%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Title: On the Bilinear Equations for Fredholm Determinants % Appearing in Random Matrices % Type: CRM/MSRI preprint %============================================================= % Author: J. Harnad % Compiler: PlainTeX % Date: April 1998 % =========================================================== %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Useful Definitions %%%%%%%%%%%%%%%%%%%%%%%%% % \def\@{{\char'100}} \def\qed{\vbox{\hrule width7pt\hbox{\vrule height7pt\hskip6.2pt \vrule height7pt} \hrule width7pt}} \long\def\abstract#1{\bigskip{\advance\leftskip by 2true cm \advance\rightskip by 2true cm\eightpoint\centerline{\bf Abstract}\everymath{\scriptstyle}\vskip10pt\vbox{#1}}\bigskip} \long\def\resume#1{{\advance\leftskip by 2true cm \advance\rightskip by 2true cm\eightpoint\centerline{\bf R\'esum\'e}\everymath{\scriptstyle}\vskip10pt \vbox{#1}}} \def\smallmatrix#1{\left(\,\vcenter{\baselineskip9pt\mathsurround=0pt \ialign{\hfil$\scriptstyle{##}$\hfil&&\ \hfil$\scriptstyle{##}$\hfil \crcr#1\crcr}}\,\right)} \def\references{\bigbreak\centerline{\sc References}\medskip\nobreak\bgroup \def\ref##1&{\leavevmode\hangindent 15pt \hbox to 15pt{\hss\bf[##1]\ }\ignorespaces} \parindent=0pt 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\if\relax#1\relax \else \expandafter\xdef\csname #1@\endcsname{{\rm(\loc@lnumber)}} \expandafter \gdef\csname #1\endcsname##1{\csname #1@\endcsname \ifcat##1a\relax\space \else \ifcat\noexpand##1\noexpand\relax\space \else \ifx##1$\space \else \if##1(\space \fi \fi \fi \fi##1}\relax \fi \eq@@{\loc@lnumber}} % \def\eq@@#1{\iflines \else \eqno\fi{\rm(#1)}} \def\m@th{\mathsurround=0pt} % % Command \display % \def\display#1{\null\,\vcenter{\openup1\jot \m@th \ialign{\strut\hfil$\displaystyle{##}$\hfil\crcr#1\crcr}} \,} \newif\ifdt@p % \def\@lign{\tabskip=0pt\everycr={}} % \def\displ@y{\global\dt@ptrue \openup1 \jot \m@th \everycr{\noalign{\ifdt@p \global\dt@pfalse \vskip-\lineskiplimit \vskip\normallineskiplimit \else \penalty\interdisplaylinepenalty \fi}}} % % Command \displayno % \def\displayno#1{\displ@y \tabskip=\centering \halign to\displaywidth{\hfil$ \@lign\displaystyle{##}$\hfil\tabskip=\centering& \hfil{$\@lign##$}\tabskip=0pt\crcr#1\crcr}} % % References % \def\cite#1{{[#1]}} % \def\ucite#1{$^{\scriptstyle[#1]}$} % \def\ccite#1{{ref.~{\bf [#1]}}} % \def\ccites#1{{refs.~{\bf[#1]}}} % \catcode`\@=\active % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Hyphenation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \hyphenation{Moerbeke} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Formatting %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \magnification=\magstep1 \hsize= 6.75 true in \vsize= 8.75 true in % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Headers %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \def\RightHeadText{Equations for Fredholm Determinants Appearing in Random Matrices} \def\LeftHeadText{J. Harnad} % %%%%%%%%%%%%%%%%%%%%%%%%% Document %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %\leftline{ \hfill MSRI 1999-031 (1999) \break} \bigskip \bigskip \bigskip \centerline{\titlefont On the Bilinear Equations for Fredholm Determinants} \centerline{\titlefont Appearing in Random Matrices \footnote{\hskip -5pt$^\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 $$ := \res_{z=\infty}\tr(X(z) Y(z)). \eq.. $$ 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 {\eightpoint % \item{\bf [AHP]} Adams, M.R., Harnad, J. and E. Previato, ``Isospectral Hamiltonian Flows in Finite and Infinite Dimensions I. 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