Psi^A, \qquad m\in \bfN.  \eq..
$$
The compatibility of these equations with \PsiAz implies the following equations
for the matrices $\{B_m, C\}$.
$$
\eqalignno{
\DD_k B_m  &= m B_{m+k-1} + [C, B_{m+k}] + [B, B_{m + k + 1}]
+ \sum_{l=0}^{m-1}[B_l,\, B_{m+k-l-1}], \eqn  DkBAm.a.  \cr
\DD_k C  &= [B,\ B_k], \qquad  k,m \in \bfN   \eqn DkCAm.b. }
$$
\nextnumber
(where the first term of \DkBAm is absent if $m+k=0$ and the last
term is absent if $m=0$).

   The strategy for deriving the hierarchy of equations for $\tau^A$ is to now choose
a  $k$-value $(k_1)$ in \DkRAm, \DkBAm, \DkCAm  and use these
equations, together with \RAmexp to express all the relevant matrix elements
of the $B_m$'s  for $m\le k$ in terms of the $R_{k}$'s  for  $k < k_1$ and the
corresponding $\DD_{k}$'s applied repeatedly to them. Equations  \DkBAm, \DkCAm for
$k=k_1$ may then be expressed entirely in terms of these quantities, and hence in terms
of repeated applications of the operators $\DD_{k}$ to $F^A=\ln \tau^A$. An
essential step in this procedure is to also eliminate the additional variables $x_0$,
$y_0$ from the equations through use of the invariant conditions \xzeroAiry, \yzeroAiry.

   For example, choosing $k_1 = 1 $, we note that  for $m=0$, eq.~\RAmexp
reduces to \RAeval while \ for $k=0,1$ and $m=0$, \DkRAm reduces to \DDzeroR, \DDoneR 
and for $k=0, m=0$, eqs.~\DkBAm, \DkCAm give \PSzeroAderiv, \xyAzeroderiv. Combining
these with the invariant relations \xzeroAiry, \yzeroAiry allows us to express the
relevant matrix elements of $C$, $B_0$ and $B_1$ as
$$
\eqalignno{
x_0 &= \DD_0 R +R^2, \quad  \phantom{R +R\DD_0^2 R + 2 (\DD_0R)^2} y_0=R, \eqn.a.\cr
Q_0 &=-4\DD_0R, \qquad  \phantom{R +R\DD_0^2 R + 2 (\DD_0R)^2} 
S_0 = -2R\DD_0 R - \DD_0^2 R, \eqn.b.\cr 
P_0 &= {1\over 2}R -{1\over 4}\DD_0^3R -R\DD_0^2 R -{1\over 2}(\DD_0R)^2
-R^2 \DD_0 R  \eqn.c. \cr
Q_1 &=-2R-6(\DD_0R)^2-\DD_0^3 R.  \eqn.d.} 
$$
\nextnumber
Substituting these in eq.~\DkCAm for $k=1$ gives \ASVAiryR. Similarly, eq.~\DkBAm
for $k=1, m=0$ and eq~\RAmexp for $m=1$ produce the following expressions for the
relevant matrix elements of $B_1$ and $B_2$.
$$
\eqalignno{
S_1 =&\DD_0R -\DD_0\DD_1 R + 2(\DD_0R)(\DD_0^2R)- R(\DD_0^3R) -2R^2 +2R\DD_1R \cr
& + R^2\DD_0^2R - 6R(\DD_oR)^2 -2R^3 \DD_0R \eqn AirySone.a. \cr
Q_2 = & -2R^A_1 -\DD_1\DD_0^2 R - 6R\DD_0R +2R\DD_0\DD_1 R
-3(\DD_0 R)(\DD_0^3 R) +{1\over 4} (\DD_0^2 R)^2 \cr
& -2 R^3  - 16(\DD_0R)^3 - R^2(\DD_0^3 R) -4 R (\DD_0 R)(\DD_0^2 R) \cr
& -2 R^3 \DD_0 R - 4R^2 (\DD_0 R)^2 - 2R^2 (\DD_0R)(\DD_0^2R) + 4R^4 (\DD_0 R). 
\eqn AiryQtwo.b. }
$$
Substitution of \AiryQtwo in eq.~\DkBAm (or \DkRAm) for $k=2, m=0$,  thus gives
$$
\eqalign{
&4\DD_2R -2R^A_1 -\DD_1\DD_0^2 R - 6R\DD_0R +2R\DD_0\DD_1 R
-3(\DD_0 R)(\DD_0^3 R) +{1\over 4} (\DD_0^2 R)^2 \cr
& -2 R^3  - 16(\DD_0R)^3 - R^2(\DD_0^3 R) -4 R (\DD_0 R)(\DD_0^2 R) \cr
& -2 R^3 \DD_0 R - 4R^2 (\DD_0 R)^2 - 2R^2 (\DD_0R)(\DD_0^2R) + 4R^4 (\DD_0 R) = 0.}  
\eq..
$$
as the next equation of the hierarchy. The remaining equations may
similarly be expressed in terms of the derivations $\DD_{k}$ acting upon $F^A$.
\smallskip

\Subtitle {3b. The Bessel kernel isomonodromic system}
\nobreak \smallskip 
The Bessel kernel case is so similar to the above that only the pertinent equations
will be given, without repeating any details of the procedure. 
Define for this case, the matrices 
$$
\eqalignno{
X^B(z) & := \wt{B} + {(C_\a -\sum_{j=1}^{2n}A_j)\over z} + \sum_{j=1}^{2n} {A_j
\over z-a_j},
\eqn XBdef.a.
\cr \wt{B}  &:=  \pmatrix{ 0 & {1\over 8} \cr 0 & 0}, \qquad
C_\a := -{1\over 4} \pmatrix{ y_0 & {1\over 2}(x_0 +\a^2) \cr 8 & - y_0}.
\eqn.b.}
$$
\nextnumber
where the $A_j$'s are again defined as in \Ajdef.

The Hamiltonian system \BesselHameqa-\BesselHameqe implies that the compatibility
conditions
$$
\eqalignno{
{\di A_j \over \di a_k} &= {[A_j, A_k] \over a_j -a_k} , \quad j\neq k,  
\eqn AjkBessel.a.\cr a_j{\di A_j \over \di a_j} &= [C_\a, A_j] 
- \sum_{k=1\atop k\neq j}^{2n}{a_k[A_j, A_k] \over a_j -a_k},  
\eqn AjjBessel.b.\cr 
{\di C_\a \over \di a_k} &= [\wt{B}, A_j] \eqn CkBessel.c.}
$$ 
\nextnumber
are satisfied  for the system 
$$
\eqalignno{
{\di \Psi^B \over \di z} &= X^B(z)\Psi^B  \eqn PsiBz.a. \cr
{\di \Psi^B \over \di a_j} &= -{A_j \over z- a_j}\Psi^B, \quad j=1, \dots 2n,  
\eqn PsiBaj.b. }
$$
\nextnumber
where $\Psi^B(z, a_1, \dots a_{2n})$ is again a $2\times 2$ matrix, invertible
where defined. This again implies the invariance of the monodromy of the operator
${\di \over \di z} - X^B(z)$ under changes in the parameters $\{a_j\}$.
In view of eq.~\GBjdef, the Fredholm determinant $\tau^B_\a$ is again an
isomonodromic $\tau$-function for the system \AjkBessel-\PsiBaj. 


Defining the sequence of $2\times 2$ matrices $\{B_m ,\ m\in \bfN\}$  as in
\Bmdef, and expanding $X^B(z)$ for large $z$ gives
$$
X^B(z)= \wt{B} + {C_\a\over z} + \sum_{m=1}^\infty {B_m \over z^{m+1}},  \eq XBexp.. 
$$
and
$$
{1\over 2}\tr((X^B)^2(z))=-{1\over 4} z + {( G^B_0-G^B_\infty+\a^2)\over
4z^2} +
\sum_{m=1}^{\infty} {R^B_{\a,m}\over z^{m+1}},
\eq..
$$
where
$$
\eqalignno{R^B_{\a,1}&={1\over 4}(G^B_\infty-G^B_0 -\a^2)+ {1\over 2}\tr(C_\a^2 
+2\wt{B} B_1), \eqn RBoneexp.a.\cr 
R^B_{\a,m} &=  \tr(\wt{B} B_{m} + C_\a B_{m-1})
 + {1\over 2}\tr\sum_{k=1}^{m-2} B_k B_{m-k-1}, \qquad m\ge 2 
\eqn RBmexp.b.}
$$
are the quantities defined in \RBmdef and $G_0^B$, $G_\infty^B$ are the conserved
quatities defined in \GBzero, \GBinfty, which vanish on the particular solutions
defined by \xjdef-\yzerodef.

 The fact that the Hamiltonian vector fields generating the $a_j$ deformations
give zero when applied to the $G^B_{\a,j}$'s, and  $R^B_{\a,m}$'s again implies that
the effect of applying the operators $\DD_k$ to the $R^B_{\a,m}$'s is to evaluate
only explicit derivatives with respect to the parameters, giving
$$
\eqalign{ \DD_k R^B_{\a,1}& = {1\over 2} \tr(\wt{B} B_k) \cr
\DD_k R^B_{\a,m} &= m\tr (\wt{B} B_{m+k-1}) + (m-1) \tr (C_\a B_{m+k-2}) +
\sum_{l=1}^{m-2} l\ \tr(B_{l+k-1}B_{m-l-1}), \quad m\ge 2} \eq DkRBm..
$$
(with the sum in the last term absent if $m=2$).

  Applying the operator $\DD_m$ to $\Psi^B$, using \Bmdef and \PsiBaj, again gives the
sequence of equations 
$$
\DD_m\Psi^B = -\sum_{k=0}^\infty {B_{m+k}\over z^{k+1}}\Psi^B, \qquad m\in \bfN.  \eq..
$$
whose compatibility  with \PsiBz implies the following equations
for the matrices $\{B_m, C_\a\}$,
$$
\eqalignno{
\DD_k B_m  &= m B_{m+k-1} + [C_\a, B_{m+k-1}] + [\wt{B}, B_{m+k}]  
+ \sum_{l=1}^{m-1}[B_l, \, B_{m+k-l-1}], \eqn  DkBBm.a.  \cr
\DD_k C_\a  &= [\wt{B},\ B_k], \qquad  k,m \in \bfN, \quad m\ge 1.   \eqn DkCBm.b. }
$$
\nextnumber

   The hierarchy of equations for $\tau^B_\a$ is derived in the same way as for
the Airy case.  For example,  eqs.~\RBoneexp and \RBmexp for $k=2$ reduce
to \DDoneRB, \DDtwoRB, while \DkRBm for $k=1,2$, $m=1$ reduces to \DDoneRone,
\DDtwoRone, and eqs.~\DkBBm, \DkCBm for $k=1,2$, $m=1$ give \QSBonederiv, \xyBzeroderiv.
Combining these with the invariant relations \xzeroBessel, \yzeroBessel allows us to
express the relevant matrix elements of $C_\a$, $B_1$ and $B_2$ as
$$
\eqalignno{
x_0 &= -4(\DD_1 R^B_{\a,1} + ( R^B_{\a,1})^2 - 4R^B_{\a,1}), 
\qquad  y_0 = -4R^B_{\a,1},  \eqn.a. \cr
Q_1 &=16\DD_1 R^B_{\a,1}, \eqn.b. \cr
S_1 &= 8R^B_{\a,1}\DD_1 R^B_{\a,1} -4 \DD_1R^B_{\a,1} + 4\DD^2_1 R^B_{\a,1}, \eqn.c.
\cr  P_1 &=  R^B_{\a,2} +\a^2R^B_{\a,1} + 4(R^B_{\a,1})^2 \DD_1R^B_{\a,1} -
4(\DD_1 R^B_{\a,1})^2 \cr
&\quad+ 4 R^B_{\a,1} \DD^2_1 R^B_{\a,1} - \DD_2 R^B_{\a,1}, 
\eqn.d. \cr
 Q_2 &=16\DD_2 R^B_{\a,1} \eqn.e.}
$$
Substituting these in eqs.~\DkBBm, \DkCBm for $k=2$ gives \ASVBessel. Similar
calculations for higher values of $k$ yield the further equations of the
Bessel hierachy.

\smallskip
\goodbreak
\Subtitle {3c. The sine kernel system}
\nobreak
\smallskip

     For this case, the quantities  defined in \xjdef-\yjdef satisfy the system of 
dynamical equations defined in \cite{JMMS, TW1} 
$$
\eqalignno{  
{\di x_j \over \di a_k} &=-{1\over 2} {(x_j y_k-y_jx_k)x_k\over a_j-a_k},
\qquad  j\neq k, \eqn SineHameqa.a.\cr
{\di y_j \over \di a_k} &=-{1\over 2} {(x_j y_k-y_jx_k)y_k\over a_j-a_k},
 \eqn SineHameqb.b.\cr
{\di x_j \over \di a_j} &={1\over 2}\sum_{k=1\atop k\neq j}^n
{(x_j y_k-y_jx_k)x_k\over a_j-a_k} +2y_j,  \eqn SineHameqc.c.\cr
{\di y_j \over \di a_j} &={1\over 2}\sum_{k=1\atop k\neq j}^n 
{(x_j y_k-y_jx_k)y_k\over a_j-a_k}-{\pi^2\over 2}x_j.
\eqn SineHameqd.d.}
$$
This is again a  compatible system of nonautonomous Hamiltonian equations, generated by
the Poisson commuting Hamiltonians $\{G^S_j\}$ defined in \GSjdef.  They imply that the
compatibility conditions
$$
\eqalignno{
{\di A_j \over \di a_k} &= {[A_j, A_k] \over a_j -a_k} , \quad j\neq k 
\eqn Ajksine.a.\cr  
{\di A_j \over \di a_j} &= [B_S, A_j] 
- \sum_{k=1\atop k\neq j}^{2n}{[A_j, A_k] \over a_j -a_k}, \quad j\neq k
\eqn Ajjsine.b.\cr }
$$ 
\nextnumber
are satisfied for the system 
$$
\eqalignno{
{\di \Psi^S \over \di z} &= X^S(z)\Psi^S,  \eqn PsiSz.a. \cr
{\di \Psi^S \over \di a_j} &= -{A_j \over z- a_j}\Psi^S, \quad j=1, \dots 2n,  
\eqn PsiSaj.b. }
$$
\nextnumber
where
$$
\eqalignno{
X^S(z) & := B_S +  \sum_{j=1}^{2n} {A_j\over z-a_j},
\eqn XSdef.a.
\cr B_S  &:=  \pmatrix{ 0 & {\pi^2\over 2} \cr -2 & 0}, \eqn.b.}
$$
with the $A_j$'s again defined as in \Bmdef. As in the previous cases, this implies the
invariance of the monodromy of the operator ${\di \over \di z} - X^S(z)$. In view of
eq.~\GSjdef, the Fredholm determinant $\tau^S$ is an isomonodromic $\tau$-function for
the system
\Ajksine-\PsiSaj. 

Expanding $X^S(z)$ for large $z$ gives
$$
X^S(z)= B_S + \sum_{m=0}^\infty {B_m \over z^{m+1}},  \eq XSexp.. 
$$
with the matrices $\{B_m ,\ m\in \bfN\}$ again defined as in \Bmdef, and
$$
{1\over 2}\tr((X^S)^2(z))=-\pi^2 + \sum_{m=0}^{\infty} {R^S_m\over z^{m+1}},
\eq..
$$
where
$$
R^S_m := \sum_{j=1}^{2n}a_j^m G^S_j =  \tr(B_S B_m)
 + {1\over 2}\tr\sum_{k=0}^{m-1} B_k B_{m-k-1}, \quad m\in \bfN. \eq RSmexp..
$$

 Applying the operators $\DD_k$ to $R^S_m$ again just differentiates
explicitly with respect to the parameters, giving
$$
\DD_k R^S_m = m\,\tr (B_S B_{m+k-1}) + \sum_{l=1}^{m-1} l\ \tr(B_{l+k-1}B_{m-l-1}) \eq
DkRSm..
$$
(with the first term absent if $k + m=0$ and the sum in the last term absent if $m=0$).


  Applying $\DD_m$ to $\Psi^S$, using \XSexp and \PsiSaj,  gives the
sequence of equations 
$$
\DD_m\Psi^S = -\sum_{k=0}^\infty {B_{m+k}\over z^{k+1}}\Psi^S, \qquad m\in \bfN,  \eq..
$$
whose compatibility  with \PsiSz implies the following equations
for the matrices $\{B_m\}$,
$$
\DD_k B_m  = m B_{m+k-1} + [B_S, B_{m+k}] + \sum_{l=0}^{m-1}[B_l, \, B_{m+k-l-1}]. 
\eq DkBSm..
$$

   The hierarchy of equations for $\tau^S$ is derived in the same way as for
the Airy and Bessel cases, except that we no longer have two conserved
quantities like $G^{A,B}_0$, $G^{A,B}_\infty$. To derive a closed system of equations,
we are obliged to include two further dependent variables $\tau^S_{\pm}$,
which we choose as the nonvanishing entries of the matrix $[B_S,B_0]\tau^S$,
$$
\tau_+^S:= (2P_0 -{\pi^2\over 2}Q_0)\tau^S, \qquad   \tau_-^S:=S_0\tau^S.  
\eq tauSpmdef..
$$
The remaining component of $B_0$, which cancels in the commutator $[B_S,B_0]$, is
$$
R^S_0=\tr(B_S B_0) = P_0 +{\pi^2\over 4}Q_0 = 0, \eq RSzero..
$$
where the first equality follows from choosing $m=0$ in  \RSmexp. This provides a single
conserved quantity that vanishes for the particular solution defined by \xjdef-\yjdef.

  To derive the hierarchy of $\tau$-function equations , we first combine
eqs.~\tauSpmdef-\RSzero, which allows us to express the matrix elements of 
$B_0$ as
$$
Q_0 = -{\tau^S_+\over \pi^2\tau^S},   
\qquad P_0 ={\tau^S_+\over 4\tau^S} ,\qquad  S_0 = {\tau^S_-\over\tau^S}.
\eq QPSzero..
$$
Eq.~\DkBSm for $k=0, m=0$ gives
$$
\DD_0 P_0=-\pi^2 S_0, \quad \DD_0Q_0= 4S_0,  \quad\DD_0S_0 =2P_0 -{\pi^2\over 2}Q_0,
\eq DzeroQPSzero..
$$
and substituting \RSzero, \QPSzero in \DzeroQPSzero gives 
$$
\eqalignno{
\DD_0\tau^S &=0, \eqn tauSzero.a. \cr
\DD_0\tau_-^S &= \tau_+^S, \quad \DD_0\tau_+^S=-4\pi^2\tau_-^S. \eqn taupmzero.b.}
$$
These equations are  the lowest ones in the sine kernel hierarchy; note that they are
linear because of the vanishing of the invariant $R^S_0$. To obtain higher,
nonlinear equations, we first note that eq.~\RSmexp for $m=1$ gives
$$
R^S_1 = P_1 +{\pi^2\over 4}Q_1 
+{1\over 4}(S_0^2-Q_0P_0),  \eq RSone..
$$
while  \DkRSm for $k=0,1$,
$m=1$ reduces to
$$
\eqalignno{
\DD_0 R^S_1 &= R_0=0, \eqn DDzeroRSone.a.\cr
\DD_1 R^S_1 &=   P_1 +{\pi^2\over 4}Q_1. \eqn DDoneRSone.b.}
$$
The first of these just gives the equation
$$
\DD_0\DD_1\tau^S=0, \eq..
$$
which already follows from \tauSzero. The second, combined with eq.~\RSone 
and eq.~\DkBSm for $k=1, m=0$ gives the further equation
$$
\tau^S\DD^2_1\tau^S - (\DD_1\tau^S)^2=\tau^S\DD_1\tau^S 
-{1\over 4} (\tau^S_-)^2 - {1\over 16 \pi^2}(\tau^S_+)^2 .  \eq..
$$
Equation \DkBSm for $k=1, m=0$ gives
$$
\DD_1S_0 = 2P_1 -{\pi^2\over 2} Q_1,
\qquad \DD_1P_0 = -\pi^2 S_1, \qquad \DD_1Q_0 = 4S_1.  \eq..
$$
Solving these, together with \DDoneRSone, gives the following expressions for the 
matrix entries of $B_1$:
$$
\eqalignno{
Q_1 &= {2\over \pi^2} {\DD_1\tau^S\over \tau^S} -{1\over 2\pi^2} \left({\tau^S_-\over
\tau^S}\right)^2 -{1\over 8\pi^4}\left({\tau^S_-\over \tau^S}\right)^2 
-{1\over \pi^2}\DD_1\left({\tau^S_-\over \tau^S}\right) ,     \eqn QSone.a.\cr
 P_1 &= {1\over 2} {\DD_1\tau^S\over \tau^S} -{1\over 8} \left({\tau^S_-\over
\tau^S}\right)^2 -{1\over 32\pi^2}\left({\tau^S_-\over \tau^S}\right)^2 
+{1\over 4}\DD_1\left({\tau^S_-\over \tau^S}\right) , \eqn PSone.b.\cr
 S_1 &= -{1\over 4\pi^2}\DD_1\left({\tau^S_+\over \tau^S}\right). \eqn SSone.c.}
$$
Combining eq.~\DkBSm for $(k=1,m=1)$ and for $(k=2, m=0)$ gives
$$
\DD_2Q_0 = \DD_1 Q_1 - Q_1, \qquad
\DD_2P_0 = \DD_1 P_1 - P_1, \qquad
\DD_2S_0 = \DD_1 S_1 - S_1, \qquad  \eq DDtwo..
$$
Substitution  of \QPSzero, \QSone-\SSone into \DDtwo gives the next equations of the
hierarchy. Repeating this procedure for higher $(k, m)$ values  similarly generates
the higher equations.

\section 4. Classical $R$-Matrix approach and relation to isospectral flows

  In  \cite{ASV1, ASV2}, a key step in deriving the hierarchies of equations 
for the Fredholm determinants $\tau^A$ and $\tau^B_\a$ was to begin with
certain bilinear equations satisfied by KP $\tau$-functions with respect to the flow
parameters $\{t_1, t_2, \dots\}$ and to then use Virasoro constraints to replace the
$t_m$-derivations at vanishing $t$-values by the operators $\DD_m$. In this
section, we show how the classical $R$-matrix  approach to the underlying
isomonodromic deformation equations developed in \cite{H} provides a direct link
with commuting isospectral flows in the loop algebra $\Lsl(2)$, without the
requirement that these arise as reduced KP flows. This fits into the
broader framework of commutative isospectral flows in loop algebras with respect to the
rational $R$-matrix Poisson  (or Adler-Kostant-Symes) structure \cite{RS, AV, AHP}
(and allows us to include the sine kernel case, which does not appear as a reduced KP
flow).

   First we recall \cite{H, HTW} that the isomonodromic deformation equations
\AjkAiry-\CkAiry, \AjkBessel-\CkBessel, \Ajksine-\Ajjsine may be viewed as Hamiltonian
equations on the space of sets $\{A_j\}_{j=1\dots 2n}$ of $\grsl(2)$ elements, with
respect to the Lie Poisson bracket, extended in the Airy and Bessel cases by the
canonical variables $(x_0, y_0)$. (The particular form \Ajdef for the $A_j$'s  just
represents a canonical parametrization on the symplectic leaves for which the
Casimir invariants $ \{\tr A_j^2\}$ all vanish.) The formulae \XAdef, \XBdef, \XSdef
define a Poisson embedding of this space into the space $\Lsl(2)^*_R$ of
rational, traceless $2\times 2$ matrices depending rationally on the auxiliary loop
variable $z$, with respect to the Lie Poisson bracket on $\Lsl(2)$ corresponding to the
Lie bracket: 
$$
[X,Y]_R :={1\over 2}[RX,Y] + {1\over 2}[X, RY],  \eq..
$$
where
$$
R :=P_+ -P_-  \eq..
$$
is the {\it classical $R-$matrix}, given by the difference of the projection 
operators 
$$
\eqalign{
P_+:\Lsl(2) &\ra  \Lslp(2)  \qquad P_+:\Lsl(2) \ra  \Lslp(2)  \cr
P_-:X &\ra  X_+  \qquad P_-:X \ra  X_-}  \eq..
$$
to the subalgebras $\Lslp(2)$, $\Lslm(2)$ consisting respectively of the nonnegative
and  negative terms in the Laurent expanson of $X(z)$ for large $z$. The space
$\Lsl(2)^*_R$ is identified as a subspace of $\Lsl(2)$ through the trace-residue
pairing
$$
<X, \, Y> := \res_{z=\infty}\tr(X(z) Y(z)). \eq..
$$

   In this setting, the isomonodromic deformation equations \AjkAiry-\CkAiry,
\AjkBessel-\CkBessel, \Ajksine-\Ajjsine may all be expressed in the form
$$
\eqalignno{
{\di X\over  \di a_j} &= -[(dG_j)_-, \, X] + {\di (dG_j)_-\over \di z},  
\eqn dXaj.a.\cr
 (dG_j)_- &=  - {A_j\over z-a_j},  \eqn dGjdef.b.} 
$$
where $X$ denotes $X^S$, $X^A$ or $X^B$, and $G_j$ denotes $G_j^S$,  $G^A_j$ or
$G^B_{\a,J}$ respectively. Viewing the Hamiltonians $\{G_j\}$ as spectral invariants
defined on the space $\Lsl(2)$, eq.~\dXaj follows from the Adler-Kostant-Symes
theorem, in view of the relations
$$
{\di_0 X \over \di a_j} = -{\di (dG_j)_-\over \di z}, \eq..
$$
where ${\di_0 X \over \di a_j}$ denotes the  derivative  with respect to the
{\it explicit} dependence on the parameters $\{a_j\}$ only.

   Rather than using the spectral invariants $\{G_j\}$ as Hamiltonians, we consider
the Hamiltonian equations generated by the linear combinations $R^S_m$, $R^A_m$ or
$R^B_{\a,m}$ defined in \RSmdef, \RAmdef \RBmdef, which are all of the form
$$
\DD_m X = -[(dR_m)_-,\, X] + {\di (dR_m)_-\over \di z},  \eq DmXnonauton..
$$
with the respective identifications for $X$ and $\{R_m\}$. These are just 
equations \DkBAm, \DkCAm, \DkBBm, \DkCBm or  \DkBSm, depending on the
identification, since
$$
R_m= {1\over 2} \res_{z=\infty}z^m \tr X^2(z),  \eq..
$$ 
and therefore $dR_m$, viewed as an element of $\Lsl(2)$, is just
$$
dR_m = z^m X(z) = \sum_{k=0}^\infty {B_{k}\over z^{k-m+1}}.  \eq dRm..
$$
implying
$$
(dR_m)_-  = \sum_{k=0}^\infty {B_{m+k}\over z^{k+1}}.  \eq dRm..
$$

   If, instead of the nonautonomous systems occurring here because of the
identifications of the $a_j$'s as multi-time parameters, we consider the autonomous
systems generated by the {\it same} set of Hamiltonians $\{R_0, R_1, \dots\}$,
denoting the corresponding flow parameters $\{t_0, t_1, \dots\}$, the resulting
equations have the isospectral form
$$
{\di X\over \di  t_m} =\pm [(dR_m)_{\pm}, \, X],  \eq DmXauton..
$$
where either of the projections $(dR_m)_{\pm}$ may be used, since the differential
$dR_m$, given by \dRm, commutes with $X$. Although these systems are generated by the
same Hamiltonians as the nonautonomous systems \DmXnonauton, they of course do {\it
not} generate isomonodromic deformations of the operator ${\di \over \di
z} - X(z)$, and in fact are not even compatible with the systems \DmXnonauton;
however, they are compatible amongst themselves, generating commuting isospectral
Hamiltonian flows. The close relationship between the autonomous and associated
nonautonomous systems implies a correspondence between the structure of the
resulting hierarchies. 

  To see this, we substitute the expressions  \XAdef, \XBdef and \XSdef for $X(z)$ and
\dRm for $dR_m$ into \DmXauton to obtain the systems
$$
\eqalignno{
{\di B_m\over \di t_m}  &= [C, B_{m+k}] + [B, B_{m + k +1}]
+ \sum_{l=0}^{m-1}[B_l,\, B_{m+k-l-1}], \eqn  DkBAmataon.a.  \cr
{\di C\over \di t_m}  &= [B,\ B_k], \qquad  k,m \in \bfN   \eqn DkCAmauton.b. }
$$
\nextnumber
for $X=X^A$,
$$
\eqalignno{
{\di B_m\over \di t_m}  &= [C, B_{m+k-1}] + [\wt{B}, B_{m+k}]  + 
+ \sum_{l=1}^{m-1}[B_l, \, B_{m+k-l-1}], \eqn  DkBBmauton.a.  \cr
{\di C_\a\over \di t_m}  &= [\wt{B},\ B_k], \qquad  k,m \in \bfN, \quad m\ge 1.   
\eqn DkCBmauton.b. }
$$
\nextnumber
for $X=X^B_\a$ and
$$
{\di B_m\over \di t_m}  =  [B_S, B_{m+k}] + \sum_{l=0}^{m-1}[B_l, \, B_{m+k-l-1}] 
\eq DkBSmauton..
$$
for $X=X^S$. These only differ from the equations  \DkBBm, \DkCAm,
\DkBBm, \DkCBm and \DkBSm by the absence of the term $m B_{m+k-1}$ in the right hand
side of \DkCAmauton, \DkCBmauton, \DkBSmauton and the replacement
$$
\DD_m \ra {\di \over \di t_m}  \eq..
$$
for the derivation on the left hand side. The procedure for deriving hierarchies 
for such systems is well known in the isospectral context  (see, e.g. \cite{FNR} for
details); the recursive procedure used in section 3 above is just the analog of
this approach applied to the isomonodromic systems \DkBBm, \DkCAm, \DkBBm, \DkCBm and 
\DkBSm.

   As a final point, it should be noted that almost nothing in the derivation of the
$\tau$-function equations of sections 2 and 3 depended on the fact that the specific
$\tau$-functions involved were equal to the Fredholm determinants \sineFredholmdet,
\AiryFredholmdet, \BesselFredholmdet. Everything just followed from the general
form of the isomonodromic deformation equations \AjkAiry-\CkAiry, \AjkBessel-\CkBessel
and \Ajksine-\Ajjsine, the only features specific to the identifications of $\tau^A$,
$\tau^B_\a$, $\tau^S$ as  Fredholm determinants being the fact that the matrix
residues $A_j$ were of rank $1$ (as seen from the parametrization  \Ajdef) and the
invariants $G^A_0$, $G^A_\infty$, $G^B_0$,  $G^B_\infty$ vanished. By allowing these
invariants, as well as the constants $\{\det A_j\}$, to take arbitrary values, an
identical procedure leads to equations for the $\tau$-functions of
the general isomonodromic systems, which only differ from the ones derived in sections
2 and 3 by the nonzero constant values of the two additional invariants 
$G^A_0$, $G^A_\infty$ or $G^B_0$ and $G^B_\infty$. For example,  eq.~\ASVAiryR is
replaced in the general case by
$$
\DD_0^3 R -4\DD_1 R + 2 R + +4(g_\infty^2-g_0)\DD_0R-2g_\infty (\DD^2_0R +2R \DD_0R) + 6
(\DD_0 R)^2 = 0,
\eq ASVAiryRgeneral..
$$
where $g_0$, $g_\infty$ are the values taken by the invariants $G_0^A$,
$G_\infty^A$, respectively. The other equations of these hierarchies may similarly
expressed in a way that allows arbitrary values for these constants. 


\goodbreak
 

\bigskip \bigskip 
%%%%%%%%%%%%%%%%%%%%%%%%%% Acknowledgements %%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\noindent\eightpoint{ {\it Acknowledgements.}
The author would like to thank P. van  Moerbeke for helpful discussions relating
to this work.} 

\bigskip \bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% References %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf References}
\medskip  
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%

}
%
\vfill \eject

\end
