\input harvmac.tex \def\CP {{\cal P }} \def\CL {{\cal L}} \def\CV {{\cal V}} \def\p {\partial} \def\CS {{\cal S}} \def\hb {\hbar} \def\inbar{\,\vrule height1.5ex width.4pt depth0pt} \def\IB{\relax{\rm I\kern-.18em B}} \def\IC{\relax\hbox{$\inbar\kern-.3em{\rm C}$}} \def\IP{\relax{\rm I\kern-.18em P}} \def\IR{\relax{\rm I\kern-.18em R}} \Title{\vbox{\baselineskip12pt\hbox{YCTP-P17-90}\hbox{}}} {\vbox{\centerline{Matrix Models of 2D Gravity} \centerline{} \centerline{and} \centerline{} \centerline{Isomonodromic Deformation$^1$}}} \centerline{Gregory Moore} \bigskip{\baselineskip14pt \centerline{Department of Physics}\centerline{Yale University} \centerline{New Haven, CT 06511-8167}} \bigskip \bigskip \bigskip \noindent We review the relation between the string equations of the matrix model approach to 2D quantum gravity and the method of isomonodromic deformation. \footnote{}{$^1$ Based on lectures given at the workshop of the Yukawa International Seminar, Kyoto, 10-19 May 1990, The Carg\`ese Workshop on Random Surfaces and 2D Gravity, May 28-June 1 1990, and the Trieste conference on ``Topological Methods in Quantum Field Theories,'' 11-15 June 1990} \draft \newsec{Introduction} One of the principle goals of the theory of 2D gravity is making sense of the formal expression \eqn\formal{ Z(\mu,\kappa;T_i)=\sum_h\int_{MET_h}dg e^{\mu\int\sqrt{g}+\kappa\int R} Z_{QFT(T_i)}[g] } where we integrate over metrics $g$ on surfaces with $h$ handles with a weight defined by the Einstein-Hilbert action ($\mu$ is the cosmological constant and $\kappa$ is Newton's constant, or, equivalently, the string coupling) together with the partition function of some 2D quantum field theory, $QFT(T_i)$. The parameters $T_i$ should be thought of as coordinates on a subspace of the space of 2D field theories, or, equivalently, coordinates for a space of string backgrounds. The expression \formal\ is of course extremely formal, especially when one includes the sum over topologies. Nevertheless, in the physical literature, rigorous definitions of this expression can be proposed using the methods of random matrix theory. Last year these concrete definitions led to a beautiful result for \formal\ for an appropriate set of quantum field theories \nref\BK{E. Br\'ezin and V. Kazakov, ``Exactly solvable field theories of closed strings,'' Phys. Lett. {\bf B236}(1990)144.}% \nref\DS{M. Douglas and S. Shenker, ``Strings in less than one dimension,'' Rutgers preprint RU-89-34.}% \nref\GM{D. Gross and A. Migdal, ``Nonperturbative two dimensional quantum gravity,'' Phys. Rev. Lett. {\bf 64}(1990)127.}% \nref\GMi{D. Gross and A. Migdal, ``A nonperturbative treatment of two-dimensional quantum gravity,'' Princeton preprint PUPT-1159(1989).} \nref\newD{M. Douglas, ``Strings in less than one dimension and the generalized KdV hierarchies,'' Rutgers preprint RU-89-51.} \nref\bdss{T. Banks, M. Douglas, N. Seiberg, and S. Shenker, ``Microscopic and macroscopic loops in non-perturbative two dimensional gravity,'' Rutgers preprint RU-89-50.}\refs{\BK{--}\bdss}. The answer depends only on a particular combination $x=x(\mu,\lambda)$ and is most simply expressed as a differential equation satisfied by \eqn\spfcht{u(x;T_i)={\p^2\over\p x^2} Z .} The differential equation satisfied by $u$ for coupling to the $(p,q)$ minimal conformal field theory \ref\bpz{Belavin, Polyakov, Zamalodchikov, Nucl. Phys.}\ was most elegantly formulated by M. Douglas in \newD\ in terms of a differential operator $L=D^q+u_{q-2}(x)D^{q-2}+\cdots u_0(x)$ where $D=d/dx$ and $u_{q-2}$ is identified with the ``specific heat'' $u(x)$ of \spfcht . The system of equations: \eqn\dgle{[L^{p/q}_+,L]=1 } where the subscript $+$ indicates the differential operator part of a pseudodifferential operator. The equations for massive models interpolating between the minimal models have the form $\sum_p T_p [L^{p/q}_+,L]=1$. The specific case of the $(2p-1,2)$ models (corresponding to single hermitian matrix models) have been most intensively studied. In this case we have simply $L=D^2+u(x)$ so that hierararchy of kdv equations $\dot L=[L^{(2p-1)/2}_+,L]$ may be written as \eqn\kdv{{\p u\over \p t}={\p\over\p x}R_p [u(x)] } where the conserved densities $R_p$ of the kdv flow are known as ``Gelfand-Dickii'' potentials in the 2d gravity literature. The string equations then take the perhaps more familiar form \eqn\onemat{\sum_j (j+\half)T_j R_j[u(x)]=x} Several interesting properties of these equations were discovered shortly after the initial formulation of the double scaling limit. One of the more remarkable results is that a solution to \onemat\ , as a function of the $T_j$ should satisfy kdv flow in the $T_j$ \bdss\ . The original argument for this was proposed from the physical point of view using the matrix model integral, which makes certain implicit assumptions about boundary conditions. Later the issue of proper boundary conditions for physically acceptable solutions of \onemat\ were clarified, again from the matrix model point of view, by Br\'ezin, Marinari, and Parisi \ref\BMP{E. Brezin, E. Marinari, and G. Parisi, ``A Non-Perturbative Ambiguity Free Solution of a String Model,'' ROM2F-90-09}. These authors argued, heuristically, that the existence of physically reasonable solutions depends on the parity of $m$, the largest index for which $T_m\not=0$. In particular, they proposed that physically reasonable solutions exist only for $m$ odd, and in this case the asymptotics fixes a unique solution of \onemat\ which is pole-free on the real axis. They backed up their arguments with a numerical solution for the case $m=3$. This development led to the very interesting papers \ref\dss{M. Douglas, N. Seiberg, and S. Shenker, ``Flow and instability in quantum gravity,'' Rutgers preprint, RU-90-19} \ref\bahnot{G. Bahnot, G. Mandal and O. Narayan, ``Phase transitions in 1-matrixc models,'' IAS preprint IASSNS-HEP-90/52}\ in which it was demonstrated, again numerically, that one cannot use kdv flow to define ``pure gravity'' (an $m=2$ solution) by flowing from the well-defined $m=3$ solution. Essentially, the solution develops a shock wave and there is no well-defined $T_2\to \infty$ limit. This paper is a review and continuation of \ref\geom{G. Moore, ``Geometry of the string equations,'' Yale preprint YCTP-P4-90}\ where the formalism of isomonodromic deformation \nref\Its{A. Its and V. Yu. Novokshenov, {\it The Isomonodromic Deformation Method in the Theory of Painlev\'e Equations,} Springer Lect. Notes Math. 1191.}\ and some related ideas were applied to the string equations. The purpose of our paper \geom\ was threefold. First, the isomonodromic deformation formalism is well suited to proving rigorously the statements regarding the properties of kdv flow and existence and uniqueness of solutions mentioned above. In particular, in section ??? below we review how this formalism can be used to prove these properties. The formalism applies equally well to the string equations associated with unitary-matrix models and with double-cut phases of the hermitian matrix models so we outline the equations for this case too. Second, through the work of the Kyoto school \nref\Jimboi{M. Jimbo, T. Miwa, K. Ueno, ``Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coefficients,'' Physica {\bf 2D}(1981)306.} \nref\Jimboii{M. Jimbo and T. Miwa, ``Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coefficients. II,'' Physica {\bf 2D}(1981)407.} \nref\Jimboiii{M. Sato, T. Miwa, and M. Jimbo, ``Aspects of Holonomic Quantum Fields Isomonodromic Deformation and Ising Model,'' in {\it Complex Analysis, Microlocal Calculus and Relativisitic Quantum Theory}, D. Iagolnitzer, ed., Lecture Notes in Physics 126} \nref\Jimboiv{M. Jimbo, ``Introduction to Holonomic Quantum Fields for Mathematicians,'' Proc. Symp. in Pure Math. {\bf 49}(1989)part I. 379.} \refs{\Jimboi{--}\Jimboiv}\ it is known that the isomonodromic deformation formalism is closely related to the quantum field theory of free fermions in two-dimensional spacetime. This is extremely suggestive since hermitian matrix models can also be written in terms of free fermions. It is worth having a good understanding of any connection between these fermions since it might be an indication of a deep connection between nonperturbative 2D gravity and ``quantum field theory on the spectral curve.'' These matters are discussed in section three below. Third, and more philosophically, interesting physics is usually related to interesting geometry, and this certainly ought to be the case for nonperturbative quantum gravity. We would like to know the underlying geometrical significance of the string equations. By this we do {\it not} mean the geometrical meaning of the terms in the asymptotic expansion of solutions to the string equations, for these have already been adequately understood from the point of view of topological field theory \ref\witt{E. Witten, ``On the structure of the topological phase of two dimensional gravity,'' preprint IASSNS-HEP-89/66} \ref\distler{J. Distler, ``2D quantum gravity, topological field theory and multicritical matrix models,'' princeton preprint PUPT-1161} \ref\DiWit{R. Dijkgraaf and E. Witten, ``Mean Field Theory, Topological Field Theory, and Multi-Matrix Models,'' IASSNS-HEP-90/18;PUPT-1166} \ref\newvsq{E. Verlinde and H. Verlinde, ``A Solution of two dimensional topological quantum gravity,'' preprint IASSNS-HEP-90/40} \ref\dvv{Dijkgraaf, Verlinde, Verlinde}. Some proposals for a geometrical meaning of the string equations can be found in section four below. Closely related matters have been discussed in many recent papers. Of these we draw particular attention to \ref\emil{E. Martinec, unpublished} \ref\witrev{E. Witten, ``Two dimensional gravity and intersection theory on moduli space,'' IAS preprint, IASSNS-HEP-90/45}\ \ref\morozovi{A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov, and A. Orlov, ``Matrix Models of 2D gravity and Toda Theory,'' P.N. Lebedev Institute preprint, July 1990} \ref\morozovii{A. Mironov and A. Morozov, ``On the origin of virasoro constraints in matrix models: lagrangian approach,'' P.N. Lebedev Institute preprint, July 1990} where, among other things, the matrix models were shown to be equivalent to the infinite Toda chain - an important integrable system- even {\it before} taking the continuum limit. \newsec{Matrix Models and 2D Field Theory} In this section we outline how a 2D field theory ``on the spectral curve'' may be seen to emerge from the matrix model integral. Our understanding of this phenomenon is woefully incomplete, but we believe some key features may be seen already at this stage. Most importantly we will see that the spectral parameter of inverse scattering theory is identified with the eigenvalue coordinate of the random matrix path integral. This point of view has also been emphasized in \morozovi\morozovii\ (although some details are different). \subsec{The level-spacing problem} The clearest example of the phenomenon we are discussing can be seen already in the large $N$ limit of hermitian matrix models. Consider the matrix model: \eqn\gauss{\eqalign{ Z_N&=\int d^{N^2}\phi e^{- N tr V(\phi)}\cr &=\int \prod d\lambda_i \Delta^2 e^{-N \sum_i V(\lambda_i)}\cr} } where $V(\lambda)$ is a polynomial in $\lambda$. and consider furthermore the probability $\tau(I;N)$ that no eigenvalue falls in the range $I=[\lambda_1,\lambda_2]$. It was shown in \ref\dnsmt{M. Jimbo, T. Miwa, Y. Mori and M. Sato, ``Density Matrix of an Impenetrable Bose Gas and the Fifth Painlev\'e Transcendent'' Physica {\bf 1D} (1980)80} that $\tau(a_1-a_2)=\lim_{N\to \infty} \tau([a_1/N,a_2/N];N)$ is the partition function of a quantum field theory of free fermions (and is, moreover the tau function for the isomonodromy problem related to the Painlev\'e V equation.) Since the context in which this was originally understood is (superficially) removed from matrix models we will show how this may be understood from the matrix model point of view. We begin by explaining the origin of the quantum field theory. Correlation functions with the measure \gauss\ can be interpreted \ref\mehta{M. L. Mehta, {\it Random Matrices} Academic Press,1967. } \ref\itzdr{See sec. 10.3 in C. Itzykson and J.-M. Drouffe, {\it Statistical Field Theory}, vol. 2, Cambridge Univ. Press. 1989} \bdss\ as expectation values in a slater determinant of fermion one-body wavefunctions given by orthonormal functions: % %$$\psi_j(\lambda)= \bigl({N\over 2\pi}\bigr)^{1/4} %\bigl({N\over j!}\bigr)^{1/2}(\lambda^j+\cdots)e^{-N\lambda^2/ 4} $$ % $$\psi_j(\lambda)= P_n(\lambda)e^{-{N\over 2}V(\lambda)} $$ where $P_n$ are orthonormal polynomials for the measure $d\lambda e^{-NV(\lambda)}$. As in \bdss\ we may pass to second quantized wavefunctions: \eqn\sec{ \eqalign{\psi(\lambda)&=\sum_{n=1}^\infty \psi_n(\lambda)a_n\cr \psi(\lambda)^\dagger&=\sum_{n=1}^\infty \psi_n(\lambda)a_n^\dagger\cr \{\psi^\dagger(\lambda),\psi(\lambda')\}&=\delta(\lambda-\lambda')\cr} } where the ground state is the Fermi sea with the first $N$ levels filled. As argued in \bdss\ the main contributions to correlation functions come from the neighborhood of the Fermi level. For even potentials we can rewrite the recursion relation for orthogonal polynomials \BK\DS\GM\ in the form: \eqn\newrec{ \eqalign{ \lambda p_{2n}(\lambda)&=\sqrt{r_{2n+1}}p_{2n+1} +\sqrt{r_{2n}}p_{2n-1}\cr \lambda p_{2n+1}(\lambda)&=\sqrt{r_{2n+2}}p_{2n+2} +\sqrt{r_{2n+1}}p_{2n} \cr} } By evaluating these at $\lambda=0$ we see that quite generally if we expect a continuum limit for the orthonormal wavefunctions themselves in the neighborhood of $\lambda=0$ we should define \eqn\sclwv{ p_{2n+1}({\lambda\over N})=(-1)^nf_1(x,\lambda) \qquad p_{2n}({\lambda\over N})=(-1)^nf_2(x,\lambda) } where $x=n/N$. Assuming $r_n$ has an expansion of the form $r_n=r(x)+\epsilon^2r_1(x)+\cdots$ where $\epsilon=1/N$ we find that \newrec\ implies \eqn\sclwvi{ \eqalign{ f_1&=(r(x))^{-1/4}sin\biggl[\lambda\int^x {dx'\over \sqrt{r(x')}}\biggr]\cr f_2&=(r(x))^{-1/4}cos\biggl[\lambda\int^x {dx'\over \sqrt{r(x')}}\biggr]\cr } } Since the dominant contributions of physical quantities come from the neighborhood of the Fermi level ($x=1$) we see that quite generally the orthonormal wavefunctions become sines and cosines. THese arguments can be checked explicitely using hermite functions in the case of a gaussian measure. From the behavior of the 1-body wavefunctions we see that $\hat \psi(\gamma;N)\equiv {1\over \sqrt{N}}\psi(\gamma/N)$ has a smooth large $N$ limit. For example, for the two-point function one easily verifies the exact formula: \eqn\extpt{ \langle N|\psi^\dagger(\lambda_1)\psi(\lambda_2)| N\rangle =\sqrt{r_{N+1}}{\psi_{N+1}(\lambda_1)\psi_N(\lambda_2)- \psi_{N+1}(\lambda_2)\psi_N(\lambda_1)\over \lambda_1-\lambda_2} } so that we have \eqn\limkernel{ \langle N|\hat\psi^\dagger(\gamma)\hat\psi(\gamma')| N \rangle \rightarrow {1\over \pi} {sin (\gamma-\gamma')\over \gamma-\gamma'} \qquad .} On the operator level we define $p=(n-2N)/2N$ and \eqn\osc{\eqalign{ a_1(p)&=\sqrt{N}(-1)^{n+N}\bigl(\hat a_{2n}- i\hat a_{2n+1}\bigr)\cr a_2(p)&=\sqrt{N}(-1)^{n+N}\bigl(\hat a_{2n}+ i\hat a_{2n+1}\bigr)\cr} } where $\hat a_n\equiv a_{N+n}$. The sum over $n$ becomes an integral $\int^{\infty}_{-1}dp$. Again, assuming that the main contributions come from the neighborhood of the Fermi level we extend this to an integral over the entire $p$ axis. Our main claim is therefore that $\hat \psi$ has a good large $N$ limit and is given by \eqn\lmfn{\eqalign{ \hat\psi(\gamma)&= e^{i\gamma}\int^\infty_{-\infty} dp~ a_1(p)e^{i\gamma p} +e^{-i\gamma}\int^\infty_{-\infty} dp~ a_2(p)e^{-i\gamma p}\cr &=e^{i\gamma}\psi_1(\gamma)+e^{-i\gamma}\psi_2(-\gamma)\cr} } and that the Fermi sea becomes the ground state defined by $a_i(p)|0\rangle=0$ for $p>0$ and $a_i^\dagger(-p)|0\rangle=0$ for $p>0$. Now let us return to the problem of the level spacing. In terms of the orthogonal polynomials $\psi_j$ we have \mehta\itzdr \eqn\level{\eqalign{ \tau(I;N) &=det\biggl[\delta_{j,k}-\int_{\lambda_1}^{\lambda_2}d\lambda \psi_j(\lambda)\psi_k(\lambda)\biggr]_{0\leq j,k\leq N-1} \cr &=\langle N|exp\bigl(-\int_{\lambda_1}^{\lambda_2} \hat\psi^\dagger(\lambda) \hat\psi(\lambda) d\lambda \bigr)|N\rangle\cr}\qquad . } In the $N\to\infty$ limit, taking $\lambda_i=a_i/N,$ and $I=[a_1,a_2]$ we obtain: $$\tau(I)=\langle 0|:exp\biggl( -\int_I\hat\psi^\dagger \hat\psi\biggr) :|0\rangle$$ where the normal-ordered exponential is evaluated by expanding in power series and point-splitting all the integrals so that the delta functions in the two-point correlation functions don't contribute. This is simply a correlation function in the theory of the one-dimensional Bose gas, also known as the nonlinear Schr\"odinger theory, in the completely impenetrable case, as studied in \dnsmt \ref\Korep{A.R. Its, A.G. Izergin, V.E. Korepin, and N.A. Slavnov, ``Differential equations for quantum correlation functions,'' preprint} \ref\Korepi{A.R. Its, A.G. Izergin, and V.E. Korepin, ``Temperature correlators of the impenetrable bose gas as an integrable system,'' ICTP preprint IC/89/120}. In particular, $\tau(I)$ is simply the Fredholm determinant $det~(1-K)$ where $K$ is the kernel defined by \limkernel\ on the interval $I$. and the integral operator $I+K$ is in the infinite dimensional group of ``completely integrable kernels'' described in \Korep\ . Following the general procedure described in \dnsmt\Korep\Korepi\ we define $\chi_1(z)=e^{iz}\psi_1(z)$ and $\chi_2(z)=e^{-iz} \psi_2(-z)$ so that $\hat\psi=\chi_1+\chi_2$ and $\psi^\dagger= \chi_1^\dagger + \chi_2^\dagger$, and consider the ``Baker-Akhiezer framing'': \eqn\levii{ \Psi_{\alpha\beta}(\lambda,\lambda')={\langle 0|\chi_\alpha^\dagger(\lambda) exp(-\int_I\chi^\dagger \chi)\chi_\beta(\lambda')|0\rangle\over \langle 0|exp(-\int_I\chi^\dagger \chi)|0\rangle} \qquad .} As we will see below this matrix satisfies a linear ODE in $\lambda$ reminiscent of the Knizhnik-Zamalodchikov equation whose study allows us to derive information on the determinant $\tau(I)$. \subsec{Multi-Cut Solutions} We now turn to an example in which we take a double scaling limit. We consider a special class of multicritical potentials \ref\molinari{Molinari} \ref\crmr{C. Crnkovic and G. Moore,``Multi-Critical Multi-Cut Matrix Models,'' Yale preprint YCTP-??-90} \eqn\mulpt{V'_m(\lambda)=k(m)\lambda^{2m+1}\bigl(1-{1\over \lambda^2} \bigr)^{1/2}|_+ } where the subscript $+$ means we keep the polynomial part in an expansion about infinity and \eqn\const{k(m)=2^{2m+1}{(m+1)!(m-1)!\over (2m-1)!} \qquad .} The potential $V_m(\lambda)$ has the shape of a double well, and the bump at $\lambda=0$ becomes progressively flatter as $m$ increases. These potentials are characterized by the property that the eigenvalue distribution at tree level consists of two separate cuts, which, at the critical point, meet at the origin. These theories have very interesting phase transitions investigated in \molinari\dss\crmr\ . We now consider the double scaling limit and the recursion relations for these models. We denote orthonormal polynomials by $\CP_k$ and we have the standard relation \eqn\reci{\lambda \CP_n=\sqrt{R_{n+1}}\CP_{n+1}+\sqrt{R_n}\CP_{n-1} } As in \ref\molinari{molinari} \dss\ we assume that $R_{2n}$ and $R_{2n+1}$ have different scaling limits: \eqn\scli{\eqalign{R_{2n}&=r_c+a^{1/m}f(z) + a^{2/m} g(z)+\cdots\cr R_{2n+1}&=r_c-a^{1/m}f(z)+a^{2/m}g(z)+\cdots\cr} } where, in the standard way, $x={2n\over N}=1-a^2 (z-z_0)$ and $Na^{2+1/m}=1$. The tree-level string equation is easily shown to be \crmr\ \eqn\tree{f^{2m}= {-z\over 2^{2m-1} (m+1)} } Since $V$ is even we can group the orthogonal polynomials into even and odd classes and the operation of multiplication by $\lambda$ maps even orthogonal polynomials to odd polynomials and vice versa. By studying the recursion relation \reci\ in the neighborhood of $\lambda\cong 0$ we expect that the orthogonal polynomials \eqn\lmply{\eqalign{ f^+(z,\lambda)&\equiv j(a,\lambda)(-1)^k\CP_{2k}(a^{1/m}\lambda)\cr f^-(z,\lambda)&\equiv j(a,\lambda)(-1)^k\CP_{2k+1}(a^{1/m}\lambda)\cr} } will have smooth limits, where we have allowed for a possible ``wavefunction renormalization'' $j$ which is $z$-independent. Thus, following the reasoning before we now consider a doublet of second quantized Fermi fields....etc. The double-scaling limit of \reci\ becomes \eqn\reciii{\lambda \vec \psi=\sqrt{8}\biggl(-i\sigma_2 {d\over dz}-\sigma_1 {f\over 4}\biggr)\vec \psi} where $\vec\psi=(f^+(z,\lambda),f^-(z,\lambda) )$. We now consider the double scaling limit of the full recursion relation \eqn\reciv{{d\over d\lambda}\vec\CP = N\bigl(V'(\lambda)_+\bigr) \vec\CP } where the subscript $+$ indicates that we keep only the upper triangular part of the matrix representation of the operator $V'(\lambda)$ in the basis of orthonormal polynomials. In the double scaling limit, with an appropriate choice of $j$ and a slight redefinition of $\vec \psi$, the recursion relations become \eqn\laxi{ \CL\vec\psi\equiv \biggl({d\over dz}+\sigma_3 \lambda+f \sigma_1\biggr)\vec\psi=0} \eqn\laxii{ \biggl({d\over d\lambda}-M_m(\lambda,f)\biggr)\vec\psi=0 } where $M_m$ is polynomial in $\lambda$ and differential polynomial in $f$. We may find $M_m$ explicitly as follows. Note that the commutator of $M_m$ with $\CL$ is just $\sigma_3$. In integrable systems theory the method of Zakharov and Shabat determines the space of matrices $M$ which are polynomial in $\lambda$ and differential polynomial in $f$ and whose commutator with $\CL$ is $\lambda$-independent. The vector space of such matrices is spanned by the matrices occuring in the Lax pairs for the modified KDV hierarchy. (See, e.g., \ref\DrS{Drinfeld and Sokolov, ``Equations of Korteweg-de Vries type and simple Lie algebras,'' Sov. Jour. Math. (1985)1975, section 3.8}\ .) The flatness condition following from \laxi\ and\laxii\ gives an ordinary differential equation for $f$. Comparing with the result \tree\ derived at tree level we find \eqn\res{M_l=(2l+1) \biggl[\bigl (U_l+(V_l'/2\zeta)\bigr)\sigma_3 +V_l\sigma_1 -(V_l'/2\zeta)i\sigma_2\biggr]\Psi =0} where \eqn\uets{\eqalign{ U_l&\equiv R_l + \zeta^2 R_{l-1}+\cdots \zeta^{2l}R_0 +{z\over 2l+1}\cr V_l&\equiv \zeta S_{l-1} +\zeta^3 S_{l-2}+\cdots \zeta^{2l-1} S_0\cr S_l&\equiv f R_l-\half R_l'\cr} } and the KDV potentials are evaluated for $u=f^2 + f'$. The compatibility conditions become $S_m+{z\over 2m+1} f=0$, which is the equation found in \ref\peri{V. Periwal and D. Shevitz, ``Unitary-Matrix Models as Exactly Solvable String Theories,'' Phys. Rev. Lett. {\bf 64}(1990)1326.} for unitary-matrix ensembles. This differential equation is the equation for a self-similar solution to the MKdV flow \eqn\mckf{ {\p f\over \p t_l}={\p \over \p x}S_l[f] } that is, if $\psi(x,t)=t^\alpha f(z)$ where $z=t^\alpha x$ and $\alpha=-1/(2l+1)$. then $f$ must satisfy the above ODE. More generally, by adding the multicritical potentials $V=\sum_\ell t_\ell a^{(2m-2\ell)/m} V_\ell$ we simply replace \eqn\repl{M_m\rightarrow \sum_\ell t_\ell M_\ell} which yields the more general string equations: \eqn\msvmd{\sum t_l S_l +{1\over 2m+1} z f=0 } By arguments analogous to those above we find that the orthogonal polynomials must also satisfy linear equations in the variables $t_\ell$: WRITE OUT LAX FOR MKDV \subsec{Single-Cut solutions} The double scaling limit was originally defined for a different universality class of phase transitions. In these cases the critical behavior arises from the integration near a nonzero value of $\lambda=\lambda_c$, and for the $m^{th}$ multicritical point we have the scaling behavior $R_n\to r_c + a^{2/m} u(z)$ for $n/N=1-a^2 (z-z_0)$, with $Na^{2+1/m}$ fixed. In this case it is natural to assume that the orthogonal polynomials have a limit of the kind: $$p_n(\lambda_c+a^{2/m}\lambda)\to j(a,\lambda)p(z,\lambda)$$ where the wavefunction renormalization will be determined below. The limit of the recursion relation becomes \eqn\schrod{\bigl({d^2\over dz^2} + u(z)\bigr)p(z,\lambda)=\lambda p(z,\lambda) } showing that the orthogonal polynomials define a Baker-Akhiezer function. Similarly, in the limit we have the quantum field $$a^{1/m}\psi(\lambda_c+a^{2/m}\lambda)\rightarrow \hat \psi(\lambda)=\int dz a(z) p(z,\lambda)$$ and the two point function is simply $$\langle z_0|\hat \psi^\dagger(\lambda_1) \hat\psi(\lambda_2)|z_0\rangle={p'(z_0,\lambda_1)p(z_0,\lambda_2)- p'(z_0,\lambda_2)p(z_0,\lambda_1)\over \lambda_1-\lambda_2}$$ If we send one field to infinity we obtain simply the Baker function, reminiscent of the situation in conformal field theory. Moreover, if we perturb around the multicritical point we add the potential \eqn\prtrb{ \delta V= N\sum t_l (\lambda-\lambda_c)^{l+1/2} a^{(2m-2l)/ m}} and the partition function now becomes \eqn\newtpt{ \langle 0|e^{\int \psi^\dagger(\lambda)\psi(\lambda) \sum t_l^{l+1/2}\lambda^{l+1/2} }|0\rangle } Thus, as a function of the $t_l$ the partition function is simply the Fredholm determinant $det(1-K)$ for the kernel defined by \eqn\kerna{\eqalign{ K(\lambda_1,\lambda_2)&=\sqrt{\vartheta(\lambda_1)}K_0(\lambda_1,\lambda_2) \sqrt{\vartheta(\lambda_2)}\cr K_0(\lambda_1,\lambda_2)&={p'(z,\lambda_1)p(z,\lambda_2)- p'(z,\lambda_2)p(z,\lambda_1)\over \lambda_1-\lambda_2}\cr \vartheta(\lambda)&=\sum t_l \lambda^{l+1/2}\cr} } Again, $1+K$ is in the class of completely integrable integral operators \Korep (in this case the convergence of the integrals is rather delicate. the only case we can analyze is the ``topological'' point $m=1$ in which case the relevent integrals are conditionally convergent). Following the procedure of \dnsmt\Korep\Korepi\ as before we consider the linear matrix equation satisfied by $\vec\psi=(p'(x,\lambda)~p(x,\lambda))$: \eqn\lopp{ \CL\vec\psi\equiv -{d\over dx} + \pmatrix{0&\lambda+u\cr 1&0\cr}\vec\psi=0 \qquad .} On the other hand, from their origin in the matrix model it is clear that ${d\over d\lambda}\vec\psi$ may be expressed as a matrix operator on $\vec\psi$ which is polynomial in $\lambda$ with $x$-dependent coefficients as in \laxii\ . As in the previous discussion we can determine $M_m$ by the method of Zakharov and Shabat \DrS\ . Recall that the $l^{th}$ KdV flow can be written as the compatibility condition $[2\p/\p t_l + \CP_l,\CL]=0\qquad ,$ where the $sl(2)$ matrix \eqn\popp{ \CP_l\equiv \pmatrix{A_l& B_l\cr C_l&-A_l\cr} } may be expressed in terms of the conserved densities $R_l$ of KDV flow \ref\Gelf{I.M. Gelfand and L.A. Dickii, ``Asymptotic Behavior of the Resolvent of Sturm-Liouville Equations and the Algebra of the Korteweg-De Vries Equations,'' Russian Math Surveys, {\bf 30}(1975)77.} via \eqn\potent{\eqalign{ C_l&=R_l+\lambda R_{l-1}+\cdots + \lambda^{l-1}R_1 + \lambda^l R_0\cr A_l&=\half C_l'\cr B_l&=(\lambda+u)C_l-A_l'\cr} } It follows that if we define \eqn\wittx{ \IP=-\hbar{d\over d\lambda}-\half\sum_j (j+\half)T_{j}\CP_{j-1} +\pmatrix{0&\hbar x-(\sum_j (j+\half) T_jR_j)\cr 0&0\cr} } (where $\CP_{-1}=0$) then the equation $[\IP_l,\CL]=0$ is equivalent to the string equation. Thus the compatibility conditions for the first pair and last pair of linear systems \eqn\linsys{\eqalign{ \IP \Psi(\lambda,x,T_j)&=0\cr \CL \Psi(\lambda,x,T_j)&=0\cr (2\hbar{d\over dT_j}+\CP_j)\Psi(\lambda,x,T_j)&=0\cr} } give the massive $(2l-1,2)$ and KdV equations, respectively. It can be shown that the compatibility conditions for the first and third equations above follow from the string and kdv equations. Similar considerations allow us to write the string equations for arbitrary $(p,q)$ in first order form. \subsec{Relation to other work} The emergence of a quantum field theory on the spectral curve has been discovered in several different guises in recent investigations into matrix models. In \ref\das{S.R. Das and A. Jevicki, ``String field theory and physical interpretation of d=1 strings,'' Brown preprint BROWN-HET-750} \ref\sengupt{A.M. Sengupta and S.R. Wadia, ``Excitations and interactions in d=1 string theory,'' Tata preprint}\ it is shown that one may derive a quantum field theoretical representation of the $d=1$ matrix model involving fields $\phi(t,\lambda)$ where $t$ is the 1-dimensional time and $\lambda$ is the eigenvalue coordinate of the string. This should be regarded as the $d=1$ version of the phenomena discussed for $d<1$ in this paper. It would be extremely interesting to related the $d=1$ theories to the $d<1$ theories by some analog of the Feigin-Fuks construction. polchinski? There has also been a good deal of fuss over the discovery that the partition function of the matrix model is annihilated by operators forming a subalgebra of the Virasoro algebra \ref\fukuma{Fukuma, Kawai, ???} \dvv\morozovi\morozovii\ . We will see later that the 2D quantum field theory on the spectral curve is related to a 2D conformal field theory, thus making the appearance of a Virasoro algebra completely natural \foot{Essentially the same remark was made in a pretty paper by A. Mironov and A. Morozov \morozovii\ .}. It would be extremely interesting to understand the origin of the field theory on the spectral curve in the sum over topologies of conformal-field-theoretic correlators. Some curious speculations along these lines are described in \ref\knizhnik{V.G. Knizhnik, ``Multiloop amplitudes in the theory of quantum strings and complex geometry,'' Usp. Fiz. Nauk. {\bf 159}(1989)401. English translation: Sov. Phys. Usp. {\bf 32}(1989)945, section 12} \morozovi\ . In this context it is interesting to note that many integrable massive field theories have correlation functions related to Painlev\'e equations \ref\mcoy{E. Barouch, B.M. McCoy, and T.T. Wu, Phys. Rev. Lett. {\bf 31} (1973)1409; T.T. Wu, B.M. McCoy, C.A. Tracy, and E. Barouch, Phys. Rev. {\bf B13}(1976)316.} \Jimboiii\ \Korep\ . \newsec{Isomonodromic Deformation} In the previous section we saw the appearance of a 2D quantum field theory on the spectral curve, and interpreted the string equations and kdv flow as compatibility conditions for linear equations satisfied by correlation functions in the theory. These linear differential equations are essentially flatness conditions, and very similar equations have appeared in conformal field theory. In conformal field theory we often meet linear differential equations, for example null vector equations or the Knizhnik-Zamalodchikov equation, which depend on parameters, e.g. the positions of various fields and the moduli of some Riemann surface, and which have interesting monodromy. It is quite typical that deforming the parameters, e.g., the moduli of the surface, leaves the monodromy unchanged. In the conformal field theory literature this is often expressed in terms of the flatness of the Friedan-Shenker vector bundles over moduli space. This flatness condition is an example of isomonodromic deformation. In this section we will see that isomonodromic deformation is the key behind the linear systems occuring in all the matrix model problems discussed above. \subsec{The Level spacing problem} Following \dnsmt\ we show that $\Delta_I$ is simply the $\tau$ function for isomonodromic deformation associated with $gl(2)$ fermions $\psi_{1,2}$, as in the previous sections. At this point one may easily follow all the steps in section seven of \dnsmt\ using the interpretation in terms of a doublet of fermions \foot{For readers of \dnsmt\ one can identify, for example, $R_I^\pm(x,x';\xi)$ with the two point function of $\hat\chi^\dagger_i(x)$ with $\hat\psi(x')$, and so on.}. In particular, we can compute the monodromy of \levii\ under analytic continuation of $x$ around $a_i$. From the operator product expansion we compute $d\Psi \Psi^{-1}$ where $d$ is exterior differentiation in $x,a_i$. This gives once more Knizhnik-Zamalodchikov type equations, interpreted here as isomonodromic deformation equations. From the general arguments of the previous section we deduce that $\Delta_I$ is the tau function for isomonodromic deformation. work out more details--motivates isomonodromy \subsec{Stokes phenomenon} The isomonodromic deformation method focuses on the monodromy of the solutions of the equation in $\lambda$. Since our equation is of the form ${d\over d\lambda}\Psi=\CA(\lambda)\Psi$ where $\CA$ is polynomial in $\lambda$ (and we consider $\lambda$ as a coordinate on $\IP^1$) it would appear that there can be no monodromy. In fact, the singularity at infinity induces a kind of monodromy in the form of Stokes phenomenon. To put Stokes phenomenon in perspective let us consider again the differential equation \eqn\lin{{d\over d\lambda}\Psi=\CA(\lambda)\Psi} where $\CA$ is meromorphic in $\lambda$. Near a regular singular point $\lambda_0$ where $\CA$ has a pole we may write a formal solution to the differential equation as \eqn\regi{\Psi(\lambda)=\hat\Psi(\lambda)e^{M~log(\lambda-\lambda_0)}} where $\hat\Psi$ is a formal power series in $\lambda-\lambda_0$. It is a nontrivial property of differential equations with regular singularities that $\hat \Psi$ is in fact a convergent power series, so we may conclude that the monodromy is $e^{2\pi iM}$. Stokes phenomenon appears when we try to find solutions to our equation and $\CA$ has a pole of order larger than one. For simplicity we assume that $$\CA(\lambda)={A_{-r}\over (\lambda-\lambda_0)^{r+1}}+\cdots$$ with $A_{-r}$ diagonalizable. In this case it can be shown \ref\Wasow{W. Wasow, {\it Asymptotic Expansions for Ordinary Differential Equations}, Interscience, 1965.} that we have a formal solution \eqn\formi{\Psi=\hat\Psi e^{T(\lambda-\lambda_0)}} with $$T={D_{-r}\over (\lambda-\lambda_0)^r}+\cdots {D_{-1}\over(\lambda-\lambda_0)}+M~log~(\lambda-\lambda_0)$$ where the $D_i$ are diagonal and commute with the ``formal monodromy'' $M$. In general $\hat \Psi$ is only an asymptotic series. It nonetheless has nontrivial analytic meaning in the following sense. There exist angular sectors in the neighborhood of the singular point as in \fig\sectors{Stokes sectors in the neighborhood of an irregular singular point. The two solutions $\Psi_{1,2}$ are asymptotic to the formal solution only in the indicated sectors.} in which there exist true solutions of \lin\ which are asymptotic to the formal solution \formi\ . The regions can be found from considering the nature of the essential singularity $e^T$. Notice that this is exponentially growing and decaying in the angular sectors where $cos r\theta$ is positive and negative, respectively, where $\theta$ is an angular coordinate around $\lambda_0$. Typically a true solution asymptotic to the formal solution can only be defined in a sector of angular width $\pi/r$ which contains regions of both growth and decay. Therefore if we compare two such solutions which are defined on regions with a nontrivial overlap, as in \sectors\ they may differ by right-multiplication by a constant matrix. Such matrices are called Stokes matrices, and, labelling the sectors by $\Omega_k$ we obtain a set of Stokes matrices $S_k$ associated with the differential equation. A basic example: Airy functions According to the works \ref\flasch{H. Flaschka and A. Newell, ``Monodromy and Spectrum-Preserving Deformations I,'' Comm. Math. Phys. {\bf 76}(1980)65.} \Jimboi\Jimboii\Jimboiii\Jimboiv\ the Stokes matrices should be considered as a generalization of the notion of the monodromy of the differential equation. This is not meant so suggest that a solution to the differential equation cannot be analytically continued to a single valued solution in an entire neighborhood of the singular point, and this happens in many classical examples. The point is, such a single-valued solution has the ``wrong'' asymptotic behavior in all but one sector. \subsec{Asymptotic analysis of the PII family} We now apply the isomonodromic deformation formalism to the compatibility conditions for the string equations. Since the equation in $\lambda$ depends polynomially on $\lambda$ it cannot have interesting monodromy in the complex plane. It does have an irregular singular point at infinity and, it turns out, physically interesting solutions to \laxii\linsys\ exhibit Stokes phenomenon. The differential equations \laxii\linsys\ depend on parameters $x,T_j,f,f_z,f_{zz},\dots$, where, for the moment, we consider $f,f_z,f_{zz},\dots $ to be independent quantities. In the case of 2D gravity, the parameters $x,T_j$ have the dual physical interpretation of coordinates in the space of 2D field theories and moduli of a generalization of a riemann surface (discussed in section 5 below). NEED BETTER MOTIVATION It is therefore natural to ask what conditions $f,f_z,f_{zz},\dots$ must satisfy if the monodromy is to be independent of the moduli. We may answer this question as follows. As we vary $z,T_j$ we obtain a family of invertible matrix solutions $\Psi(\lambda,z,T_j)$ to \eqn\lasaga{{d\over d\lambda} \Psi=\sum t_{\ell} M_\ell\Psi } so we may consider the quantity ${\p \Psi\over \p z}\Psi^{-1}$, which is a rational matrix in $\lambda$. The only singularities can be at $\lambda=\infty$, so we need only know the behavior of a solution to \lasaga\ in this limit. Therefore we perform an asymptotic solution of \lasaga\ by relating the matrix elements of \res\ to the resolvent $R(x,\lambda^2)$ (??square?) of the Schr\"odinger operator $L=D^2+u$. The result is that $\Psi=\hat\Psi e^T$ where $$T=\half\biggl(\sum_l{2m+1\over 2l+1}t_l \lambda^{2l+1}\biggr)\sigma_3 $$ and $$\eqalign{ \hat\Psi&=1+{\hat \Psi_1\over \lambda}+{\hat \Psi_2\over \lambda^2}+\cdots\cr \hat\Psi_1&={-if\over 2}\sigma_2+H\sigma_3\cr}$$ where $H'=\half f^2$. Since the Stokes data is unchanged under deformation of $z$ we can compute $\p_z\Psi \Psi^{-1}$ by substituting the asymptotic expansion and we find the condition \laxi\ . From this condition it follows that, as functions of $z$, $f_z$ must be the derivative of $f$ etc., and $f(z)$ must satisfy the string equation. Similarly, one may compute $\hat\Psi{\p T\over\p t_l}\hat\Psi^{-1}mod (1/\lambda)$ to obtain the linear condtion ????. This shows that if one considers a solution to \msvmd\ as a function of the $t_\ell$ then, if the stokes data are fixed, $f$ must satisfy mkdv flow. \subsec{Asymptotic analysis of the PI family} A very similar computation can be carried out for the equation $\IP\Psi=0$. There is one (important) technical change. Since the highest power of $\lambda$ does not multiply an invertible matrix one must make a transformation \Jimboii\ \ref\Kapaev{A. Kapaev, ``Asymptotics of solutions of the Painlev\'e equation of the first kind,'' Differential Equations, {\bf 24}(1988)1107.} $\lambda=\zeta^2$ and \eqn\zee{\eqalign{ \Psi(\lambda)&=\zeta^{1/2}\pmatrix{1&1\cr 1/\zeta&-1/\zeta\cr} W(\zeta)\cr W(\zeta)&=\zeta^{-1/2}\half \pmatrix{1&\zeta\cr 1&-\zeta\cr}\Psi(\lambda)\cr} } The leading singularity in the equation for $W$ now multiplies $\sigma_3$ and we may easily find $W\sim \hat W e^{T/\hbar}$ where \eqn\asymp{ \eqalign{ T&= \biggl(-{1\over 4} \sum_{j=1} T_j \zeta^{2j+1} +\zeta x\biggr) \sigma_3\cr \hat W&=1+ {H_1\over \zeta}\sigma_3 + {u\over 4\zeta^2}\sigma_1 +\CO(1/\zeta^3)\cr} } Again, this may be proved by relating the matrix elements in the differential equation to the resolvent of the Schr\"odinger operator $L=-D^2+u$. As before, if we deform $x$ keeping the Stokes data fixed then we can evaluate the expression ${\p \Psi\over \p x}\Psi^{-1}$ from its asymptotics. (In this case one must take into account that the equation for $W$ has a regular singularity at zero.) The result is just: \eqn\laxi{{\p\Psi\over \p x}\Psi^{-1}=\pmatrix{0&\lambda+u\cr 1&0\cr} } and comparing with section 3.1 we see that this means $u(x)$ must satisfy the string equation. Similarly we can ask for the condition that deformations in $T_j$ keep the Stokes data fixed, and we find remaining linear equations in \linsys\ so that $u(x;T_j)$ satisfies KDV flow. We will indicate below how the physical asymptotic conditions on $u(x)$ as a function of $x$ fix the Stokes data uniquely. Thus we have proven that $u(x;T_j)$ satisfies kdv flow. The original proposal of Banks, Douglas, Seiberg, and Shenker \bdss\ , was based on more physical (but entirely reasonable) arguments. The following argument for KDV flow has also been proposed in \emil\witrev\morozovi\ . In the matrix model, before the continuum limit is taken, the jacobi matrix representing multiplication by $\lambda$ in the space of orthogonal polynomials satisfies toda flow. It was proposed some time ago \ref\moser{J. Moser, ``Finitely many mass points on the line under the influence of an exponential potential--an integrable system,'' in Lecture Notes in Physics {\bf 38}, J. Moser, ed., p. 467} that the continuum limit of toda flow should give (m)kdv flow. \subsec{$\tau$ functions} One of the beautiful results of the Kyoto school on isomonodromic deformation was the definition of the $\tau$ function for isomonodromic deformation, which motivated the perhaps better known tau function of the KP hierarchy. Applying \Jimboi\Jimboii\ to our case we see that \eqn\clsdfrm{ \omega=Res_{\zeta=\infty} tr\Biggl[ \bigl(\hat W^{-1}{d\hat W\over d\zeta}\bigr) d T\Biggr] } is a closed one-form on the space of deformation parameters. Since $\omega$ is closed one can then define (locally, in the space of deformation parameters) the {\it tau function} via $\omega=d(log \tau)$. We have just seen that $dH_1/dx=u$, and substituting into the equation for $d(log\tau)$ we get $d(log\tau)=-2H_1$, hence \eqn\freeeng{ u(x;T_j)=-{d^2\over dx^2}log \tau } and hence the partition function of the matrix model (whose logarithm is the partition function of 2D gravity) is simply the tau function. Since the tau function for isomonodromic deformations is known always to be holomorphic \ref\Miwa{T. Miwa, ``Painlev\'e property of monodromy preserving deformation equations and the analyticity of $\tau$ functions,'' Publ. Res. Inst. Math. Sci. {\bf 17} (1981)703}\ we see that the {\it only} singularities of a solution $u$ to the string equations are second order poles. This result may be easily extended to the entire hierarchy of $(p,q)$ equations. If we apply the definition \clsdfrm\ to the asymptotic expansion for the PII family we find \eqn\tauf{f^2=2H'={\p^2\over \p z^2}log~ \tau \qquad .} On the other hand, computing the connected two-point function of $tr\phi^2$ one can show \peri\crmr\ \eqn\twopt{ \langle tr\phi^2~tr\phi^2\rangle_c =R_N\bigl(R_{N+1}+R_{N-1}\bigr)\to {1\over 8} - a^{2/m} f^2 } so that the partition function of the unitary matrix model is just the tau function. (WE HAVE USED $g=f^2$ WHICH WE ONLY PROVED TO TREE LEVEL SO FAR!) \subsec{Stokes matrices for the PI family} In order to obtain physical solutions to the string equation we need to choose proper boundary conditions. In the isomonodromic deformation literature it is shown that the ``initial conditions'' of a solution to the nonlinear compatibility condition (the string equation) may be taken to be the stokes matrices of the associated linear problem. In this section we discuss the stokes data for the PI family for the ``physical'' solutions to the string equations. The issue of the correct choice of boundary conditions for a physically acceptable solution to the string equation is dictated by the origin of the specific heat in the matrix model integral \BMP \ref\David{F. David, ``Phases of the large N matrix model and non-perturbative effects in 2d gravity,'' Saclay preprint SPhT/90-090}. For example, in the case of the m=3 member of the PI family, i.e., $R_3[u(x)]=x$, the physical asymptotics are given by $u\sim \CO(x^{1/3})$ as $x\to\pm\infty$. Since each asymptotic condition fixes two boundary conditions we expect that these asymptotics uniquely specify the solution. From the numerical integration in \BMP\ it appears that the solution is pole-free. We will show that the above asymptotics uniquely determines the Stokes parameters in the monodromy problem associated to $R_3=x$. Since one can reconstruct a solution from the stokes data (via the * operator, see below) it follows that the solution is completely unique. This unique solution is real for all $x$. We investigate the direct monodromy problem following closely the treatment in \ref\Itsi{See reference \Its, especially, chapter 5.} \Kapaev . Recall that, after the transformation \zee\ our equation can be written as \eqn\dforz{\eqalign{ \hbar {d W\over d\zeta}&=\Biggl[(B+\zeta^2 C+\Delta_l) \sigma_3-(B-\zeta^2 C+\Delta)i\sigma_2+(2\zeta A -{\hbar \over 2\zeta})\sigma_1\Biggr]W\cr\cr &=\biggl[(\zeta^{2l+2}+{u^2\over 8}\zeta^{2l-2}+\cdots)\sigma_3 + (-{u'\over 4}\zeta^{2l+1}+\cdots)\sigma_1 + (-{u\over 2}\zeta^{2l}+\cdots)(-i\sigma_2)\biggr]W \cr} } where $$\eqalign{ C\equiv -\half\sum_j (j+\half)T_{j}C_{j-1}& \qquad \qquad \Delta=\hbar x -\sum (j+\half)T_jR_j\cr A_l= \half C' &\qquad\qquad B=(\lambda+u)C-A'\cr} $$ Equation \dforz\ has an irregular singularity of order $2l+3$ at infinity and a regular singularity at the origin. We consider some general properties of the stokes matrices for these equations. For the $(2l-1,2)$ string equation we will have $4l+6$ Stokes sectors $\Omega_k$ each containing a unique ray $\theta={\pi\over 4l+6}(2k-1)$, $k=0,\dots 4l+5$ along which $cos[(2l+3)\theta]=0$, thus we may take neighborhoods of infinity defined by: \eqn\sectors{ \Omega_k\equiv \{\zeta| {\pi\over 4l+6}+ {\pi\over 2l+3}(k-2) < arg \zeta < {\pi\over 4l+6}+ {\pi\over 2l+3}k\} } for $k=0,\dots, 4l+5$. We will consider $k$ as an integer defined modulo $4l+6$. The Stokes sectors have the property that in $\Omega_k$ there is a unique solution $W_k$ to \dforz\ with the asymptotic behavior $\hat W e^{T/\hbar}$. On the overlap $\Omega_k\cap\Omega_{k+1}$ the two solutions $W_{k+1}$ and $W_k$ must be related by Stokes matrices \eqn\sto{W_{k+1}=W_k S_k \qquad. } Since $W_{k+1}$ and $W_k$ have the same asymptotic expansion in their respective sectors, the $S_k$ are constrained to be triangular. In fact \eqn\sto{ S_{2k}=\pmatrix{1&s_{2k}\cr 0&1\cr}\qquad\qquad S_{2k+1}=\pmatrix{1&0\cr s_{2k+1}&1\cr} } The stokes parameters $s_i$ are not all independent but satisfy the constraints: \noindent 1. $s_{k+2l+3}=s_k$. \noindent 2. \eqn\mncnst{S_1\cdots S_{2l+3}=-i \sigma_1 } \noindent 3. $s_{2l+3-k}=-\bar s_k$ if $u$ is real. We may prove these properties as follows. For property 1, we use the symmetry of equation \dforz\ to conclude that $W_{k+2l+3}(\zeta)=\sigma_1W_k(-\zeta)\sigma_1$ and hence that $S_{k+2l+3}=\sigma_1 S_k\sigma_1=S_k^{tr}$, from which 1 follows. Next, for 2 we remark that the original equation in $\lambda$ is regular throughout the $\lambda$ plane. The solution is simply $Pexp\int^\lambda A(\lambda')d\lambda'$ for an appropriate matrix $A$ hence the only singularities in $\Psi$ can occur at infinity. Thus, near $\zeta=0$ we have \eqn\nrzr{ W(\zeta)\cong \half\zeta^{-1/2}\pmatrix{1&\zeta\cr 1&-\zeta\cr} } By \nrzr\ , if we analytically continue $W_1$ from $\Omega_1$ then $W_1(-\zeta)=-i\sigma_1W_1(\zeta)$ from which we obtain 2. Finally if $u$ is real then all the coefficients of powers of $\zeta$ in $\IP$ are real so that $\bar W_k(\zeta)=W_{-k}(\bar\zeta)$ implying 3. Since the determinant of \mncnst\ is automatically satisfied, \mncnst\ only imposes three independent constraints on the Stokes matrices so we have $2l+3-3=2l$ independent Stokes parameters. Note that this is the number of initial conditions in the string equation. In fact, the two sets of parameter spaces may be regarded as the same. Let us now consider the implications of physical asymptotics. Consider the equation $\IP\Psi=0$ as $x\to \pm\infty$. We must distinguish several cases. Since $R_l=\kappa_l u^l +\cdots $ with $$\kappa_l=(-1)^l{(2l-1)!!\over 2^{l+1}l!}$$ we see that perturbation theory predicts that the solution to the string equation is given by \eqn\ascses{\eqalign{ u\sim \bigl(-1/2\kappa_{l+1}\bigr)^{1/(l+1)}x^{1/(l+1)} &\qquad l\quad {\rm even},\quad x\to\pm\infty\cr u\sim \pm\bigl(1/2\kappa_{l+1}\bigr)^{1/(l+1)}(-x)^{1/(l+1)} &\qquad l\quad {\rm odd}, \quad x\to -\infty\cr} } where $l$ is the largest index for which $t_{l+1}\not=0$. In fact, perturbation theory tells us to take the $+$ root for $u$ in the case of odd $l$, but we may easily examine both cases at once. Rescaling variables and only keeping leading order terms we may rewrite the equation $\IP \Psi=0$ as \eqn\limeqt{ {dW\over d\xi}=\tau\Biggl[ (2\xi^2\pm 1)p_l^\pm(\xi)\sigma_3 \mp p_l^\pm(\xi)i\sigma_2 -{1\over 2\xi \tau}\biggl({8l\kappa_{l+1} \over l+1}\xi^2 p_l(\xi)+1 \biggr) \sigma_1 +\CO(1/\tau^2)\Biggr]W } where $p_l^\pm(\xi)=\sum_{p=0}^l (\pm 1)^p\kappa_p\xi^{2l-2p}$. EXPLAIN WHY LOWER l's IRREL For $l$ odd we obtain this equation with $x\to -\infty$, where: $$\zeta=(-x/2\kappa_{l+1})^{1/(2l+2)}\xi \qquad\qquad \tau=(-x/2\kappa_{l+1})^{(2l+3)/(2l+2)}\qquad .$$ For $l$ even we obtain this equation with $\pm x\to +\infty$, where $$\zeta=(\pm x/(-2\kappa_{l+1}))^{1/2l+2}\xi \qquad\qquad \tau=(\pm x/(-2\kappa_{l+1}))^{(2l+3)/(2l+2)}\qquad .$$ The evaluation of the Stokes matrices is carried out by doing a WKB analysis in the $\tau\to\infty$ limit as in \Its . Thus one obtains true solutions to \limeqt\ which are asymptotic as $\tau\to\infty$ to the WKB ansatz: $$ W^{WKB}\sim T exp\Biggl[\tau\int^\zeta\Lambda(\zeta')d\zeta' \Biggr] $$ where $T$ diagonalizes \limeqt to $\Lambda=\mu\sigma_3$. There are several WKB solutions in different regions defined by the turning points and conjugate Stokes lines. The turning points are simply the roots $\xi_i$ of $\mu$, and the conjugate Stokes lines are the lines defined by the vanishing real part: \eqn\skis{ \Re\int_{\xi_i}^{\xi}\mu(\xi')d\xi'=0 } The general procedure for finding the stokes matrices is described in \Its . A simple consequence of this procedure allows us to obtain certain necessary conditions on the Stokes matrices which, in some cases, fix the parameters uniquely. The main observation is that if, at a turning point which is a root of $p_l$ the conjugate Stokes lines form three large regions each abutting an open region infinity as in fig. 6, then the Stokes matrix for the transition function associated with the middle region $\tilde \Omega_2$ is trivial \geom\ . Our first task is therefore to describe the Stokes lines. For the the case $l=2$ we have fig. 9 and fig. 10. From the limit $x\to +\infty$ we obtain that $s_1=s_6=0$. From the limit $x\to -\infty$ we obtain $s_2=s_5=0$, and from the monodromy constraints we get $s_0=s_3=s_4=-i$. Unfortunately, if we move on to higher $l$ we meet a technical problem which is that the configuration of Stokes lines becomes too complicated to use our observation immediately to set half the Stokes parameters to zero. Nevertheless, studying small $l$ leads to a natural guess for the stokes parameters in the general case. For $l$ even, comparing the constraints from the physical asymptotics at either end of the axis will fix two disjoint sets of Stokes parameters. We expect that $s_1=s_2=\cdots =s_l=0$ while $s_{l+1}=s_{l+2}=s_{2l+3}=-i$ so that the solution is unique, and the stokes data is always concentrated on the wedges abutting the $x$ and $y$ axes in the $\zeta$ (=$\sqrt{\lambda}$) plane. We have not used the reality constraints in a very essential way. An interesting solution of the Painlev\'e equation is the triply truncated solution, characterized by $u\sim +(-x)^{1/2}$ for $x\to-\infty$ and $u\sim\pm i x^{1/2}$ for $x\to \infty$. Using the above technique one easily shows that in this case the constraints $s_2=s_3=-i$, $s_0=s_5=0$, $s_1+s_4=0$ are supplemented by $s_1=0$ or $s_4=0$, depending on the sign of the imaginary part. In either case we confirm that the solution is unique. An analog of the BMP solutions exists for the PII family \ref\Hast{S.P. Hastings and J.B. McLeod, ``A boundary value problem associated with the second painlev\'e transcendent and the Korteweg-de Vries equation,'' Arch. Rat. Mech. and Anal. {\bf 73}(1980)31} \ref\cdm{\v C. Crnkovi\'c, M. Douglas, and G. Moore, ``Physical solutions for unitary-matrix models,'' Yale preprint YCTP-P6-90.}. In this case physical asymptotics specifies that $u(x)$ grows algebraically at one end of the axis and decays exponentially at the other end \dss\cdm\ . Applying the above techniques one finds that in this case (at least for the first two members of the PII family) the stokes data is concentrated entirely on the $y$-axis, but the necessary conditions leave a single parameter undetermined. In this case, however, the inverse monodromy problem is equivalent to an integral equation (the self-similar Gelfand-Levitan-Marchenko equation) which can be examined directly. This is done in \cdm\ where it is shown that the physical solution is unique. Again it is interesting examine the Stokes data. We may again apply our lemma for the first two cases in the series. In the first two cases in the PII hierarchy these two constraints lead to a configuration of Stokes lines from which we may conclude {\it necessary} conditions that, together with the monodromy constraints only determine the Stokes parameters up to a single real parameter. The nontrivial Stokes matrices are again concentrated the y-axis. \subsec{KdV orbits are disconnected} The string equation and kdv flow are compatible equations. This does not mean that, as we change the $T_j$ the solution automatically satisfies kdv flow. As we have seen above, if the stokes data are held fixed then we do have kdv flow. If we consider the kdv orbits of solutions of the string equation we may wonder whether they are connected. Certainly they are formally connected. For example, consider the equation \eqn\flowi{ \half(m_1+\half)R_{m_1}+\half(m_2+\half)T R_{m_2}=x } with $m_1>m_2$. As discussed in \bdss \ref\cgmr{C. Crnkovic, P. Ginsparg, and G. Moore, ``The Ising model, the Yang-Lee edge singularity, and $2D$ quantum gravity,'' Phys. Lett. {\bf 237B}(1990)196.}\ if one scales a solution $u(x;T)$ to \flowi\ using $v(y;T)=T^{-2/(2m_2+1)}u\bigl(T^{-1/(2m_2+1)}y;T\bigr)$, then the large $T$ limit $v(y)=\lim_{T\to\infty}v(y;T)$ must be a solution of the lower order string equation $\half(m_2+\half)R_{m_2}[v]\sim x$, provided the limit exists. However, the existence of this limit is a very delicate issue. Indeed, in \dss\ convincing evidence is presented that the flow from $m=3$ to $m=2$ does not exist. Since, as we have seen, physical asymptotics fixes the stokes data, and since kdv flow is isomonodromic, we are in a position to investigate analytically the result in \dss\ . Suppose the $T\to\infty$ limit does exist. Then we can scale $\zeta\to T^{-1/(2m_2+1)}\zeta$ in \dforz\ to obtain a connection with a smooth $T\to\infty$ limit. Since solutions can in principle be obtained from the path-ordered exponential of the connection, solutions to \dforz\ will also have smooth $T\to\infty$ limits. Moreover, from the asymptotics in $\zeta$ we see that the coefficients have smooth $T\to\infty$ limits and in fact approach the asymptotics of the lower order equation. Thus, fundamental solutions smoothly approach fundamental solutions for the lower order equation, although they will be defined on small regions of angular width ${2\pi\over 4m_1+2}$ and hence only define part of a fundamental solution for the lower order equation which is defined on the larger regions of angular width ${2\pi\over 4m_2+2}$. Because of this we find two rules governing flows: \noindent 1. A large region associated with a trivial Stokes matrix for the ``$m_2$ equation'' cannot contain a small region with a nontrivial Stokes matrix for the ``$m_1$ equation.'' \noindent 2. A large region associated with a nontrivial Stokes matrix for the ``$m_2$ equation'' must contain at least one small region with a nontrivial Stokes matrix for the ``$m_1$ equation.'' Note that for even and odd $l$ the nontrivial Stokes data disagrees on the real axis. Hence, by rule 1 it is impossible to use KdV flow to go from an even $l$ to an odd $l$ model, confirming the result of \dss\ . Note that flow from an even $l$ to a smaller even $l$ is consistent with rules 1 and 2. COMPARE DAVID'S RESULTS!!! CONTRADICTION??? \newsec{Isomonodromy and Free Fermions} In this section we will interpret the isomonodromy problem connected with the string equations in conformal-field-theoretic terms. Our paradigm will be the solution of the Riemann-Hilbert problem for the case of regular singular points given ten years ago by the Kyoto school \ref\holoii{M. Sato, T. Miwa, M. Jimbo, ``Holonomic Quantum Field Theory II,'' Publ. RIMS {\bf 15}(1979)201}. We review their construction first, in the light of subsequent developments in CFT. Then we consider the case of irregular singular points. Developing further some work of Miwa \ref\miwai{T. Miwa, ``Clifford operators and Riemann's monodromy problem,'' Publ. Res. Inst. Math. Sci. {\bf 17}(1981)665}, we find that the theory of irregular singular points can be included at the expense of the introduction of a new kind of operator. In a way we are making a nontrivial extension of conformal field theory by expanding the class of functions admitted in the theory from analytic functions with algebraic singularities to analytic functions with essential singularities. This is reflected in the need to expand the class of operators from twist operators to star operators. \subsec{Regular Singular Points} The basic idea of \holoii\ is that the solution to an $m\times m$ matrix differential equation \eqn\diffl{ {d\Psi\over dz}=A(z)\Psi(z) } may be characterized uniquely by its monodromy properties. More precisely, suppose $A$ has only simple poles at points $a_\nu$ and the residue can be diagonalized to $L_\nu$. Then the matrix $\Psi$ can be uniquely characterized by the requirement that (i.) $\Psi(z_0)=1$ (ii.) $\Psi(z)$ is holomorphic in $z\in\IP^1-\{a_1,\dots , a_n\}$ (iii.) $\Psi(z)=\hat \Psi^{(\nu)}(z) e^{L_\nu log(z-a_\nu)}$ for, $z\cong a_\nu$ where $\hat \Psi^{(\nu)}$ is holomorphic and invertible in a neighborhood of $a_\nu$. Conversely, any such matrix defines a rational matrix $A=\Psi_z\Psi^{-1}$ with at most simple poles. Thus, if one can construct appropriate ``twist operators'' $\varphi_i$ such that the correlation function \eqn\corr{\Psi_{\beta\alpha}(z_0;z)= (z_0-z) {\langle \bar \psi_\beta(z_0)\psi_\alpha(z) \varphi_n(a_n)\cdots \varphi_1(a_1) \rangle\over \langle \varphi_n(a_n)\cdots\varphi_1(a_1)\rangle } } has the correct monodromy properties, then it must be a solution of \diffl . Thus we have reduced the global Riemann-Hilbert problem to the {\it local} problem of finding conformal fields $\varphi_{L}(a)$ with the operator product expansion \eqn\twis{ \psi_\alpha(z)\varphi_L(a)\sim \bigl[\CO^0_{L,\alpha}(a)+ (a-z)\CO^1_{L,\alpha}(a)+\cdots\bigr](z-a)^{L_\alpha} } The construction of these operators proceeds by choosing a basis of curves $\gamma_\nu$ circling once around $a_\nu$ and generating the fundamental group $\pi_1(\IP^1-\{a_\nu\};z_0)$ and defining: \eqn\gentwis{ \varphi_{M}(a)=exp\Biggl[\int_\CC^a tr\biggl\{log(M)J(y)\biggr\} {dy\over 2\pi}\Biggr] } where $J_{\beta\alpha}=\bar\psi_\beta\psi_\alpha$ and $\CC$ is a contour (a branch cut for $\Psi$) emanating from $a$. Consider now \corr\ with such operators inserted. As we analytically continue $z$ around $a$ the simple pole in the ope of $\psi$ with $J$ gives rise to the monodromy $\psi_\alpha\to \psi_\gamma M_{\gamma\alpha}$ in the Fermi field. Therefore \corr\ solves conditions ({\it i-iii}). Let us now consider isomonodromic deformation of \diffl\ . It is clear from locality of the ope that changing the $a_\nu$ leaves the monodromy data unchanged. Necessary and sufficient conditions for isomonodromic deformation are given by: \eqn\schles{\eqalign{ {\p \Psi\over \p z_0}&=-\sum {A_\nu\over z_0-a_\nu}\Psi\cr {\p \Psi\over \p a_\nu}&=\biggl(-{A_\nu\over z-a_\nu}+-{A_\nu\over z_0-a_\nu} \biggr)\Psi\cr} } The compatibility conditions for these linear equations give the nonlinear Schlesinger equations \foot{For an appropriate choice of matrices, for example, these equations reduce to PVI \Jimboii .}. From \corr\ we see that the linear equations of isomonodromic deformation theory should be thought of as transport equations on moduli space, analogous to the Knizhnik-Zamalodchikov equations, so that the theory of isomonodromic deformation for regular singular points fits nicely into the framework of Friedan-Shenker modular geometry. According to the general theory of isomonodromic deformation \Jimboi\Jimboii\Jimboiii\ there is a tau function associated to the deformation parameters $a_\nu$. In this case it is given by \eqn\taufn{ d~log~\tau(a_1,\dots, a_n)=-\sum_\nu Res_{z=a_\nu}tr \Biggl[ (\hat \Psi^{(\nu)})^{-1} {\p \hat \Psi^{(\nu)}\over\p z} d \biggl(log(z-a_\nu)L_\nu\biggr)\Biggr] } where the $d$ is a differential in the parameters $a_\nu$. In fact the $\tau$ function is given by $\tau=\langle \varphi_1\dots\varphi_n\rangle$\ \Jimboi\Jimboii\Jimboiii\ . We will now rederive this using general principles of conformal field theory \foot{Exactly the same observation was made 3 years ago in \knizhnik\ , ch. 4. I was unaware of this work when I published \geom .}. We have normalized \corr\ so that it is equal to $\delta_{\alpha\beta}$ at $z=z_0$. Taking the operator product expansion as $z\to z_0$ and matching this with an expansion of a solution to \diffl\ around $z_0$ we find \eqn\corrii{\eqalign{ {\langle J_{\beta\alpha}(z_0) \varphi_n(a_n)\cdots \varphi_1(a_1) \rangle\over \langle \varphi_n(a_n)\cdots\varphi_1(a_1)\rangle } &=-A(z_0)_{\beta\alpha} \cr {\langle \CT(z_0) \varphi_n(a_n)\cdots \varphi_1(a_1) \rangle\over \langle \varphi_n(a_n)\cdots\varphi_1(a_1)\rangle } &=\half tr A^2(z_0)\cr} } where $\CT$ is the stress energy tensor. Since $L_{-1}$ always takes a derivative with respect to position we have $${\p\over\p a_\nu}log[\langle\varphi(a_1)\cdots\varphi(a_n)\rangle] = \oint_{a_\nu}dz_0 {\langle \CT(z_0) \varphi_n(a_n)\cdots \varphi_1(a_1) \rangle\over \langle \varphi_n(a_n)\cdots\varphi_1(a_1)\rangle } = \oint_{a_\nu}dz_0 \half trA^2(z_0)$$ On the other hand, substituting the local expansion $\Psi=\hat\Psi(z-a)^L$ we get \eqn\lcexp{\hat \Psi^{-1}\hat \Psi_z + L/(z-a) =\hat \Psi^{-1}A\hat \Psi\qquad .} Squaring this equation we find \eqn\sqaring{ {\p\over\p a}\bigl(log~\tau\bigr)=Res~tr~\biggl(\hat\Psi^{-1}\hat\Psi_z {L\over z-a}\biggr)=\half Res~tr~A^2 \qquad ,} and hence the tau function is simply the correlation function of twist operators. \subsec{Irregular Singular Points} Let us now attempt to repeat the previous discussion for the case of a differential equation \diffl\ where $A$ is rational but can have irregular singularities. Our treatment is the same in spirit as the discussion of T. Miwa \miwai\ , although there are some differences of detail. Recall that at an irregular singular point we divide up a neighborhood of the point into sectorial domains, each containing a fundamental solution with asymptotics \eqn\stoii{ \Psi\sim\biggl(\sum_{l\geq 0}\hat\Psi^{(l)}(z-a)^l\biggr)(z-a)^L e^{T(z-a)} } where $L$ and $$T(z-a)=\sum_{i=1}^{r}{T_r\over (z-a)^r}$$ are diagonal, and $\hat \Psi^{(0)}$ is invertible. In particular, the analytic continuation of $\Psi_1$ will have the asymptotic expansion \eqn\stoki{ \Psi_1\sim\biggl[\sum_{l\geq 0} \hat\Psi^{(l)}(z-a)^l\biggr](z-a)^L e^T(S_1\cdots S_{k-1})^{-1} } in the sector $\Omega_k$. A solution to the differential equation can be uniquely characterized by the conditions $(i)-(iii)$ above except that $(iii)$ must be replaced by the requirement that $\Psi$ have the asymptotic expansions \stoki\ . Assume there is only one irregular singular point and define $\tilde \Psi\equiv \Psi_1 e^{-T(z-a)}$. We now search for quantum field operators $V_{S,T,L}(a)$, which we call ``star'' operators, such that \eqn\corstar{ \tilde\Psi_{\beta\alpha}(z_0;z)= (z_0-z) {\langle \bar \psi_\beta(z_0)\psi_\alpha(z) V_{S_1,T_1,L_1}(a_1)\cdots \rangle\over \langle V_{S_1,T_1,L_1}(a_1)\cdots\rangle } } Evidently, a star operator is characterized by its operator product expansion with $\psi$, $\bar\psi$, e.g., for $z\in \Omega_k$ we have \eqn\starope{ \psi_\alpha(z)V_{S,T,L}(a)\sim \bigl[\CO^0_{\gamma}(a)+ (a-z)\CO^1_\gamma(a)+\cdots\bigr](z-a)^{L_\gamma}\bigl[ e^T(S_1\cdots S_{k-1})^{-1}e^{-T}\bigr]_{\gamma\alpha} } From this description it looks very unlikely that star operators exist \foot{For example, it is often claimed that operator product expansions in CFT are convergent. Note that \starope\ is only asymptotic. The reason for this is ultimately to be found in the fact that the string coupling has become dimensionful \DS . Note that it is $x$ and the masses $T_j$ which multiply the terms giving the essential singularity at infinity. }. We now give at least a formal construction of these operators. Consider a ray $\CC$ emanating from a point $a$. Consider the product of operators $$exp\bigl[\int^a_\CC dy\psi_\alpha(y) M_{\alpha\beta}(y)\bar\psi_\beta(y) \bigr]\psi_\alpha(z)$$ where $M$ is some matrix defined along the line. If we analytically continue in $z$ through the curve $\CC$ and compare with the other operator ordering it is a simple consequence of Cauchy's theorem and the operator product expansion that we have the exchange algebra: \eqn\galg{\eqalign{ exp\bigl[\int^a_\CC dy\psi_\alpha(y) M_{\alpha\beta}(y)\bar\psi_\beta(y) \bigr]\psi_\alpha(z-\epsilon)&=\cr\cr \psi_\gamma(z+\epsilon) (e^{M(z)})_{\gamma\alpha}& exp\bigl[\int^a_\CC dy\psi_\alpha(y) M_{\alpha\beta}(y)\bar\psi_\beta(y) \bigr]\cr} } where $z$ is a point on $\CC$ and $z\pm\epsilon$ are points above and below $z$. Thus, defining $\CS_k\equiv e^T S_k e^{-T}$ we may define, at least formally, \eqn\stardef{ V_{S,T,L}(a)=\varphi_L(a)\prod_k exp\Biggl[\int^a_{\CC_k} tr\biggl\{(log \CS_l(y))J(y)\biggr\}dy \Biggr] } where we choose contours $\CC_k$ in $\Omega_k$ such that the matrix $\CS_k$ approaches the identity rapidly. In \miwai\ T. Miwa obtained formulae for star operators using a slightly different formalism. Comparing his formulae in terms of infinite expansions evaluated by Wick's theorem we obtain the same result. As shown in \miwai\ contours of integration can be defined so that for small enough Stokes data the integrals make sense, thus giving a more precise definition to the star operator. It follows from locality of the operator product expansion that the differential equation satisfied by \corstar\ has the property that the monodromy data $S,L$ are preserved if we vary the parameters $a_i,T_i$. Just as the tau function for isomonodromic deformation of an equation with regular singular points is given by the correlation function of twist operators, the tau function for isomonodromic deformation in an equation with irregular singular points is given by the correlation function of star operators \miwai\ . One may give a formal argument for this following steps analogous to those leading from 4.7 to 4.9. The analog of \corrii\ is \eqn\stress{\eqalign{ {\langle J_{\beta\alpha}(z_0) V_{S_1,T_1,L_1}(a_1)\cdots \rangle\over \langle V_{S_1,T_1,L_1}(a_1) \cdots\rangle } &=-A(z_0)+T'(z_0)\cr {\langle \CT(z_0) V_{S_1,T_1,L_1}(a_1)\cdots \rangle\over \langle V_{S_1,T_1,L_1}(a_1)\cdots \rangle } &=\half tr (A-T')^2(z_0)\cr} } where $\CT$ is the stress-energy tensor. Using formal manipulations with the ope one can show that \eqn\gobpe{ -Res_{z_0=a} tr\delta T(z_0) A(z_0)= {\sum_k\int^a_{\CC_k}dy tr\biggl(\delta T(y){\delta\over \delta T(y)}\biggr)\langle V_{S,T,L}\cdots\rangle\over \langle V_{S,T,L}(a)\cdots \rangle} } Putting together \stress\ and \gobpe\ we then find \eqn\taustar{ {d\over da}~log~\langle V\cdots V\rangle= \half Res_{z_0=a} tr~A^2(z_0) ={d\over da}~log~\tau } where the second equality follows from an argument similar to the case of regular singularities. Similarly, one can show that the dependence of $log~\tau$ and $log\langle V\cdots V\rangle$ on other parameters is the same. \subsec{$\tau$ functions for 2D gravity} As a special case of the above formalism we can express the solution $u$ of the string equations in terms of a fermion correlation function. We represent the solutions $W$ to \dforz\ and $\Psi$ to \lasaga\ as fermion two-point functions in the presence of star operators. For the $\tau$ function of the PII family we may define $t(y)\equiv \half \sum_\ell {2m+1\over 2\ell +1}t_\ell \lambda^{2\ell+1}$, so that $\tau(t_\ell)=\langle V(\infty;s_k,t_\ell)\rangle$ where \eqn\srptw{\eqalign{ V(\infty;s_k,t_\ell)=\prod_k & exp\biggl[s_{2k+1}\int^a_{\CC_{2k+1}}e^{2t(y)}\psi_1\bar\psi_2(y)\biggr]\cr &\qquad exp\biggl[s_{2k}\int^a_{\CC_{2k}}e^{-2t(y)}\psi_2\bar\psi_1(y)\biggr]\cr} } For the PI family we have a fermion twist operator at the origin (from the regular singularity in \dforz\ ). If we bosonize the two fermions $e^{i\phi_i}=\psi_i$ then the twist operator at the origin is just $e^{i\omega/\sqrt{2}}$ where $-i\sqrt{2}\p\omega=\bar\psi\sigma_3\psi$. The $\tau$ function is now \eqn\srpon{ \tau(t_\ell)=\langle V(\infty;s_l,t_\ell) e^{i\omega/\sqrt{2}} \rangle} where the star operator is the same as in the PII case but $t(y)=-{1\over 4}\sum t_j\zeta^{2j+1}$. These expressions are, of course, rather formal. It would be worthwhile making rigorous sense of them since, at least formally, they make transparent some interesting properties of the 2D gravity partition function. For example, a corollary of the operator formalism is that a $\tau$ function satisfies certain Virasoro constraints. Applying \stress\ to this case with $A,T$ obtained from \dforz\lasaga\ we find: \eqn\vircni{ \eqalign{ L_n\tau&={1\over 4}\delta_{n,0}\tau\qquad n\geq-1\qquad{\rm for~ PI}\cr L_n\tau&=0\qquad \qquad n\geq-1\qquad{\rm for~ PII}\cr} } Similarly since commutation with $J=\bar\psi\sigma_3\psi$ rotates the fermions $\psi_{1,2}$ oppositely we may imagine that there is an identity like \eqn\cohi{ e^{\int_\CC\bigl(t^{(2)}(y)-t^{(1)}(y)\bigr)J(y)dy} V(a;s,t^{(1)}_\ell) e^{-\int_\CC\bigl(t^{(2)}(y)-t^{(1)}(y)\bigr)J(y)dy} =V(a;s,t^{(2)}_\ell)} and hence we would have \eqn\cohii{ \tau(t_\ell)=\langle t_\ell-\bar t_\ell|\Omega_{\bar t}\rangle} where $|\Omega_{\bar t}\rangle$ is the state created by the star operator at $\bar t$, and $\langle t_\ell-\bar t_\ell|$ is a coherent state for the scalar $\omega$ where only the odd oscillator modes are excited. (This follows since $t(y)$ involves only odd powers of $y$.) Combining with \vircni\ we may obtain, formally, expressions similar to those in \fukuma\dvv\morozovi\morozovii\ . Finally, let us compare with the fermion formalism of the matrix model described in section 2. There we found that the partition function at couplings $t_\ell$ can be expressed in terms of a correlation function in the ground state for couplings $\bar t_\ell$ according to: \eqn\mtrxtaus{ \eqalign{ \tau_{PI}&=\biggl\langle e^{\int_\IR\sum_{\ell}\bigl(t_\ell-\bar t_\ell\bigr)\lambda^{\ell +1/2}\psi^\dagger\psi(\lambda) d\lambda} \biggr\rangle\cr \tau_{PII}&=\biggl\langle e^{i\int_\IR\sum_{\ell}\bigl(t_\ell-\bar t_\ell\bigr)\lambda^{2 \ell +1}\psi^\dagger\psi(\lambda) d\lambda} \biggr\rangle\cr} } DERIVE? natural explanation of stokes data etc . For the $(2l-1,2)$ models we have a star operator at infinity and a twist operator at the origin. We may choose the contours of the star operator to converge to the branch cut for the twist operator at the origin. Then applying \gobpe\ to the case where we vary the parameter $x$, (one of the parameters in $T(\zeta)$) we see that the tau function for the string equations can be computed using the star operator, \eqn\grvstr{\eqalign{ V_{S,x}(a)=e^{i(\phi_1(a)+\phi_2(a))/2}\prod_k & exp\biggl[s_{2k+1}\int^a_{\CC_{2k+1}}e^{2T(y)}\psi_1\bar\psi_2(y)\biggr]\cr &\qquad exp\biggl[s_{2k}\int^a_{\CC_{2k}}e^{-2T(y)}\psi_2\bar\psi_1(y)\biggr]\cr} } and a twist operator $\varphi$, as the two-point function: \eqn\taustr{ \tau(x)=\langle V_{S,x}(\infty)\varphi(0)\rangle } Very similar considerations hold for the string equations in the PII family, the main difference being that we do not have to take a squareroot of the spectral parameter, hence there is no twist field. \subsec{Virasoro action, heuristic origin from matrix model} \newsec{Grassmannians, Krichever's construction, and all that} When the connection to the KDV hierarchy was discovered in 2D gravity the theory of the KDV hierarchy, as presented in \ref\BMN{Dubrovin, Matveev, and Novikov, ``Non-Linear Equations of Korteweg-De Vries Type, Finite-Zone Liner Operators, and Abelian Varieties,'' Russian Math Surveys, {\bf 31} (1976)59} \ref\qpKdV{E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, ``Transformations Groups for Soliton Equations,'' I. Proc. Japan Acad. {\bf 57A}(1981)342; II. Ibid., 387;III. J. Phys. Soc. Japan {\bf 50}(1981)3806;IV. Physica {\bf 4D}(1982)343;V. Publ. RIMS, Kyoto Univ. {\bf 18}(1982)1111;VI. J. Phys. Soc. Japan {\bf 50} (1981)3813;VII. Publ RIMS, Kyoto Univ. {\bf 18}(1982)1077.} \ref\segal{G. Segal and G. Wilson, ``Loop Groups and Equations of KdV Type,'' Publ. I.H.E.S. {\bf 61}(1985)1.} , was already familiar to string theorists. Indeed, one of the main points of the so-called operator formalism \foot{Representative papers include...} is the equivalence of the Krichever theory with the theory of free fermions on an algebraic curve. In the previous section we related the $\tau$ function of 2D gravity to a correlation function of free fermions. In this section we will see that the two theories are closely connected, but because of stokes phenomenon we require an extension of the the theory in \BMN\qpKdV\segal\ . \subsec{Quasiperiodic kdv flow and isomonodromy} We begin by explaining a remark of M. Jimbo and T. Miwa that the tau function of the quasiperiodic solutions to the kdv equations is a special case of the isomonodromic tau function \Jimboii\ . Recall that quasiperiodic kdv flow is straightline motion along $Pic_{g-1}(X)$ for a riemann surface $X$ of genus $g$, and, fixing an origin $\CL_0$ for $Pic_{g-1}$ the tau function is just \eqn\qptau{tau={Det~\bar\p_{\CL}\over Det~\bar\p_{\CL_0}} \qquad .} If $\CL\otimes\CL_0^{-1}$ has divisor $P_1+\cdots P_g-Q_1-\cdots-Q_g$ then by the insertion theorem \ref\bost{Alvarez-Gaum\'e, Bost, Moore, Nelson, Vafa} we have \eqn\qpti{\tau=\langle\psi(P_1)\cdots\psi(P_g)\bar\psi(Q_1)\cdots \bar\psi(Q_g)\rangle_{(X,\CL_0)} \qquad .} Choosing a point $P_\infty$ ``at infinity'' and a local coordinate $1/z$ near $P_\infty$ the Baker function is essentially \eqn\qpbki{\langle\bar\psi(P_\infty) \psi(z)\psi(P_1)\cdots\psi(P_g)\bar\psi(Q_1)\cdots \bar\psi(Q_g)\rangle_{(X,\CL_0)} \qquad .} Now suppose $\pi:X\to \IP^1$ is an $m$-fold branched covering. As is well-known from the theory of orbifolds \ref\zam{Al. B. Zamalodchikov, ``Conformal scalar field on the hyperelliptic curve and critical Ashkin-Teller multipoint correlation functions,'' Nucl. Phys. {\bf B285}(1987)481.} \ref\berrad{M. Bershadsky and A. Radul, ``Conformal field theories with additional $Z_N$ symmetry,'' Int. Jour. of Mod. Phys. {\bf A2}(1987)165.} \ref\lance{L. Dixon, D. Friedan, E. Martinec and S. Shenker, ``The conformal field theory of Orbifolds,'' Nucl. Phys. {\bf B282}(1987)13.} \ref\vafa{ S. Hamidi and C. Vafa, ``Interactions on Orbifolds,'' Nucl. Phys. {\bf B279}(1987)465} \knizhnik\ we can represent one weyl fermion on $X$ by $m$ weyl fermions $\psi_\alpha,\bar\psi_\alpha$ on $\IP^1$, where $\alpha=1,\dots, m$ labels the sheets, in the presence of twist operators at the branch points. For example, denoting the branch points on $\IP^1$ by $b_i$ we have the twist field correlator $Det~\bar\p_{\CL_0}=\langle \prod\varphi_i(b_i)\rangle\qquad .$ Similarly, if $P_i$, $Q_i$ lie on branches $\alpha_i,\beta_i$, respectively then the Baker function descends to the ``Baker framing'' \eqn\qpbkfr{\tilde Y_{\alpha\beta}(\lambda)= \langle\bar\psi_\alpha(\infty) \psi_\beta(\lambda)\psi_{\alpha_1}(\pi(P_1))\cdots \psi_{\alpha_g}(\pi(P_g))\bar\psi_{\beta_1}(\pi(Q_1))\cdots \bar\psi_{\beta_g}(\pi(Q_g))\prod_i\varphi_i(b_i) \rangle_{\IP^1} \qquad ,} where $\lambda=\pi(z)$. The actual Baker framining $Y$ differs from $\tilde Y$ by an invertible diagonal matrix with an essential singularity at infinity of the form $\sim exp\bigl(\sum t_j z^j\bigr)$. Regarding the fermion insertions as special cases of twist operators and following the reasoning of the previous section we see that $Y$ satisfies a liner ODE in $\lambda$ with regular singularities at $\pi(P_i)$. Clearly we have isomonodromic deformation in these parameters, and, as we have seen, the $\tau$ function is just \eqn\qptii{ \langle \psi_{\alpha_1}(\pi(P_1))\cdots \psi_{\alpha_g}(\pi(P_g))\bar\psi_{\beta_1}(\pi(Q_1))\cdots \bar\psi_{\beta_g}(\pi(Q_g))\prod_i\varphi_i(b_i) \rangle_{\IP^1} \qquad ,} but this is the same as \qpti\ which is the tau function of the krichever theory. From this discussion we conclude that the required generalization of the krichever theory is the generalization from twist operators to star operators. In the next section we will arrive at the same conclusion from a different point of view. \subsec{Noncommutative Burchnall-Chaundy-Krichever theory} BCK THEORY: based on [P,Q]=0. here [P,Q]=1. noncommutative rs? put hbar, take limit to relate the two. impossible to diagonalize simultaneously. must look to another formulation. go to lax pair. Compare us, novikov, is his conjecture right? how do we see his transcendental equation? other papers. {\subsec Grassmannians} general discussion of tau functions-- is it the most general solution to kdv hierarchy? also--compare with segal-wilson tau function, argue that it is NOT in their class because of the baker function. note that already you can see this in pure gravity, where ``baker function'' is just an airy function and exhibits stokes phenomenon. nevertheless, free fermion interpretation in terms of ``star'' operators etc. shows that there must be a sense in which this tau lies on some kind of completion of the orbit. It is well known that in the case of almost periodic solutions to the KdV hierarchy the tau function is properly thought of as a section of a determinant line bundle \segal\ and that this determinant line bundle is just the vacuum bundle for free fermions defined on the algebraic curve associated to the solution $u(x)$ via the BCK theory. We can argue that, with some modifications, this picture continues to hold for the tau functions associated to the isomonodromy problems discussed above. Recall that in the operator formalism we choose a disc surrounding some point $P$ and define a local Hilbert space on a circle surrounding that point. The partition function of fermions on the surface may be regarded as a function of the line bundle $\CL\to\Sigma-\{P\}$ of which the fermion wavefunctions are holomorphic sections. By considering the restriction of these sections to a circle surrounding $P$ we obtain a subspace $W\subset L^2(S^1)$ defining an element of the Grassmannian. Denoting by $\Omega$ the fermion vacuum created by the geometry $\CL\to\Sigma-\{P\}$, that is, $\Lambda^{max}W$ we obtain: \eqn\taudet{ \tau_W=\langle 0|\Omega_W\rangle=\int_W d\psi d\bar \psi e^{\int \bar\psi\bar\p\psi}=Det\bar\p_W } As explained in \ref\wittgr{E. Witten, ``Conformal field theory, Grassmanians, and algebraic curves,'' Commun. Math. Phys. {\bf 113}(1988)189}\ we may easily incorporate correlation functions into this picture through use of the multiplicative Ward identities. Recall that if we want a correlator of $\langle \prod\psi(P_i)\prod\bar\psi(Q_i)\rangle$ when the fermion field lives in the line bundle $\CL\to\Sigma$ associated to $W\in Gr$ we proceed as follows. Note that the Lagrangian is invariant under $\psi\to f\psi$, $\bar\psi\to f^{-1}\bar\psi$. This does not imply that the expression \taudet\ is invariant, since the allowed space of sections is drastically changed. First of all, the boundary conditions are modified by $W\to f\cdot W$. Secondly, at zeroes and poles of $f$ the $L^2$ condition on the wavefunctions becomes an unusual normalizability constraint. Using a local analysis of the Hilbert space on a circle surrounding such a pole one can show \wittgr\ that this unusual constraint can be replaced by the insertion of an appropriate operator in the presence of ordinary fermions. As an example, let us take $f$ to have a single zero and pole. Taking into account the conformal weight one half of fermions we get \eqn\multwrd{[(df^{-1}/dz)_P(df/dz)_Q]^{-1/2} \langle\bar\psi(P)\psi(Q)\rangle_{f\cdot W}={Det\bar\p_{W}\over Det\bar\p_{f\cdot W}} } As pointed out in \wittgr\ this is the physical explanation for the relation between the Baker function and the tau function. Suppose now we have an $m$-tuplet of fermions $\psi_\alpha$ defined on a riemann surface $\Sigma$. Removing a neighborhood of a point $P\in\Sigma$, the boundary conditions on $\vec\psi$ define a subspace of $W\subset Gr(H^m)$ where $H=L^2(S^1)$ as usual \foot{The relation between the quantum field theory of a fermion on a surface $\Sigma$ which is an $m$-fold cover of the plane, and the quantum field theory of an $m$-tuplet of fermions on the plane in the presence of branch points gives a nice explanation of the construction on pp. 34-35 of \segal\ . Moreover, it shows directly why their association of a space $W$ in $Gr^{(n)}$ to an $n^{th}$ order operator coincides with the space constructed via the BCK theory in the case that the operator satisfies the stationary generalized KdV equations.}. The partition function is again $Det\bar\p_W$ as usual, and the bottom row of the Baker-Akhiezer framing describes a holomorphic section of a holomorphic $m$-plane bundle over $\Sigma-P$. The theory of multiplicative Ward identities again holds, but with some interesting generalizations. the transformations: \eqn\genmt{ \eqalign{ \vec\psi(\zeta)&\longrightarrow Y(\zeta)\vec\psi(\zeta) \cr \vec{\bar\psi}(\zeta)&\longrightarrow (Y^{-1})^{tr}(\zeta)\vec{\bar\psi}(\zeta) \cr} } preserve the action. The multiplication by $Y$ will again change the boundary conditions (by the obvious multiplicative action of $Y(\zeta)$ on $H^m$) and if $Y$ has singularities then the $L^2$ condition on the fermi wavefunctions is modified in the neighborhood of these singularities. Again, we can replace these conditions by operator insertions. If $Y$ has poles or zeroes then the operators are simply products of $\psi_\zeta$, $\bar\psi_\alpha$ and their derivatives. If $Y$ has branch cuts then we have inserted twist operators $\varphi_L$. A rigorous discussion of construction of the associated $\bar\p$ operator and its determinant for the insertion of twist operators on $\IP^1$ has been given in \ref\Palmer{J. Palmer, ``Determinants of Cauchy-Riemann operators as $\tau$-functions,'' Univ. of Arizona preprint; ``The tau function for Cauchy-Riemann operators on $S^2$,'' unpublished letter to C. Tracey.}. Finally, if $Y$ has essential singularities associated to an irregular singular point of a differential equation then we have inserted a star operator. In all cases we have $$\langle V\dots\rangle_W\sim {Det\bar\p_{Y^{-1}\cdot W}\over Det\bar\p_W} $$ Moreover, and quite generally, if we consider the multiplicative WI's for a correlation function of twist and star operators obtained by multiplying the Fermi fields by a function with a single zero and pole we obtain a ``discrete'' relation between the fermion two point function in the presence of these operators and a ratio of tau functions. We expect that one can explain in conformal-field-theoretic terms all the relations obtained by use of ``Schlesinger transformations'' in \Jimboii\ . (Schlesinger transformations themselves can be viewed as replacing a twist operator by its normal ordered product with $\psi,\bar\psi$.) From these remarks we see that the KdV flow is again induced by the infinite dimensional abelian group $\Gamma_+$ of \segal . In this case the flow has the interesting interpretation of being the flow of the renormalization group. \newsec{Conclusions} main points. future directions. I am very grateful to T. Banks, E. Br\'ezin, C. Crnkovic, M. Douglas, V. Drinfeld, I. Frenkel, H. Garland, A. Its, M. Jimbo, V. Korepin, A. Morozov, D. Pickrell, D. Sattinger, N. Seiberg, R. Shankar, S. Shatashvili, S. Shenker, and G. Zuckerman for helpful discussions. This work was supported by DOE grant DE-AC02-76ER03075. thanks cargese,kyoto,trieste \listrefs \bye \ref\frsh{D. Friedan and S. Shenker, ``The analytic geometry of two-dimensional conformal field theory,'' Nucl. Phys. {\bf B281} (1987)509; D. Friedan, ``A new formulation of string theory,'' Physica Scripta T {\bf 15}(1987)72.}. The conformal blocks of a correlation function are horizontal sections of a flat vector bundle over the moduli space of curves. A horizontal section satisfies a differential equation which essentially follows from the idea that the stress energy tensor defines a connection on the bundle. If we discuss nontrivial (nonrational) conformal field theories, e.g., those associated with nonlinear sigma models with Calabi-Yau spaces as targets, the flatness of the connection is the condition that the spacetime equations of motion are satisfied, i.e., that the appropriate generalizations of Einstein's equations are satisfied. We propose that a similar picture holds in the case of $2D$ gravity. We begin by writing the string equations as flatness conditions. These conditions are compatibility conditions for transport equations in a space parametrized by $x,T_j$, the cosmological constant and the masses associated to the $2D$ gravity model. The parameters $x,T_j$, together with the initial conditions for the nonlinear differential equations known as the ``string equations,'' are identified with the moduli of a certain class of meromorphic gauge fields on $\IP^1$. This moduli space is given a further interpretation in section five as a generalization of the moduli space of curves. The analogy to conformal field theory is developed further in section six where we interpret the transport equations in $x,T_j$ as Knizhnik-Zamalodchikov-type equations for a free fermion construction of current algebra. The novel element is that the correlators in question involve operators (dubbed ``star operators'') which are not normally considered in conformal field theory. In section seven we suggest how one might establish a direct connection between the formalism of this paper and the random matrix formulation of $2D$ gravity. It is well-known that the quantum field theory of free fermions on a curve provides an elegant framework for understanding much of the theory of the quasiperiodic solutions of the generalized KdV hierarchies. Following some observations of Gross and Migdal \GM\ , Douglas emphasized the importance of the generalized KdV hierarchies in \newD\ . This led to the suggestion \newD\bdss\ that the partition function of the matrix model might be a tau function in the sense of \ref\qpKdV{E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, ``Transformations Groups for Soliton Equations,'' I. Proc. Japan Acad. {\bf 57A}(1981)342; II. Ibid., 387;III. J. Phys. Soc. Japan {\bf 50}(1981)3806;IV. Physica {\bf 4D}(1982)343;V. Publ. RIMS, Kyoto Univ. {\bf 18}(1982)1111;VI. J. Phys. Soc. Japan {\bf 50} (1981)3813;VII. Publ RIMS, Kyoto Univ. {\bf 18}(1982)1077. } While not strictly true, we show that this conjecture is essentially correct: the partition function of 2D gravity is given by the tau function for an isomonodromic deformation problem closely related to that of the stationary KdV equations. The tau function in the quasiperiodic case admits an interesting interpretation as a function on an orbit of a loop group \qpKdV \ref\igor{I. Frenkel, ``Representations of affine Lie algebras, Hecke modular forms, and Korteweg-de Vries type equations,'' Proceedings of the 1981 Rutgers Conference on Lie Algebras and related topics. Lecture Notes in Mathematics 933, 71. Springer 1982.}\ , and it would be very interesting to find an analogous interpretation in this case. In an effort to demonstrate that the above picture is not merely useless reinterpretation of known results we have shown in appendix A how the present formalism can be used to establish some properties of the string equations which have recently become interesting in connection with the so-called ``nonperturbative violation of universality'' in matrix models. It has been repeatedly emphasized by Atiyah, Hitchin, Ward, and Witten that low-dimensional integrable differential equations and field theories should be related to higher dimensional gauge theory. The four-dimensional self-dual Yang-Mills equations are expected to play a central role in such a formulation. In appendix B we sketch some connections between those ideas and the ideas of this paper. After we completed most of this work we found that some ideas similar to those of sections 2,3 and 5, in the context of the MKdV hierarchy and the associated PII equation, have been discussed by Flaschka and Newell \ref\flasch{H. Flaschka and A. Newell, ``Monodromy and Spectrum-Preserving Deformations I,'' Comm. Math. Phys. {\bf 76}(1980)65.}. V. Korepin also pointed out to us some overlap between the remarks of section seven and those of \ref\Korep{A.R. Its, A.G. Izergin, V.E. Korepin, N.A. Slavnov, ``Differential Equations for Quantum Correlation Functions,'' Australian National University preprint; Trieste preprints, IC/89/120,107,139}. We have been informed by E. Martinec of similar progress, especially in relating the gravity partition function to a tau function \ref\martinec{E. Martinec, private communication.}. \newsec{String Equations as Flatness Conditions} Let us recall how M. Douglas wrote the general $(p,q)$ string equations in \newD\ . If $L=D^q+u_{q-2}D^{q-2}+\cdots + u_0$ is the continuum limit of a multiplication operator $f(\lambda)\to \lambda f(\lambda)$ on the orthogonal polynomials $f$ in a matrix chain model then, he argued, the continuum limit of the conjugate derivative operator $f(\lambda)\to {d\over d \lambda} f(\lambda)$ must be of the form $P=L^{p/q}_+$, where the subscript indicates we keep only the differential operator part of a pseudodifferential operator. The nonlinear differential equations $[P,L]=1$ should define nonperturbative 2D quantum gravity coupled to the $(p,q)$ minimal model of conformal field theory. Similarly, the equations for massive models coupled to 2D gravity are of the form $[P,L]=1$ where $P=\sum_p t_p L^{p/q}_+$ and the $t_p$ are the ``masses'' in the theory. Our first task will be to rewrite these equations in first order matrix form. {\it The $(2l-1,2)$ equations:} The fact that a solution to $\sum(j+\half)T_jR_j=\hbar x$ satisfies the KdV flow in $T_j$ \bdss\ is extremely surprising to those familiar with the almost periodic solutions of KdV, where analogous parameters play the role of moduli of an associated Riemann surface, while the KdV flows are (straightline) flows along the Jacobian of that surface. We will comment on this relation further below. For now we content ourselves with the following consistency check \bdss\ on \linsys\ , using the notation of Gelfand-Dickii \Gelf . Taking derivatives with respect to $x,T_k$ and assuming the KdV flow in $T_k$ we have \eqn\consis{ \eqalign{ \hbar&=\sum (j+\half)T_jR_j'\cr 0&=(k+\half)R_k'-{1\over \hbar}\sum_j(j+\half)T_j\xi_jR_{k+1}'\cr} } where $\xi_j$ are the vector fields generating KdV flow \Gelf\ and we have used commutativity of the flows. The first equation implies $\hbar \delta/\delta u=-\sum_j(j+\half)T_j\xi_j$, and substitution into the second equation gives $0=(k+\half)R_k'+ (\delta/\delta u)R_{k+1}'$, a true identity. This verifies consistency. One argument for the KdV flow was given in \bdss\ , we will give another argument below. Note especially that the argument fails for $\hbar=0$. \bigskip {\it The $(p,q)$ equations} In this case we will be somewhat less detailed. Let $L=D^q+u_{q-2}D^{q-2}+\cdots$ and consider the generalized KdV flow $[L^{p/q}_+,L]=dL/dt$. We may rewrite this, following Drinfeld-Sokolov, using the operator $$\CL=D+\Lambda+\pmatrix{0&\dots&-u_0\cr .&\dots&.\cr .&\dots&.\cr .&\dots&.\cr 0&\dots&-u_{q-2}\cr 0&\dots & 0\cr} $$ where $\Lambda$ has entries $1$ along the lower diagonal and $\lambda$ in the $1,q$ matrix element. Using the methods of \DrS\ one may construct $\CA_{q,p}(\lambda)=\Lambda^p+\cdots$ such that the generalized KdV flows are equivalent to $${\p\CL\over \p t}=[\CA(\lambda),\CL]$$ in particular, there are potentials $R_{q,p;i}$, $i=2,3,\dots q$ generalizing the $R_l$'s used above such that $$[\CA_{q,p},\CL]=\pmatrix{0&\cdots&-R_{q,p;q}'\cr 0&\cdots&-R_{q,p;q-1}'\cr .&\dots&.\cr .&\dots&.\cr 0&\dots&-R_{q,p;2}'\cr 0&\dots&0\cr} $$ Thus, as before, we may write the string equations as the flatness conditions $[\IB_{q,p},\CL]=0$ where $$\IB_{q,p}=\hbar {d\over d\lambda}+\CA_{q,p}+ \pmatrix{0&\cdots&\hbar x -R_{q,p;q}\cr 0&\cdots&\CH_{q-1}-R_{q,p;q-1}\cr .&\dots&.\cr .&\dots&.\cr 0&\dots&\CH_2-R_{q,p;2}\cr 0&\dots&0\cr} $$ and the $\CH_i$ are constants, analogous to the magnetic field of the Ising model. We may formulate the equations for massive models in the obvious way by taking linear combinations of the $\IB_{q,p}$. \bigskip {\it Unitary matrix models} {\it General Semisimple Lie algebras} In \DrS\ Drinfeld and Sokolov wrote analogues of the KdV equations associated to any Lie algebra $g$. These are again of the form $\dot\CL=[\CP,\CL]$ where $\CL={d\over dx} + q + \Lambda$, $q$ is a function taking values in the Lie algebra and $\Lambda$ is a standard element in the affine Kac-Moody algebra $\hat g$. When this algebra is realized as a loop algebra we may again modify $\CP\to \tilde\CP$ to obtain some equations of the form $0=[{d\over d\lambda} + \tilde \CP , \CL]$ and these should be the string equations for some matrix model \foot{In \newD\ Douglas suggested they would be associated to the nondiagonal minimal models. This idea has been studied in detail in \ref\kutas{D. Kutasov and Ph. Di Francesco, ``Unitary minimal models coupled to 2D quantum gravity,'' Princeton preprint, PUPT-1173.}. }. %MORE DETAIL. %ARE THE MODIFIED EQS AGAIN SELF-SIMILAR? % %HAMILTONIAN STRUCTURE, W-ALGEBRAS Flatness conditions arise very often in physics. The above interpretations suggest, e.g., that possibly one should think about a pure (holomorphic) Chern-Simons theory along the lines of \ref\witjones{E. Witten, ``Quantum Field Theory and the Jones Polynomial,'' Commun. Math. Phys. {\bf 121}(1989)351} with a suitable restriction on the fields. Such an interpretation yields a nice interpretation, e.g., of the $(p,q)$ actions of \ref\paulact{P. Ginsparg, M. Goulian, M.R. Plesser, and J. Zinn-Justin, ``$(p,q)$ String Actions,'' Harvard preprint HUTP-90/A015;SPhT/90-049} in terms of Wilson loops. However it is difficult to see why the gauge symmetry should be broken. We will comment again on this below. \newsec{String Equations and Isomonodromic Deformation} The compatibility conditions of the previous section arise naturally in a very interesting problem known as the isomonodromy problem. The theory of isomonodromic deformation has been adequately reviewed in So we confine ourselves here to a very brief description of the method. Consider a linear homogeneous differential equation \eqn\irreg{ {d\Psi\over d z}=A(z)\Psi } At an irregular singular point $a$ of order $r$ we can write $$A(z)=\sum_{n=-r}^\infty A_n (z-a)^{n-1}\qquad .$$ Assuming $A_{-r}$ is diagonalizable one can show that there is a formal solution to \irreg\ of the form \foot{We state this more carefully in the following section.}. The analytic meaning of the formal solution \stoii\ is that we can divide up a neighborhood of $a$ into sectorial domains $\Omega_k=\{d_k1$ theories. For example we can consider star operators in WZW theories. As we saw, the isomonodromic deformation equations can be interpreted as KZ equations, but these equations were at level $1$. It would be interesting to write down and study the higher level equations. The free field realizations of WZW might facilitate the construction of star-like operators in these models. It would also be very interesting to understand if there is a connection between the above ideas and the topological field theory approach of \witt\DiWit\distler\newvsq . In the latter approach the topological origin of KdV flows has proven elusive. In the present approach they are interpreted as flows defined by Knizhnik-Zamalodchikov-type equations on a certain moduli space. Perhaps some kind of asymptotic expansion of a star operator in negative powers of $x$ will provide the missing link between these formalisms. Be all this as it may, our work has implications beyond the framework of 2D gravity. We have been led to a generalization of modular geometry. Firstly, one should include the notion of star operators in general conformal field theories. Thus, the basic monodromy data of such a theory will include not only matrices representing the braid group and operator product expansions (``$B$ and $F$ matrices'') but also Stokes matrices occuring in more exotic exchange algebras than have been hitherto examined. We might therefore expect that these monodromy data will define some generalization of a modular tensor category \ref\ms{G. Moore and N. Seiberg, ``Lectures on RCFT,'' preprint RU-89-32; YCTP-P13-89}. Secondly, as we have described, the introduction of star operators leads to a notion of quantum Riemann surface and quantum moduli space. Just as there are analytical and topological aspects to modular geometry (the latter corresponding to the the modular functor point of view), one might expect that the analytic aspects associated to this generalized MTC would be related to some geometry of quantum moduli space. The geometric category representing this extended notion of a MTC might well be useful in understanding the analog of modular geometry for integrable but nonconformal models. From what we have managed to glean of the geometrical meaning of this extension we may expect to find a rich but peculiar combination of algebraic and analytic geometry. \bigskip \centerline{\bf Acknowledgements} \appendix{A}{An Application: The BMP Solutions} \appendix{B}{Twistor Correspondence} Mason and Sparling \ref\mas{Mason and Sparling, ``Nonlinear Schrodinger and Korteweg-De Vries are reductions of self-dual Yang-Mills,'' Phys. Lett. {\bf 137A}(1989)29.} have shown that solutions to the KdV equations are in one-one correspondence with holomorphic vector bundles over minitwistor space ($=T\IP^1$) which posess an extra symmetry. Since solutions to the string equations define, in particular, solutions to the KdV equations we may apply the observation of \mas\ to associate corresponding holomorphic vector bundles over $T\IP^1$. %The question is: what distinguishes these solutions of %the KdV geometrically? % %As shown in \mas\ the holomorphic vector bundle can %be characterized by defining its space of sections as follows. Consider the equations: \eqn\linsysi{\eqalign{ %\IP \Psi(\lambda,x,T_j)&=0\cr \CL \Psi(\lambda,x,T_1)&=0\cr (-2{\p\over \p T_1}+\CP_1)\Psi(\lambda,x,T_j)&=0\cr} } which are compatible when $u(x,T_1)$ satisfies the KdV equation. By taking a linear combination of these conditions and changing the framing by $$\Psi\to\pmatrix{1&0\cr H&1\cr}\Psi$$ where $H'=u$ we obtain the linear system which can be regarded as the twice dimensionally-reduced system equivalent to the SDYM's equations in a space of signature $(2,2)$ \mas . This is similar to the relation to inverse scattering theory pointed out in \ref\Belav{A. Belavin and V. Zakharov, ``Yang-Mills Equations as Inverse Scattering Problem,'' Phys. Lett. {\bf 73B}(1978)53}. Using standard twistor methods these can be regarded as defining a holomorphic structure on a two-dimensional complex vector bundle over $T\IP^1$. (In brief, pulling back the connection to $\IP^3(\IC)-\IP(\IC)$ via the standard twistor fibration, the self-duality condition becomes an integrability condition for a holomorphic 2-plane bundle, and dividing out by the lift of the symmetry $\IR^4\to\IR^4/\IR=\IR^3$ brings us down to minitwistor space $T\IP^1$. See \ref\atiyah{M.F. Atiyah, {\it Collected Works}, vol. 5, Clarendon Press, Oxford, 1988} for details.) In this paper a crucial role was played by the ``equation in $\lambda$,'' $\IP\Psi=0$. Such equations arise naturally in the twistor construction - essentially they embody the statement that the bundle on twistor space is the pullback of a bundle on spacetime. %A very direct construction of solutions to the bogomolnyi %equations is by solving the scattering problem along the %line. describe mason-sparling's extra symmetry, fixes sphere %but acts nontrivially in fibers. % % %Thus the Baker framing $\Psi$ can also be %regarded as a framing for a holomorphic vector bundle on %($T\IP^1$ ?) The key point is that $\Psi$ also satisfies %a differential equation in $\lambda$. SHOW: This equation %effectively expresses the fact that a unitary bundle has been %pulled back from spacetime up to twistor space. % %%is x on the tangent space?-no, x is not a coordinate in minitwistor %space. complexified x is coordinate on the space of sections of %%tangent bundle % %what distinguishes geometrically the stationary solutions of KdV? %(these can also be written as isomonodromy!) %role of tau function in twistor theory? \vfill\eject \bigskip \centerline{\bf Figure Captions} \noindent Fig. 1. A schematic drawing of the Riemann surface and the Krichever line bundle defined by the BA framing, in the neighborhood of a branch point. \bigskip \noindent Fig. 2. A schematic drawing of a framing defined by the equation in $\hb$ in the neighborhood of a branch point. The oscillations are along the directions in $V_\lambda$ defined by the framing in fig. 1. \bigskip \noindent Fig. 3. An example of conjugate Stokes lines in the case where the branch points are all real. In this case the surface has genus two. \bigskip \noindent Fig. 4. A schematic view of a choice of contours for the star operator. \bigskip \noindent Fig. 5. A simple Fermi sea. \bigskip \noindent Fig. 6. Three large regions abutting infinity and a turning point $\xi_i$. \bigskip \noindent Fig. 7. Stokes lines for the case $l=1$, $u\sim +(-x)^{1/2}$ for $x\to-\infty$. We have also indicated the Stokes parameter associated with each region. \bigskip \noindent Fig. 8. Stokes lines for $l=1$, $u\sim -(-x)^{1/2}$, $x\to-\infty$. \bigskip \noindent Fig. 9. Stokes lines for $l=2$, $u\sim x^{1/3}$, $x\to+\infty$. \bigskip \noindent Fig. 10. Stokes lines for $l=2$, $u\sim x^{1/3}$, $x\to-\infty$. \bigskip \noindent Fig. 11. Stokes lines for $l=4$, $u\sim x^{1/5}$, $x\to+\infty$. \bigskip \noindent Fig. 12. Stokes lines for $l=4$, $u\sim x^{1/5}$, $x\to-\infty$. \listrefs \bye