%Paper: hep-th/9309068
%From: "Simon Dalley" <dalley@puhep1.Princeton.EDU>
%Date: Sat, 11 Sep 93 20:25:18 EDT
%Date (revised): Sun, 12 Sep 93 13:35:09 EDT
%Date (revised): Sun, 12 Sep 93 13:55:25 EDT

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%    The Kosterlitz-Thouless Phenomenon on a Fluid Random Surface.
%    author - Simon Dalley.
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\input phyzzx


\sequentialequations

\def\rk#1{$^{#1}$}                        %Real Nucl. Phys. format
\overfullrule=0pt
\catcode`\@=11
\def\mm{matrix model}
\def\ns{non-singlet}
\def\dsl{double-scaling limit}
\def \F{\phi}
\def \PH{\hat \psi}
\def \PE{\psi^\epsilon}
\def \E{\epsilon}
\def \NU{\nu}
\def \L{\lambda}
\def \B{\beta}
\def \T{\theta}
\def \P{\psi}
\def \D{{\Delta}}
\def \DR{{\Delta}_{R}}
\def \W{\omega}
\def \S{\sigma}
\def \sh{\; {\rm sh} \;}
\def \ch{\; {\rm ch} \;}
\def \th{\; {\rm th} \;}
\def \cth{\; {\rm cth} \;}
\def \DM{ {\partial \over {\partial \mu}}}
\def \O{{\cal O}}
\def \CO{{\cal O}}
\def \G{\Gamma}
\def \tT{{\tilde T}}
\def \tq{{\tilde q}}
\def\NP{{\it Nucl. Phys.\ }}
\def\AP{{\it Ann. Phys.\ }}
\def\PL{{\it Phys. Lett.\ }}
\def\PR{{\it Phys. Rev.\ }}
\def\PRL{{\it Phys. Rev. Lett.\ }}
\def\CMP{{\it Comm. Math. Phys.\ }}
\def\JMP{{\it J. Math. Phys.\ }}
\def\JTP{{\it JETP \ }}
\def\JP{{\it J. Phys.\ }}
\def\IJMP{{\it Int. Jour. Mod. Phys.\ }}
\def\Mod{{\it Mod. Phys. Lett.\ }}
\def\NC{{\it Nuovo Cimento \ }}
\def\PRep{{\it Phys. Rep.\ }}
%
\def\CO{{\cal O}}
\def\e{\epsilon}
\def\td{two-dimensional}
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\def\KT{Kosterlitz-Thouless}
\def\eqaligntwo#1{\null\,\vcenter{\openup\jot\m@th
\ialign{\strut\hfil
$\displaystyle{##}$&$\displaystyle{{}##}$&$\displaystyle{{}##}$\hfil
\crcr#1\crcr}}\,}
\catcode`\@=12


\REF\bkta{J. M. Kosterlitz and D. Thouless, \JP {\bf C6} (1973) 1181.}
%
\REF\bktb{V. L. Berezinskii, \JTP {\bf 34} (1972) 610.}
%
\REF\bktc{J. M. Kosterlitz, \JP {\bf C7} (1974) 1046.}
%
\REF\bal{S. Balibar and B. Castaing, Surf. Sci. Rep. {\bf 5} (1985) 87.}
%
\REF\lush{M. Lusher, G. Munster, and P. Weisz, \NP {\bf B180} (1981) 1;
C. Itzykson, M. E. Peskin, and J. B. Zuber, \PL {\bf B95} (1980) 259;
A. Hasenfratz, E. Hasenfratz, and P. Hasenfratz, \NP {\bf B180} 353.}
%
\REF\ian{B. Sathiapalan, \PR {\bf D35} (1987) 3277; I. Kogan, JETP Lett.
{\bf 45} (1987) 709.}
%
\REF\bish{D. J. Bishop and J. R. Reppy, \PRL {\bf 40} (1978) 1727.}
%
\REF\burt{W. K. Burton, N. Cabrera, and F. C. Frank, Phil. Trans. Roy. Soc.
(London) {\bf 243A} (1951) 299.}
%
\REF\dav{F. David, \NP {\bf B257} (1985) 45; V. Kazakov, I. Kostov, and
A. Migdal, \PL {\bf B157} (1985) 295.}
%
\REF\kaza{V. Kazakov, \PL {\bf A119} (1986) 140; V. Kazakov and D. Boulatov,
\PL {\bf B186} (1987) 379.}
%
\REF\moo{G. Moore, hepth/9203061, YCTP-P1-92.}
%
\REF\hsu{D. Kutasov and E. Hsu, \NP {\bf B396} (1993) 93.}
%
\REF\jos{J. V. Jos\'e, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson,
\PR {\bf B16} (1977) 1217; D. J. Amit, Y. Y. Goldschmidt, and G. Goldin,
\JP {\bf A13} (1980) 585.}
%
\REF\mea{S. Dalley, \Mod {\bf A7}, 1651 (1992)}
%
\REF\wein{D. Weingarten, \PL {\bf 90} (1980) 280.}
%
\REF\stan{S. Leibler and L.Peliti, Phys. Rep. {\bf C184} (1989) 101 and
references therein.}
%
\REF\pola{A. Polyakov, \NP {\bf B268} (1986) 406.}
%
\REF\mig{D.~J.~Gross and A.~A.~Migdal, \PRL {\bf 64} (1990) 717;
M. Douglas and S.~Shenker, \NP {\bf B335} (1990) 635;
E.~Br\'ezin and V.~Kazakov, \PL {\bf 236B} (1990) 144.}
%
\REF\oth{E. Marinari and G. Parisi, \PL {\bf 240} (1990) 375; {\bf B247}
(1990) 537; J. Ambjorn, J. Greensite, and S. Varsted, \PL {\bf B249} (1990)
411; S. Dalley, C. Johnson, and T. Morris, \NP {\bf B368} (1992) 655.}
%
\REF\ind{S. Das, S. Naik and S. Wadia, \Mod {\bf A4} (1989) 1033;
J. Polchinski, \NP {\bf B324 } (1989) 123;
T. Banks and J. Lykken, \NP {\bf B331} (1990) 173;
A. Tseytlin, \IJMP {\bf A5} (1990) 1833.}
%
\REF\GKa{D. J. Gross and I. R. Klebanov, \NP {\bf B344} (1990) 475;
\NP {\bf B354} (1991) 459.}
%
\REF\KS{I. R. Klebanov and L. Susskind, \NP {\bf B309}, 175 (1988).}
%
\REF\igo{I. Klebanov, in {\em String Theory \& Quantum Gravity '91} (World
Scientific 1992) p. 30.}
%
\REF\polb{A. Polyakov, \PL {\bf B103} (1981) 207.}
%
\REF\kpz{V. Knizhnik, A. Polyakov, and A. Zamolodchikov, \Mod {\bf A3} (1988)
819; F. David, \Mod {\bf A3} (1988) 1651; J. Distler and H. Kawai, \NP {\bf
B321} (1989) 509.}
%
\REF\sch{C. Schmidhuber, hepth/9212075, CALT-68-1817.}
%
\REF\fro{J. Ambjorn, B. Durhuus, and J. Frohlich, \NP {\bf B257} (1985) 433.}
%
\REF\kazc{V. Kazakov and A. Migdal, \NP {\bf B311} (1989) 171.}
%
\REF\kazb{D. Boulatov and V. Kazakov, \IJMP {\bf A8} (1993) 809.}
%
\REF\GKb{D. Gross, I. Klebanov, and M. Newman, \NP {\bf B350} (1991) 621.}
%
\REF\ivana{I. Kostov, \PL {\bf 297} (1992) 74. }
%
\REF\ivanb{I. Kostov, \NP {\bf B376} (1992) 539 and references therein.}
%
\REF\gin{P. Ginsparg, LA-UR-91-999-mc.}
%
\REF\bipz{E. Br\'ezin, C. Itzykson, G. Parisi, and J-B. Zuber, Comm. Math.
Phys. {\bf 59} (1978) 35.}
%
\REF\meb{S. Dalley, unpublished.}
%
\REF\gau{M. Gaudin, unpublished.}
%
\REF\mus{N. Muskhelishvili, {\em Singular Integral Equations} (Nordoff NV,
Groningen, Netherlands, 1953).}
%
\REF\mat{G. Moore, N. Seiberg, and M. Staudacher, \NP {\bf B362} (1991) 665.}
%
\REF\kazd{V. Kazakov and A. Migdal, PUPT-1322.}
%
\REF\bou{D. Boulatov, \Mod {\bf A8} (1993) 557.}
%
\REF\mec{S. Dalley, unpublished (to appear).}
%



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$\displaystyle{##}$&$\displaystyle{{}##}$&$\displaystyle{{}##}$\hfil
\crcr#1\crcr}}\,}
\catcode`\@=12
%
%
\def\EPI{euclidean path integral}
\def\sp{\,\,\,\,}
\def\oh{{1 \over 2}}
\def\b{\beta}
\def\a{\alpha}
\def\l{\lambda}
\def\om{\omega}
\def\m{\mu}
\def\s{\sigma}
\def\ap{\alpha'}
\def\tf{two-point function}
\def\rs{random surfaces}
\def\qg{quantum gravity}
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\def\r{\rho}
\def\DRT{{\tilde{\Delta}}_{R}}
\def\MT{\tilde{M}}

\nopagenumbers

{\baselineskip=12pt
\line{\hfil PUPT-1417}
\line{\hfil hep-th/9309068}
%\line{\hfil August 1993}
}
\title{The Kosterlitz-Thouless Phenomenon on a Fluid Random Surface}
\author{Simon Dalley\foot{Address after September 1993: Department of Physics,
Theoretical Physics, Oxford University, Oxford OX1 3NP, United Kingdom.}}
\JHL
\abstract
%{\baselineskip=12pt
The problem of a periodic scalar field on a two-dimensional lattice with
fluctuating local connectivity is studied, using matrix model techniques, with
the inclusion of vortices in the action. This represents, for example, $c=1$
string theory at finite temperature or (under duality) the SOS approximation
for thermal fluctuations of a fluid surface in a periodic potential. By
reduction to an eigenvalue problem, an exact solution is obtained at a
specific radius in the vortex-plasma phase on genus zero as as a function of
the vortex chemical potential --- it represents the first exact solution of a
non-critical string theory with non-tree-like embeddings ---
vortex--anti-vortex pairs forming cuts in the scalar field which disorder it
in the infra-red limit $(\gamma_{\rm str} = -1/2)$. The system represents a
Coulomb gas of charges in the presence of 2D quantum gravity. The class of
matrix models employed here on the circle, of Weingarten
type, is expressible as a lattice gauge theory with matter in the adjoint
representation and easily generalizes to $c>1$ string theory and fluid
membranes with rigidity; the prospects for exact solutions are briefly
addressed.
\endpage


\pagenumbers
\vsize=8.9in
\hsize=6.5in
\centerline{\caps 1. Introduction}
\bigskip

The Kosterlitz-Thouless (KT) phenomenon [\bkta, \bktb, \bktc]
has had far reaching implications
for our understanding of effectively two-dimensional (2D) physical systems.
Its myriad applications include those to 2D superfluids, crystal growth [\bal],
lattice gauge theories [\lush], and superstring theories [\ian].
The two principle pictures for the KT transition, related by
duality, are those of {\em vortex unbinding} and {\em surface roughening}. It
is through the former that KT theory received its first quantitative
experimental verification in 2D ${\rm He}^3$ [\bish], while roughening of solid
surfaces has been known for many years to materials scientists [\burt]
and is an
important technological consideration in determining rate and mode of crystal
growth.


While this phenomenon is usually studied on a fixed regular lattice,
it is known
that many aspects of 2D statistical mechanics become more tractable when
studied on lattices for which local
connectivity is itself a dynamical variable,
principally because they may be mapped to solvable large-$N$ matrix field
theories [\dav].
In the language of two-dimensional quantum field theory this corresponds
to the introduction of
2D quantum gravity. The lack of crystalline structure makes these systems
idealised models of 2D fluids or fluid surfaces. They seem to display the same
critical phenomena as on fixed regular lattices but with mild modification of
critical exponents; for example, the Ising model transition is third order
rather than second [\kaza]. The possibility of finding some non-perturabtive
results for the KT vortex plasma motivates a study of this  phenomenon on such
fluid random surfaces. In this paper, using the example of a circular scalar
field, it is shown by exact solution in the continuum limit how the gas of
unbound vortices leads to
topological order in two dimensions, making no assumptions about diluteness,
save the restriction to vortex winding number $\en =1$ (the most relevant).
Recently a series solution of the sine-Gordon model coupled to 2D gravity
has been proposed by Moore [\moo]. This solution has not yet been proved
(G.Moore, private communication) but is very likely correct. On a fixed area
surface it is convergent below the zero sine-Gordon coupling limit of the
roughening temperature, and can be summed in the infra red (IR) limit  to show
that the sine-Gordon field decouples [\hsu].
The key improvement of these (dual) models over those
defined on regular lattices is that the renormalised vortex/sine-Gordon
coupling need not be assumed small as it must be in the approximate
renormalisation group
approach [\bktc, \jos].
In the unbound/smooth phase the renormalised coupling becomes
larger with increasing length scale, while even experimentally the periodic
potential governing crystal growth is, with the exception perhaps of
Helium crystals [\bal], not small\foot{At the roughening temperature itself,
the renormalised coupling equals the bare (atomic) coupling.}.

Another, more direct motivation for studying random surface models, and that
closest to the author's, is provided by non-critical string theory. The
one-dimensional matrix models that will be employed to study KT on the circle
have higher dimensional generalisations [\mea], extensions of Weingarten's
complex matrix model [\wein],
which represent $c>1$ string theory. They seem more
promising as regards solvability than the traditional Hermitian matrix models
 -- the model solved in this paper represents the first exact solution
obtained for non-critical string theory with non-tree-like embeddings. They
may be re-formulated as certain lattice gauge theories and also facilitate
the coupling to extrinsic curvature which is of interest in the study of
amphiphilic membranes [\stan] and perhaps QCD [\pola].
Matrix models can also be used
as a starting point for the study of non-perturbative string theory [\mig,
\oth] and,
although only the genus zero case is dealt with here, one can in principle ask
about the non-perturbative effects on the KT phenomenon.

It is well-known that non-critical string theory in one dimension is like
critical string theory in two dimensions, the conformal factor playing the
role of an extra dimension [\ind]. As such, strings cannot exhibit transverse
oscillations, there being only one field theoretic degree of freedom (centre
of mass), and the situation is not much different from particle physics. To
probe the fact that one is dealing with an extended object the simplest thing
to do is either compactify the embedding coordinate or discretize it (or both).
Compactifying introduces winding modes which cause the KT vortex unbinding
transition discussed in later sections. An intrinsically continuous string
embedded on a lattice is also different from a particle on a lattice
because, while each bit of the string is discretely constrained, the centre
of mass is not; the sine-Gordon potential analogues a lattice and the KT
(de-)roughening transition occurs if the lattice spacing is made {\em too}
large [\KS, \GKa].

This paper is organised as follows: in section 2 some introductory remarks
on the Liouville and Hermitian matrix model formulations are given to set
notation and basic of notions scaling etc on the fluctuating geometry;
section 3 introduces the matrix
model solved
in this paper and recalls what is already known about it; the solution in the
vortex-plasma phase on genus zero (at
half the self-dual radius, $r=\half$) is obtained in section 4 by an
eigenvalue saddlepoint analysis; section 5 deals with extensions of the work,
telling of the extent to which $r \neq \half$ and winding number $\en >1 $ can
 be treated -- what
results exist are limited to the infra-red (IR) limit only -- and discusses
the $c>1$ models and extrinsic geometry; summary
and conclusions
are given in section 6.
\bigskip
\centerline{\caps 2. Scalar Field + 2D Quantum Gravity.}
\bigskip

This section introduces the Liouville, dynamical lattice and matrix model
 descriptions of
2D quantum gravity coupled to a periodic scalar field with momentum and winding
mode operators present in the action\foot{A useful introduction to $c=1$ 2D
gravity is ref.[\igo].}.
The continuum limit of the lattice with fluctuating local connectivity and
total number of vertices is 2D quantum general relativity on a fixed topology,
being described by a (Euclidean) intrinsic metric $g_{\mu \nu} (z_{1}, z_{2})$
and, in the present case, a compact scalar field $X(z_{1}, z_{2}) \equiv X
+2\pi r$, which in conformal gauge on genus zero, $g_{\mu \nu} =\hat{g}_{\mu
\nu} {\rm e}^{\F /\sqrt{2}}$, obey the Liouville-type action [\polb]
for some fixed
metric $\hat{g}_{\mu \nu}$; $S=S_{L} + S_{M}$ where
$$\eqalign{S_{L}& = \int d^2 z
\sqrt{\hat{g}} \left[ {1\over 8\pi}(\nabla \F)^2
+ {\DR \over 16\pi} {\rm e}^{\sqrt{2}\F} + {\sqrt{2}\over 4\pi}\F
{R}(\hat{g}) \right]\ ,\cr
S_{M}& = \int d^2 z \sqrt{\hat{g}} \left[ {1\over 4\pi} (\nabla X)^2
+\sum_{n} m_{n}^{R} {\rm e}^{{\b}_{n} {\F}} {\rm e}^{inX/r}
+ \sum_{\en} {\cm}_{\en}^{R} {\rm e}^{{\tilde{\b}}_{\en} \F} {\rm e}^{i\en
\tilde{X} r} \right]\ .\cr}\eqn\lou$$
$\DR$ is the (renormalised) cosmological constant coupling to the total area
$\int {\ e}^{\sqrt{2} \F}$ and can be thought of as a chemical potential for
constituent molecules in the fluid picture. $\tilde{X}$ is the dual scalar
field $\partial_{a} \tilde{X} = \epsilon_{ab} \partial_{b} X$, ${\cm}^{R}$'s
couple to winding (vortex) modes, $\b_{n} = (\sqrt{2} - |n|/r\sqrt{2})$ gives
the gravitational screening [\kpz] of the momentum modes\foot{The conformal
dimension of ${\rm e}^{inX/r}$ is $\D_{0} = n^2 /4r^2$ and gravitational
dimension $\D_{g} = \sqrt{\D_{0}}$.}, etc. The duality transformation
is non-tilde $\to$ tilde and $r \to 1/r$. As recently emphasised in ref.[\sch],
 the action $S$ is actually only generally covariant to leading order in
$m$ and $\cm$. Higher order requires the addition of further operators which
in the matrix model, because it is covariant by construction, are automatically
included.

If we consider the topology of a disc or a sheet with p.b.c. (torus) then $X$
may be thought of as the height fluctuation of the surface above a flat
configuration, similar to the solid-on-solid (SOS) approximation. (The fact
that $X$ is periodic makes no difference if all $\cm^{R} =0$ since one just
counts $X$ cumulatively). On a sphere one may still use this picture
meaningfully to talk of bulk quantities. In the SOS sense the $(\nabla X)^2$
term represents the increase in energy due to surface tension, set to one
by convention in \lou\ , for a small out-of-plane fluctuation. Of course the
in-plane positions of molecules in a fluid surface described by \lou\ are
ambiguous and furthermore there is no perpendicular crystaline structure
to play the role of sine-Gordan potential in real interfaces -- fluid-fluid
interfaces are always rough while fluctuation of fluid-solid interfaces are
dominated by the solid.

As a representation of the fluctuating lattice itself, it is convenient
 to use a simplicial complex [\dav, \fro] (fig.1), say triangulation, with the
usual identification of curvature and area per vertex, which in the present
case is embedded in the line parametrized by $X$. The Feynman diagrams
generated by the $N$x$N$ Hermitian matrix quantum mechanics with action
$$ S_{\Phi} = \int D^{N^2}\! \Phi (X)\ {\rm exp}\left( -N\int dX \Tr
\left[ \half {\partial \Phi \over \partial X}{\partial \Phi \over \partial X}
+ \half \Phi^2 - {g\over 3!} \Phi^3 \right] \right)\ \eqn\hmm$$
enumerate the dual lattices, with $\log{g_{c}/g} \sim \D$ where $g_{c}$ is the
critical coupling at which the sum over diagrams at a given order in the
$1/N$ expansion diverges in $g$ --- the continuum limit [\kazc].
An operator coupling
to $m_{n}$ is
$$ \int dX {\rm e}^{inX/r} \Tr [\Phi^2 (X)]\ ,\eqn\punc$$
where $\Tr \Phi^2$ creates a puncture at an arbitrary link on Fig.1. An
operator coupling to $\cm_{\en}$ is not so straightforward in the Hermitian
matrix model and this has been the main reason preventing the use of matrix
models to study the KT phenomenon on the circle. This problem will be solved
in the next section by using a more appropriate matrix model.
In comparing answers with the renormalisation group calculations on a fixed
regular lattice, one must note that matrix models are unable to probe
the local properties of the surface. There is no intermediate renormalisation
group scale in the answers it gives, one having essentially integrated over
all modes from the UV cut-off $\d$ (lattice spacing) to the IR cut-off
given by the area $A$ of the compact surface. The  continuum limit
\lou\ is achieved by tuning $\D \to 0$ so that large lattices `dominate' the
ensemble of all lattices, while at the same time tuning the matter to
criticality in the conventional sense. For applications to high-energy
physics it makes sense to go further, introducing renormalised parameters
which are held fixed as $\d \to 0$: $\D = \DR \d^2; A=A_R /\d^2 ; \cm_{\en} =
\cm^{R}_{\en} \d^{\en r -2}$. The power of matrix models is that they
yield exact answers as a function of $\DR$ and $\cm^R$ in the continuum limit;
one may also calculate the finite $\d$ corrections if desired.
\bigskip
\centerline{\caps 3. Matrix Model of Vortices.}
\bigskip

In the presence of winding modes $\cm$, as the radius is reduced to
$r=2$ the $\en = \pm 1$ modes become relevant and, according to the KT
picture, are tightly bound in $\pm$ pairs for $r>2$ but form a plasma for
$r<2$. In the Hermitian matrix model \hmm\ , which implicitly sums over all
$\en$, the non-singlet states under the $SU(N)$ symmetry $\Phi \to
\Omega^{\dagger} \Phi \Omega$ of the action have been suggested as the vortex
configurations [\GKa]. The evidence for this is that the singlet sector has no
phase transition for all $r$, while the adjoint sector can be argued to
describe configurations with one $\en =1$ vortex--anti-vortex pair and
becomes relevant at $r=2$ [\GKa, \kazb]. The precise way in which the higher
representations account for multiple pairs and $\en \neq 1$ is unknown. Given
the far greater power matrix models tend to display over the continuum
Liouville approach, it is a little disappointing that here they appear unable
to surpass it. Technically speaking the problems arise from having to perform
angular integrals at large $N$, the precursor to similar problems occuring
at $c>1$, and the situation seems quite hopeless. For these reasons a
different matrix model was introduced in ref.[\mea], for which identification
and manipulation of vortex operators is much easier. It is based on the idea,
first suggested by Weingarten for Nambu-Goto strings [\wein], of linearizing
lattice gauge theory. The definition is easily given for the $c$-dimensional
matrix model. On a $c$-dimensional hypercubic lattice with oriented links
$\pm l$ one associates a complex $N$x$N$ matrix $M(l)$, $M(-l) = M^{\dagger}
(l)$, to each link $l$. The original Weingarten partition function reads
$$ Z_{W} = \int_{-\infty}^{+\infty} \prod_{l} \prod_{i,j=1}^{N} {1\over \pi}
d[{\rm Re}M_{ij}(l)]d[{\rm Im}M_{ij}(l)] {\rm e}^{-S_{W}}\ ,\eqn\wein$$
where
$$S_{W}= \sum_{l} \Tr [M^{\dagger}(l)M(l)] - {\lambda \over N} \sum_{\rm plaq}
\Tr [M(l_{1})M(l_{2})M^{\dagger}(l_{3})M^{\dagger}(l_{4})]\ .\eqn\weing$$
These two terms are illustrated in fig.2 for $c=2$. `plaq' indicates a
sum over all oriented plaquettes on the lattice. The action has the usual
gauge symmetry associated with unitary transformations $U(l,l+1)$ at the
sites connected by the links, $M(l) \to U^{\dagger}(l-1,l)M(l)U(l,l+1)$.
When one expands \wein\ in $\lambda$ and performs the Gaussian $M$ integrals
one finds the result
$$ Z_{W} = \sum_{\cal{S}} N^{\chi} \lambda^A\ ,\eqn\exp$$
where $\cal{S}$ denotes all closed surfaces made from plaquettes, these
having been glued together pairwise along edges. $\chi$ is the Euler number
and $A$ the number of plaquettes. In this way \exp\ is a discrete version of
the partition function of first-quantized Nambu-Goto strings. Evidently this
model cannot be used for $c=1$ since the plaquettes require and embedding
dimension $c\geq 2$.

One can endow the worldsheets of this model with an intrinsic metric by adding
the following piece to the action
$$S_{C} = - \sum_{l} {\kappa_{1}\over N} \Tr [M^{\dagger}(l)M(l)
M^{\dagger}(l)M(l)] - \sum_{<l'l>} {\kappa_{2} \over N} \Tr [M^{\dagger} (l)
M(l)M(l')M^{\dagger}(l')]\ ,\eqn\col$$
where $l$ and $l'$ share a site, and whose terms are also illustrated in
fig.2. $S_{C}$ thus consists of collapsed plaquettes. If one now expands in
$\lambda, \kappa_{1}$, and $\kappa_{2}$, contracting plaquette edges
pairwise as before, the new feature is the presence of collapsed plaquettes
sewn into the surface. One may interpret the latter effect as the insertion
of surface elements of extrinsic area zero but intrinsic area one; the
intrinsic and extrinsic areas of plaquettes in $S_{W}$ are both one. More
precisely, the new expansion in surfaces is equivalent to a dynamical
quadrangulation (square simplices), representing the fluctuating intrinsic
metric $g_{\mu \nu}$, with the matter fields at each vertex being integer
height variables $(h_{1}, \ldots ,h_{c})$. Since every plaquette represents
a unit element of intrinsic area, the parameter  obtained by
scaling $\lambda \to s\lambda$ and $\kappa_i \to s\kappa_i $ is the
bare worldsheet cosmological constant $\D \sim -\log{s}$. Another parameter
should set the target space scale given by surface tension.

Let us note that this matrix model may be reformulated as a more conventional
lattice gauge theory with adjoint matter, under the decomposition $M=VH$,
for unitary $V$ and Hermitian $H$. Since the $c$-dimensional hypercubic
lattice has the ratio sites:links of $1:c$, for $c>1$ a fraction of the unitary
link variables must be constrained to one by hand. The Hilbert space is also
restricted to states made from the combination $M$. At $c=1$ the model is just
the usual lattice gauge theory with positive adjoint matter $H^2$ (the
measure is correct ${\cal D}M \propto {\cal D}V\ {\cal D}H^2$). A suitable
scheme for $c=2$ uses the {\em medial} lattice, that is the lattice whose
links join centres of adjacent perpendicular links of the cubic lattice; every
other Wilson plaquette gets constrained to 1 in this scheme.

Here we now specialize to $c=1$, for which only the collapsed plaquettes
are retained;
$$ \eqalign{S_{1} =\sum_{l= -\infty}^{\infty} & \Tr [M^{\dagger}(l)M(l)]
-{\kappa_{1} \over N} \Tr [M^{\dagger}(l)M(l)M^{\dagger}(l)M(l)] \cr
& -{\kappa_{2} \over N} \Tr [M^{\dagger}(l)M(l)M(l+1)M^{\dagger}(l+1)]\
.\cr}\eqn\one$$
Note that in the SOS picture the out-of-plane fluctuations are now
constrained to lie on a one-dimensional lattice, of spacing determined
by the parameters $\kappa_1$ and $\kappa_2$ for some fixed physical scale.
In ref.[\mea] it was proven that the double scaling limit [\mig]
of this model is
equivalent to that of the singlet sector of \hmm\ for discrete $X$, which
in turn is known [\GKa] to agree at the level of partiton function and
punctual operators with \hmm\ defined for continuous $X$ provided one makes
an appropriate rescaling of parameters and the lattice spacing is not too
large. The methods used to prove these facts are not important here for they
will not be useful later. However, one finds that the critical range of
parameters for the $c=1$ continuum limit is $\kappa_1 \in [-\kappa_2 ,
\kappa_2 ]$, where $\kappa_1 = -\kappa_2$ correponds to zero lattice
spacing and $\kappa_1 = \kappa_2 $ coincides with the marginality
of the most relevant momentum operator $n=1$ in an equivalent sine-Gordon
model. In units specified by \lou\
this means that the lattice spacing can vary from zero to $\pi$. At
$\kappa_1 = \kappa_2 = \kappa$ there is a marginal operator in the action,
induced from the lattice, which presumably has a coupling given by some
generic number since there is no free parameter in \one\ other than the
bare cosmological constant $\D = \log{\kappa_{c}/\kappa}$; it is at one of the
special tachyon  momenta [\GKb].

To study the compact $c=1$ case one can make the target lattice periodic
and, since the lattice spacing can be finite and still maintain
criticality, restrict to a finite number of lattice points;
$\sum_{l=-\infty}^{\infty} \to \sum_{l=1}^{L}$, $L+1 \equiv L$. The vortex
operators are easy to identify, being the analogue of Polyakov loop variables
in lattice gauge theory. For vortex number $p$ they are
$${\cal O}_{p} \propto \Tr \left[ \prod_{l=1}^{L} M(l) \right]^p\ ,\eqn\vor$$
with anti-vortices ${\cal O}_{p}^{*}$.
They cut a hole in the fluid surface which winds $p$ times around the circle.
The problem is to solve the theory with action
$$S =  S_{1} + \sum_{\en} \cm_{\en} {\cal O}_{\en}\ ,\eqn\new$$
${\cal O}_{p}^{*} = {\cal O}_{-p}$. The correspondence with \lou\ is
$$\eqalign{ \kappa_1 < \kappa_{2} & \Leftrightarrow m^{R}_{n} =0 ,
\cm^{R}_{\en} \neq 0
\cr \kappa_1 = \kappa_2 & \Leftrightarrow m^{R}_{n\neq 1,-1} = 0,
m^{R}_1 = m^{R}_{-1} =
{\rm const.} , \cm^{R}_{\en} \neq 0\ .\cr}\eqn\cor$$
It is too difficult to solve in full generality so some justifiable
simplifications will be made: set $\kappa_1 = \kappa_2= \kappa$;
set all but one
$\cm_{\en}$ to zero. The first restriction means that the $c=1$ fixed
point is not the usual Gaussian one but a marginal deformation of it. In order
to describe the physics of topological order etc this is a rather  inocuous
modification since the UV limit will still turn out to have $c=1$
exponents.\foot{It means however that duality cannot be used to
compare (or checked) with the results of Moore [\moo] in any detail.} It also
allows a reformulation of the model, suggested by I.Kostov [\ivana],
through the
introduction of an ``inducing field'', given by Hermitian matrices $\F (l,l+1)$
at the sites;
$$\sum_{l} \Tr[M^{\dagger}(l)M(l)] + \Tr [{\F}^2 (l,l+1)]
- \sqrt{2\kappa \over N} \Tr[(M^{\dagger}(l)M(l) + M(l+1)M^{\dagger}(l+1))
\F (l,l+1)]\ .\eqn\ind$$
Performing the Gaussian integrals in $\F$ for action \ind\ gives $S_1$
\one\ . On the other hand one may integrate over $M$ to find an effective
action for $\F$. For the vortex-free case $\cm_{\en}=0$ the resulting
theories have been developed in detail by Kostov and collaborators [\ivanb] as
a loop gas formulation of $c\leq 1$ matter on a random surface. However, if
the vortex operators are at most quadratic in $M$'s then the integral over
$M$ is still Gaussian and may be performed exactly. For the rest of this
section and the next we will therefore consider
$$L=1, \cm_1 =\cm_{-1} = h, \cm_{\en \neq 1,-1} =0\ ,\eqn\spec$$
corresponding to a single periodic link\foot{This partition function  at
$h=0$ may also be used as a description of the F-model on a dynamical
random lattice, this being the most general vertex model one can define on
the fluctuating lattice. The correspondence, and indeed the use of the
$\F$ matrix, was first noted in lectures of Ginsparg [\gin].}. This is
non-critical string theory on a circle of radius $r=\half$ in the presence
of vortex winding number $\en =1$ (the most relevant). Note that this is
deep in the purported plasma phase.
$$\eqalign{Z& = \int DM\ {\rm exp}\left( -\Tr [M^{\dagger}M] + {\kappa \over N}
\Tr [M^{\dagger}MM^{\dagger}M + MMM^{\dagger}M^{\dagger}] + h\sqrt{N}
\Tr [M+M^{\dagger}] \right)\cr
& \propto \int DM\ D\F\ {\rm exp} \left( -\Tr [M^{\dagger}M] -\Tr {\F}^2 +
\sqrt{2\kappa \over N} \Tr [M^{\dagger}M\F + MM^{\dagger}\F ]
+ h\sqrt{N}\Tr [M+M^{\dagger}]\right)\cr\ .}\eqn\part$$

To integrate out $M$ first diagonalize $\F = U^{\dagger} \lambda U$
and shift $M \to UMU^{\dagger}$. Since $M$ may be diagonalized uniquely
as $M= V^{\dagger} m W$ for unitary $V,W$ and positive diagonal $m$, with
measure $DM \propto DW\ DV\ J(m)Dm$, $U$ is absorbed by shifts of the Haar
measures $DV\ DW$. Performing the Gaussian integrals over $M_{ij}$ $(i,j=
1,\ldots ,N)$,  redefining $\sqrt{2\kappa} = 1/4b_{0}$, and rescaling $\lambda
 \to 2\lambda \sqrt{N}$ and $h^2 \to 8h^2/b_{0}$, gives
$$\eqalign{Z(h,b_{0}) = \int \prod_{i=1}^{N} d\lambda_{i}\ {\rm exp}
 ( &-N\sum_{i} 2\lambda_{i}^{2} + \sum_{i\neq j} \log{(\lambda_{i}
-\lambda_{j})}  -\sum_{i,j} \log{(2b_{0} - \lambda_{i} -\lambda_{j})} \cr
& +4Nh^2 \sum_{i} (b_{0} - \lambda_{i})^{-1})\ .\cr}\eqn\gas$$
The vortex--anti-vortex pairs appear as an extra singular term in the potential
for this eigenvalue problem, which at large $N$ may be treated by saddle-point
methods [\bipz].
Indeed, the operator coupling to $h$ was identified already in the
loop gas approach [\ivanb] and has a clear geometrical meaning. If one expands
$Z$ \part\ in Feynman diagrams, a typical piece of planar diagram will
appear as in fig.3. A ``bug'' crawling on the surface goes $2\pi$ around the
circle every time it crosses an arrowed propagator $<M^{\dagger}M>$. Closed
loops of these (the loop gas) delineate regions of constant $X$ which, since
the target lattice has only one link, takes only one value. Vortex--anti-vortex
pairs appear connected by cuts in the $X$ field. If the length of a cut is
$l$, then in terms of the $\F$ field it acts like a hole of length $2l$ on
the surface with one marked point on the boundary. The other marked point,
given by the other vortex, is not independent since the two are constrained
to be diametric. This is the origin of the macroscopic loop operator\foot{The
idea of using a marked macroscopic loop operator to represent a polymer
on the random surface may be used to solve the polymer gas problem as a
function of monomer bond energy and chemical potential [\meb].} coupling to
$h^2$ in \gas\ .

Thus the Boltzmann weights on a globally connected planar lattice take the form
$$\log{Z}(b_{0},h)
= \sum_{\rm graphs} h^{\# {\rm vortices}} b_{0}^{-\rm total \
length\ of\ loops\ \& \ cuts}\ .\eqn\bol$$
One may therefore think of the arrowed lines as flux tubes which, because
of the peculiar single valuedness of the target lattice, carry a unit
charge's worth of electric flux for a Coulomb field. $-\log{h}$ is the
chemical potential for vortices, with small $h$ corresponding to the dilute
regime. Flux tubes densely populate the surface and the continuum limit for
area is achieved when they become infinite in length and acquire the
Hausdorf dimension two, as $b_{0}$ is tuned to its critical value [\ivanb].
\bigskip
\centerline{\caps 3. Saddle-Point Solution.}
\bigskip

The leading order of the large $N$ limit of $Z(h,b_{0})$ describes surfaces
of spherical topology. There exists a systematic approach to computing the
$1/N$ corrections [\ivana],
corresponding to higher genus surfaces, but here we will
concentrate on genus zero. Introducing an eigenvalue density
$\rho (\l)$
for the problem \gas\ , the saddle-point equation for $\r$ is
$$\int_{a}^{b} d\mu\ \r (\mu) \left( {\rm P} {1\over \l -\mu} + {1\over
2b_{0} - \l -\mu} \right) = 2\lambda - {2h^2 \over (b_{0} -\l )^2 }\
,\eqn\eqm$$
where the support of $\r$ is on the interval $[a,b]$. This integral equation
has been solved for $h=0$ by Gaudin [\gau].
The same method will be used to solve
$h \neq 0$ also. To transform \eqm\ to an integral equation with Cauchy
kernel make the following redefinitions
$$\l \to b_{0} - \sqrt{A +B\l}\ , A+B=(b_{0} -a)^2 \ ,A-B= (b_{0}-b)^2
\ .\eqn\red$$
Then
$$\int_{-1}^{1} d\mu\ \r (\mu)  {\rm P} {1\over \l -\mu}
= 2(\sqrt{A+B\l} -b_{0}) +  {2h^2 \over A+b\l }\ .\eqn\neqm$$
The inversion formula for this problem is [\mus]
$$\r (\l) = -{2\over \pi} \sqrt{1-{\l}^2 } \int_{-1}^{1} {d\mu \over
\sqrt{1-{\mu}^2 }} {\rm P} \left( {1\over \l -\mu} \right)
\left[ \sqrt{A + B\mu} - b_{0} + {h^2 \over A+B\mu} \right]\ .\eqn\inv$$
$A$ and $B$ are determined by the normalisation and positivity conditions
for $\r$;
$$\eqalign{ {B\over 2} \int_{-1}^{1} d\mu \ {\r (\mu)\over \sqrt{A+B\mu}}
&= 1 \cr
\int_{-1}^{1} d\mu {1\over \sqrt{1-{\mu}^2}} \left( \sqrt{A+B\mu} -b_{0}
+{h^2 \over A+B\mu} \right) & = 0\ .\cr}\eqn\cond$$

To study the critical behaviour one need only know the form of $\r$ \inv\
for $\l \to -1$, though it may actually be found completely in terms of
elliptic integrals. The continuum limit of surfaces is achieved as the
bare cosmological constant is tuned to its critical value, given by the
condition $b=b_{0}$, when the end of the support of $\r$ meets the
singularities in \eqm\ . After some algebra, and performing the $h$-term
integral, one can rewrite the solution \inv\ in the form
$$ \r (\l) = {4\over {\pi}^2} \sqrt{1-{\l}^2} \left( BK(k) -(A+B\l )\Pi
(\l ) -{\pi Bh^2 \over 2(A+B\l )\sqrt{A-B} }\right)\ ,\eqn\sol$$
where
$$\Pi (\l ) = \int_{0}^{\pi /2} {d\F \over \sqrt{1-k^2 \sin^{2}{\F}}}
{\rm P} \left( {1 \over \l -\cos{2\F}} \right)\ ,\eqn\elt$$
an elliptic integral of the third kind, while $k^2 = 2B/(A+B)$, $\mu =
\cos{2\F}$, and $K$ $(E)$ denotes the complete elliptic integral of the
1st (2nd) kind. From the positivity condition one finds
$$2E(k)\sqrt{A+B} -\pi b_{0} + {\pi B h^2 \over \sqrt{A^2 - B^2}} =0\
.\eqn\pos$$
Using \sol\ , the normalisation condition requires a little more manipulation
before it can be brought to the form
$$-{2\over {\pi}^2}(A+B)((1-k^2 )K^2 (k) -E^2 (k)) - {2h^2 \over \pi
\sqrt{A-B}} (K(k) + {\rm n.s.} ) =1\ .\eqn\nor$$
n.s. stands for terms non-singular in the limit $b \to b_{0}$ $(k \to 1)$,
which will be unimportant for the continuum limit. To approach it one
introduces the cut-off $\d \to 0$ and renormalised parameters $M$ and $\xi$
as follows [\ivanb]:
$$\eqalign{ {A\over B}-1 &=  M^2 {\d}^2 \cr
\sqrt{{A\over B} + \l} & = \xi \d\ .\cr}\eqn\rep$$
$M^{-1}$ is  the renormalised dieletric constant of the Coulomb gas,
$M$ the renormalised boundary cosmological constant coupling to the length of
cuts in $X$. Returning to \sol\ and evaluating it perturbatively in $\d$
one eventually finds $(\xi > M)$
$$\r (\xi , M) = -{16 \sqrt{A} \over {\pi}^{2}} \d (\log{M\d}) \sqrt{\xi^2
-M^2} - {h^2 \over 2\pi {\d}^2 } {\sqrt{\xi^2 -M^2} \over \xi^2 M} +
\ldots \eqn\ssol$$
where $\ldots$ stands for terms subleading as $\d \to 0$. The bare vortex
parameter $h$ couples to a relevant operator and therefore must be tuned
to zero for finite renormalised parameter. The scaling law follows from
\ssol\ by requiring the 2nd term to be of the same order as the first
$$ h^2 = - h_{R}^{2} {\d}^3 \log{M\d}\ .\eqn\law$$
This agrees with KPZ scaling for the $\en =1$ operator at $r=1/2$ but
there is a logarithmic scaling violation; such violations are well-known at
$c=1$, occurring also for the cosmological constant [\kazc]. The latter is
determined in terms of $M$ by expanding \pos\ and \nor\ in $\d$;
$$\eqalign{ \pi b_{0} & = 2\sqrt{2A}
( 1 - M^2 {\d}^2 \log{M\d}) + {\pi h_{R}^{2}
 \over M\sqrt{2}}{\d}^2 \log{M\d} + \ldots \cr
1 & = {4A \over \pi^2 } (1- 2M^2 \d^2 \log^{2}{M\d}) -{\sqrt{2}h_{R}^{2}
\over A\pi M} \d^2 \log^2{M\d} + \ldots \cr}\eqn\pcond$$
which, omitting $+ \ldots$ now, determines $A=\pi^2 / 4$ and
$$\DR = (M^2 + {\sqrt{2} h_{R}^{2} \over \pi^2 M}) \log^{2}{M\d}
\ ,\eqn\cosm$$
where $b_{0} = \sqrt{2} + \d^2 \DR$ defines the renormalised cosmological
constant. $\d$ is thus the physical cutoff. One must now eliminate $M$ in
$\r$ \ssol\ in terms of the quantities $\DR, h_{R}$. Apart from the logarithms,
\cosm\ is cubic in $M$ and it is convenient to use the parametric solution
$$M = 2\sqrt{\Lambda \over 3} \cos{\pi - \T \over 3}\ ,\eqn\mmm$$
where
$$ \cos{\T} = {h_{R}^{2} \over \sqrt{2} \pi^2 } \left( {3\over \Lambda}
\right)^{3/2} \ ,\ \Lambda = {\DR \over \log^{2} {M\d}}\ .\eqn\subm$$
The equation \cosm\ is sketched in fig.4. \mmm\ corresponds to the largest
positive root, this being the physical one such that $M \to \infty$ as
$\DR \to \infty$. \ssol\ and \mmm\ constitute the solution for the scaling
part of $\r$, from which physical quantities may be calculated in the
continuum limit.

The first thing to notice is that \cosm\ , which is the positivity
condition for $\r$, has no solution for $h_{R}^{2} > \sqrt{2}\pi^2
(\Lambda /3)^{3/2}$. In this case the singular term in the potential \eqm\ has
become too strong and there is no smooth large $N$ limit. This is easy to
understand if one recalls the results for the sine-Gordon model [\moo, \hsu].
Vortices contribute
vacuum energy which, moreover, is negative at $r=\half$ (though not
otherwise expected to be universal). Thus the partition function at fixed
cosmological constant  does not converge in area $A_{R}$ if $\DR$ is too
small ($h_{R}$ too large).

As an intermediate step to calculating the partition function of spherical
surfaces, it is convenient to first derive the macroscopic loop
expectation at genus zero i.e. the partition function for surfaces with
the disc topology.
$$ \tilde{\omega} (z) = \int_{0}^{\infty} dl\ \omega (l)
 {\rm e}^{-zl}\ ,\eqn\disc$$
where $\omega (l)$ is the renormalised partition function for discs of
renormalised perimeter length $l$ with one marked point on the boundary.
Using \rep\ and \ssol\
$$\eqalign{\tilde{\omega} (z)& =
\int_{M}^{\infty} {d\xi \over z+\xi } \r (\xi ,M) \cr
& = -\log{M\d}\left( {4\over \pi} \sqrt{z^2 - M^2} \left( \log{z + \sqrt{z^2 -
M^2} \over M}\right) \left[ \sqrt{2} - {h_{R}^{2} \over \pi^2 M z^2 } \right]
+ {4h_{R}^{2} \over \pi^3} \left( {\pi \over 2z^2} - {1\over zM} \right)
\right)\ .\cr}\eqn\loop$$
In deriving \loop\ individual terms in the leading order which are analytic
in $z$ have been omitted since they correspond to surfaces of infinitesimal
area bounded by finite perimeter [\mat]. \loop\ for $h_{R} =0$ was given in
[\ivanb].
As a useful check one may verify that $\tilde{\omega}$ correctly solves
the finite difference equation derived by Kostov directly from the continuum
limit of \eqm\ (see e.g. ref.[\ivana]). The partition function for spherical
surfaces may be derived from $\tilde{\omega}(0) = d \log{Z}/dh^2$,
$$\eqalign{ \tilde{\omega}(0) & = 2\sqrt{2} (\log{M\d}) \left( M + {h_{R}^{2}
\over 2\pi^2 M^{2}} \right) \cr
& = 2\sqrt{2\DR \over 3} \left( \cos{\pi -\T \over 3} + \half {\cos{\T} \over
\cos^{2}{\pi - \T \over 3}} \right)\ ,\cr}\eqn\dee$$
by integrating w.r.t. $h^2$ at constant $b_{0}$. The constant of integration
is $c \sim \DR^2$ by dimensions, up to logarithms.  Using \subm\ it is
simpler to integrate w.r.t. $\T$, $dh^2 = -(\Lambda /3)^{3/2} \sqrt{2} \pi^2
\sin{\T} d\T + \ldots $
$$\log{Z} \propto {\DR^2 \over \log^2{\d^2 \DR}} \left( \sin^{2}{\T -\pi \over
3} + {1\over 8} \log{\cos{\T - \pi \over 3}} -\half \sin^{4}{\T -\pi \over 3}
\right) + c\ .\eqn\sph$$

\noindent This represents the continuum limit where all subleading
terms $+ \ldots$
have been dropped. Equations \loop\ and \sph\ are the main results of this
paper, representing the continuum limit on genus zero surfaces for the
vortex gas with arbitrary coupling.

One can use these results to investigate the number of effective degrees
of freedom in various regimes. In particular one can use \sph\ to find the
string susceptibility $\gamma_{\rm str}$, given by the fixed area partition
function $\log{Z(A_{R})} \sim A_{R}^{-3 + \gamma_{\rm str}} {\rm exp}(\DR^c
A_{R})$.
The UV limit $A_R \to 0$ occurs for $\DR \to \infty$, in which case \sph\
yields $\gamma_{\rm str} =0$, as appropriate for a massless scalar field. As
the scale $A_R$ is increased above that set by $h_{R}^{-4/3}$ the screening
effects of the vortex condensate come into play and the behaviour changes. To
determine $\gamma_{\rm str}$ more generally one must identify the effective
cosmological constant, given by the sum of contributions from gravity $(\DR)$
and matter $(-\DR^c )$, which is conjugate to area. This constant is zero
at the critical point of \cosm\ allowing identification of $\DR^c$. Since
$\T_{c} =0$ one finds
$$\eqalign{ M_{c}^{2} \log^{2}{\d M_{c}}& = {\DR^c \over 3} \cr
{h_{R}^{2} \over \sqrt{2} \pi^2 } \left( {3 \over (\DR^{c})^2 } \right)^{3/2}
\log^{3}{\d M_c} & = 1 \cr
\Rightarrow M_c & =  \left( {h_{R}^{2} \over \sqrt{2} \pi^2} \right)^{1/3}
\ .\cr}\eqn\mct$$
As usual, the presence of logarithms complicates $\DR^c$ which moreover
satifies a transcendental equation. $\DR^c = 3 M_{c}^{2}$ neglecting
 logarithms.

To find $\gamma_{\rm str}$ in the IR limit $A_{R} \to \infty$ one can
proceed analytically. Defining $\DR = \DR^c + \DRT$ and $M = M_c + \MT$,
for small $\T$ one may expand all formulae for $\DRT \ll \DR^c , \MT \ll M_c$
and find expressions for the small variables; even keeping track of logarithms
is not too tedious. Expanding about \mct\ one finds
$$\eqalign{\MT = & {1\over \log{M_c \d}}
\left( {\DRT \over 6} \sqrt{3\over \DR^c}
+ {\T \over 3} \sqrt{\DR^c \over 3} + \ldots \right) \cr
-\half \T^2 + {\T^4 \over 4!} - \ldots = & -{3\over 2} \left( {3\over \DR^c}
\right)^2 M_{c}^{2} \DRT \log^{2}{\d M_c} + {2\MT \over M_{c} \log{M_c \d}}
+ \ldots \cr}\eqn\exa$$
and expanding $\DRT$ in $\T$ gives
$$\T = 3\sqrt{\DRT \over \DR^c} + O(\DRT^{3/2})\ .\eqn\thet$$
Expanding the partition function \sph\ in $\T$ one finds
$$\log{Z} = {(\DR^c )^2 \over \log^{2}{\d^2 \DR^c}} \left( \a + \b \T^2
+ \gamma \T^5 + \ldots +(\epsilon + \nu \DRT + \eta \DRT^2 + \ldots )\right)\
,\eqn\near$$
where $\a , \b ,\gamma ,\ldots$ are known constants and the constant of
integration $c= \epsilon +\ldots $ is analytic  as $\DRT \to 0$ since
it does not depend upon $h$, only $\DR$. Thus
$$\log{Z} = {\rm analytic} + {{\rm const.} \over \log^{2}{\d^2 \DR^c}}
{\DRT^{5/2} \over \sqrt{\DR^c}} + O(\DRT^{7/2})\ ,\eqn\far$$
confirming that $\gamma_{\rm str} = -1/2$ in the IR limit, where the cuts
in $X$ disorder it to leave pure gravity. Note that the
analytic terms as $\DRT \to 0$ come from the lower cutoff on the area
integral, most importantly they do not contain logarithms of $\DRT$.

The change in behaviour from the UV to the IR can be understood from the
Coulomb gas in terms of screening effects [\bkta]. At small areas $\DR \to
\infty$
and the renormalised $M^{-1} \to 0$ \cosm\ , all flux lines being of the order
of the UV cutoff on the surface; this is the $c=1$ regime where vortices are
strongly bound to anti-vortices. At large areas $\DR \to \DR^c$ and $M \to
M_{c}$ \mct\ . When $h_{R}=0$, since the surface is dense with flux loops,
to achieve large area the dielectric constant is tuned to $\infty$ ($M_c =0$).
In the presence of the charge gas ($h_{R}\neq 0$) the inverse permittivity
is $M-M_c$, the (non-universal) shift $M_c$ being due to dielectric
polarization. It clearly required the non-perturbative treatment to see
this since \mct\ is not analytic as $h_R \to 0$.
\bigskip
\centerline{\caps 5. Extensions.}
\bigskip

Unfortunately it appears difficult to conceive of a general method for
obtaining
exact results for $r \neq 1/2$ or $\en >1$. However for $r=1/2$ and $\en =1,
2$ the integrals are still Gaussian. Adding the $\en =2$ operator $\cm_2
\Tr [MM + M^{\dagger}M^{\dagger}]$ to \part\ yields
$$\eqalign{Z = \int \prod_{i} d\l_i \ {\rm exp} (& \sum_{i\neq j} \log{(\l_i
-\l_j)} - N\sum_{i} 2\l_{i}^{2} -\half \log{((2b_{0} -\l_{i} - \l_{j})^2
-(\cm_2)^2 )} \cr
& +8N\cm_1 \sum_{i} (b_{0} -\half \cm_2 -\l_i )^{-1})\ .\cr}\eqn\newz$$
instead of \gas\ .
The geometrical meaning of the last two terms in the action \newz\ is
illustrated in terms of flux lines in fig.5. The saddle-point equation to
be solved is
$$\int_{a}^{b} d\mu \ \r (\mu) \left( {\rm P} {1\over \l -\mu} +\half {1\over
2b_{0} -\cm_2 -\l -\mu } +\half {1\over 2b_{0} + \cm_2 -\l -\mu }\right)
= 2\l - {2({\cm_1})^2 \over (b_{0} -\half \cm_2 -\l )^2 }\eqn\newsp$$
but this is more difficult than \eqm\ ; the author has not managed to solve
it to the extent that useful information can be gleaned, for example on
questions of multicriticality as $\cm_2$ is tuned as a function of $\cm_1$.
Note that not suprisingly, the same saddle point problem results,
after replacements $\cm_1 \to 0$ and $\cm_2 \to \cm_1$, for the case $\en=1$ at
$r=1$
under the assumption of a translation invariant $\F$-matrix for the two-link
target lattice. One may however give a hand-wave argument that the IR limit
is pure gravity. Consider $\cm_1=0$ and $\cm_2 \neq$ in \newsp\ for simplicity.
According to the discussion of the previous section, the scaling regime
is described by $(\l'  +\half \cm^{R}_{2})\d  = b_{0} - \l $ and
$$\int_{b'}^{\infty} d\mu' \r (\mu') \left( {\rm P} {1\over \l' -\mu'} -
\half {1\over \l' + \mu'} - \half {1\over 2\cm^{R}_{2} + \l' + \mu'}\right)
= 2(\d \l' + \d \half \cm^{R}_{2} - b_0 )\ ,\eqn\sca$$
where $\cm^{R}_{2} = \cm_2 /\d$ is consistent with KPZ scaling. The IR limit is
equivalent to $\cm^{R}_{2} \to \infty$ in which case the 3rd term in the kernel
of \sca\ may be dropped, leaving the saddle point problem of the O(1) model
in Kostov's classification [\ivanb] i.e., pure gravity.

Returning to the $c$-dimensional model \wein\ , one can ask about the extent
to which similar methods of solution may be applied, in particular the
introduction of induction fields to obtain plaquettes from an action
quadratic in $M$. Similar ideas have been applied by Kazakov and Migdal [\kazd]
to a certain lattice gauge theory with some success. Unfortunately in the
present case this procedure tends to introduce unwanted vortices (open
loops in the action) which are difficult to eliminate and leave the model
solvable; such vortices would disorder the embedding coordinate as we have
seen. The success of the Kazakov-Migdal model seems to be due to the
fact that it is still a tree-like string theory [\bou], while in \wein\ one is
dealing with plaquettes of non-zero extrinsic area. The latter must condense
on the worldsheet if one is to find a non-tree-like string theory in higher
dimensions.

The introduction of an extrinsic curvature smooths badly behaved surfaces
[\stan, \pola]. This may be included in the
Weingarten models by introducing new flavours of link matrix. For example
at $c=2$, two flavours $M,N$ suffice since the worldsheet normals are $\pm 1$.
The extrinsic curvature, given by $\half (1-\cos{\T}) = 0,1$ for parallel
($\T = \pi$) or anti-parallel ($\T = 0$) neighboring plaquettes where $\T$
is the angle subtended, can be included by using an action
$$\sum_{l} \Tr [M^{\dagger}(l)M(l) + N^{\dagger}(l)N(l)] + K \Tr[N^{\dagger}
(l)M(l) + M^{\dagger}(l)N(l)] - \kappa \sum_{\rm plaq} \Tr[P_M + P_N]\
,\eqn\ext$$
where $P_M$ is the product of $M$'s around  plaquettes of orientation $+$
and $P_N$ the product of $N$'s around  plaquettes of orientation $-$; $K$
is the bare extrinsic curvature coupling. The $c$-dimensional generalisation
has $Z_c$ symmetry and $c(c-1)$ flavours, and one assumes that the
continuous symmetry would be restored at any critical point. These models are
apparently still just as difficult to solve in full generality, but simplified
versions which still incorporate extrinsic curvature can be reduced to
eigenvalue problems. This will be dealt with in a future publication [\mec]
\bigskip
\centerline{\caps 6. Conclusions.}
\bigskip

In this paper the matrix model of random surfaces introduced in ref.[\mea]
has been further investigated and applied to the two-dimensional vortex
plasma in the presence of gravity. The exact solution on genus zero was
found at half the self-dual radius as a function of renormalised
cosmological constant and vortex chemical potential \loop\ \sph\ . In
general these matrix models seem to give new possibilities for exact
solution where the traditional Hermitian models are too difficult, the
principle improvements being:
\item
{1.} Vortex operators are manifest and easily manipulated.
\item
{2.} Exact solutions are possible even in the presence of non-tree-like
embeddings.
\item
{3.} Extrinsic curvature can be trivially introduced and, in certain
cases [\mec], leads also to solvable models.
\item
{4.}  The matrix models have gauge symmetry and are closely related to
conventional non-linear lattice gauge theories, for which there is an
abundance of expertise available.

The accompanying geometrical picture confirms the KT hypothesis of
unbound vortices, which in the renormalised scheme are joined by Coulomb
flux tubes, cuts in the periodic scalar field that disorder it at large
length scales. Hopefully this approach can be further developed to extend
the remarkable non-perturbative answers which matrix models furnish.
\bigskip


\ack
I have benefitted from discussions with I.Klebanov, I.Kostov, M.Martin-Delgado,
and M.Staudacher.
Financial support was provided by  S.E.R.C.(U.K.) post-doctoral fellowship
RFO/B/91/9033.
\endpage

\centerline{FIGURE CAPTIONS}
\bigskip

\item
{\bf Fig.1.} A triangulation of spherical topology.
\item
{\bf Fig.2.} Propogator (l), Wilson Plaquette ($\l$), and collapsed
plaquettes ($\kappa_1 ,\kappa_2 $) contributing to the Weingarten action
for non-critical string theory in $c=2$ dimensions.
\item
{\bf Fig.3.} A typical piece of planar Feynman graph for action \part\ . A
single vortex--anti-vortex pair and three ``vacuum'' flux loops are shown
arrowed. Dotted line is $<\F \F >$ propogator.
\item
{\bf Fig.4.} A sketch of the cubic equation \cosm\ .
\item
{\bf Fig.5.} Vortex flux configurations corresponding to the expansion
of action \newz\ in powers of $\cm_1$ and $\cm_2$.
\endpage

\refout


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