\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