The Large N Expansion in Quantum Field Theory and Statistical Physics

The following sections are included:

• CONTENTS

• FOREWORD

• ACKNOWLEDGEMENTS

The study of spin systems in the large N limit began with the paper of Stanley [1], who studied the N vector model using saddle point techniques. The leading order behaviour matches with the result of Berlin and Kac [2] for the spherical model. Subsequently K. G. Wilson proposed on the basis of universality the application of the large N method to the study of critical phenomena [3] and this proposal was carried out in the work of Ma [4], who calculated 1/N corrections to the critical exponents for T > T_c, and by Brézin and Wallace [5], who calculated the correction to the exponents for T < T_c and the equation of state. The 1/N correction to the exponents was also calculated by Abe [6]. 1/N² order corrections were performed by Okabe et al. [7] and Vasiliev et al. [8]. The large N limit of N vector models is the only model that has been solved in any dimension of space and historically speaking it revealed a number of features which were understood in their generality only much later: triviality above four dimensions, vanishing critical temperature in two dimensions and nontrivial critical exponents in between. Calculations in continuum field theory were performed in [9] and nonlinear sigma models in terms of pion fields were studied in [10] and [11]. The CP(N − 1) models were studied by D'Adda, Lüscher and Di Vecchia [12]. Ma has used the 1/N expansion to explain the Wilson renormalization group ideas [13].

The Berlin-Kac spherical model (or “spherical approximation to the Ising model”)

The critical behavior of a classical Heisenberg ferromagnet is studied in the limit where the spin dimensionality N is large. Corrections of order 1/N to the spherical model are obtained as functions of a continuous dimension d, 2 < d < 4. Particular attention is given to the behavior near the coexistence curve. The divergence of the magnetic susceptibility below T_c as the external field vanishes is discussed through a nonlinear realization of the O(N) symmetry, as well as in the 1/N and 4-d expansions.

The basic idea of the renormalization group is introduced and illustrative examples are presented. Emphasis is put on the application to the theory of critical phenomena. This article is prepared for pedagogical purposes. It is written at a level that a second-year graduate student in physical sciences can understand. No previous knowledge of critical phenomena or field theory is needed. We make no attempt to survey the field or cover a wide range of subjects. On the contrary, we limit the scope to the most basic aspects. We choose to elaborate at length to make the basic idea clear and the definitions precise, and to go through the examples very carefully. We feel that once these basic aspect are understood, there will be no difficulty in confronting the rapidly expanding literature on this subject.

By means of the 1/n expansion method, the critical exponent η is discussed for the three-dimensional isotropic n-vecter model with short-range interaction. Terms up to 1/n² are obtained and the result is compared with the numerical estimate based on high temperature expansion and with Wilson's expansion.

The long-distance properties of classical Heisenberg ferromagnets below the transition point are related to a continuous-field theory, the nonlinear σ model. The renormalizability of this model in two dimensions and its ultraviolet asymptotic freedom are used to derive renormalization-group equations valid above d = 2. It is argued that this model is renormalizable up to four dimensions. The scaling properties which incorporate critical and Goldstone singularities follow. Explicit calculations of exponents and of correlation functions in powers of d − 2 are given. A technique is proposed to make calculations in the symmetric phase applicable even in two dimensions.

We formulate and discuss in detail the recently discovered ℂP^{n − 1} non-linear σ models in two dimensions. We find that the fundamental particles in these theories are confined by a topological Coulomb force.

The subject of the large N limit of U(N) gauge theories was initiated in a pioneering paper by 't Hooft [1]. There are basically two themes that run through this subject and both are present in this paper. The first theme is the relation between gauge theories and the string model and the second is the fact that 1/N affords a small expansion parameter in theories where the usual perturbative expansion reveals very little information about the spectrum of the theory, for example in four-dimensional QCD…

A gauge theory with colour gauge group U(N) and quarks having a colour index running from one to N is considered in the limit N → ∞, g²N fixed. It is shown that only planar diagrams with the quarks at the edges dominate; the topological structure of the perturbation series in 1/N is identical to that of the dual models, such that the number 1/N corresponds to the dual coupling constant. For hadrons N is probably equal to three. A mathematical framework is proposed to link these concepts of planar diagrams with the functional integrals of Gervais, Sakita and Mandelstam for the dual string.

Two-dimensional QCD affords an excellent opportunity to study various dynamical questions of gauge theories. Many of its qualitative features are also valid in four dimensions. Even so it is still an insoluble problem except in the planar limit, 't Hooft [1] obtained a closed integral equation for the meson wave function to the leading order in the 1/N expansion, with solution corresponding to an infinite number of bound states. Subsequently Callan, Coote and Gross [2], Einhorn [3], Brower et al. [4], Cardy [5], Einhorn et al. [6], and Shei and Tsao [7] explored this model laboratory for questions relating to quark confinement, hadrons, scattering amplitudes and high energy behaviour. They also clarified issues related to gauge invariance and infrared cutoff. In this connection we also note the papers of Bars and Green [8] and Kikkawa [9]…

A recently proposed gauge theory for strong interactions, in which the set of planar diagrams play a dominant role, is considered in one space and one time dimension. In this case, the planar diagrams can be reduced to self-energy and ladder diagrams, and they can be summed. The gauge field interactions resemble those of the quantized dual string, and the physical mass spectrum consists of a nearly straight “Regge trajectory”.

We analyze the structure of two-dimensional Yang-Mills theory as a model of quark confinement, 't Hooft's solution, in the large-N limit, is extended to investigate the consistency and the properties of the model. We construct the hadronic color singlet bound-state scattering amplitudes. We show that they are unitary, that colored states cannot be produced, and that all long-range interactions are absent. Current amplitudes are constructed, and we show that the theory is asymptotically free and the quark mass sets the scale of mass corrections. The properties of bound states of heavy quarks are discussed, and a dynamical basis for the Okubo-Zweig-lizuka rule is suggested. We show how confinement can occur with an infrared prescription that leads to finite-mass quarks which decouple from physical states and discuss the dependence of gauge-variant amplitudes on the cutoff procedure. Higher-order effects in 1/N are shown not to change the qualitative features of the model.

We clarify the gauge invariance, infrared structure, and completeness of 't Hooft's solution for the meson sector of two-dimensional quantum chromodynamics. Electromagnetic form factors of mesons are then shown to obey an asymptotic power law, whose power is dynamically determined and is not related to the short-distance behavior of the theory. Following a review of the total annihilation cross section for producing hadrons, we discuss deep-inelastic lepton scattering. As expected, Bjorken scaling is obtained, but we show how the sum over hadronic final states reproduces the parton model precisely. The Drell-Yan-West relation and Bloom-Gilman duality are fulfilled for the relation between the scaling function and form factors. We conclude by speculating on the applicability of our picture of form factors to the real, four-dimensional world. We argue that this is a viable alternative to dimensional scaling and, phenomenologically, the differences between our predictions and the dimensional counting rules are slight for light quarks. Finally, we attempt to abstract those features of the model which may guide us toward a solution to the four-dimensional problem.

Two-dimensional gauge theories, both abelian and non-abelian, are formulated in terms of gauge invariant path ordered operators (POO). The generators of the Poincaré group are constructed with POO's. An exact equation of motion for POO's is derived and is shown to reduce to the 't Hooft eigenvalue equation in QCD in the large N limit. Nowhere appear infrared problems.

The study of the asymptotic properties of the planar perturbation series was initiated by 't Hooft [1],[2]. The aim was to rigorously construct a convergent calculational scheme even in this approximation for strongly interacting theories like QCD. The Borel summability of an asymptotically free, massive and planar euclidean field theory, for sufficiently small coupling, was proved by 't Hooft [2],[3],[4],[5] and by Rivasseau [6]. These proofs are based on a derivation of bounds on euclidean Feynman graphs in asymptotically free theories. Other references in this connection are [7],[8]. An example of a completely convergent planar theory is the SU(∞) gauge theory with massive higgs and fermionic degrees of freedom that allow for asymptotic freedom. In massless theories the proofs encounter difficulties due to infrared divergences at very high orders and hence there are no conclusive results for gauge theories like QCD…

Borel summability of any renormalizable euclidean field theory which is planar, asymptotically free, massive, and whose coupling constant is sufficiently small can be proven. The simplest example is massive λ Tr φ⁴ in the N → ∞ limit, with the “wrong” sign of λ. An outline of the proof is given. Our methods also yield further information on the analyticity of the Borel function for massless theories.

Asymptotically free quantum field theories with planar Feynman diagrams [such as SU(∞) gauge theory] are considered in 4 dimensional Euclidean space. It is shown that if all particles involved have non-vanishing masses and if the coupling constant(s) λ (or g²) are small enough (λ ≦ λ^crit), then an absolutely convergent procedure exists to obtain Green functions that uniquely solve the Dyson-Schwinger equations.

We use the methods of [1] to show that the planar part of the renormalized perturbation theory for -euclidean field theory is Borel-summable on the asymptotically free side of the theory. The Borel sum can therefore be taken as a rigorous definition of the N → ∞ limit of a massive N × N matrix model with a + trgϕ⁴ interaction, hence with “wrong sign” of g. Our construction is relevant for a solution of the ultra-violet problem for planar QCD. We also propose a program for studying the structure of the “renormalons” singularities within the planar world.

The work of 't Hooft established a connection between gauge theories and string theory by showing that Feynman diagrams generate 2-dim. surfaces in the 1/N expansion. Wilson's lattice gauge theory [1], which was developed around the same time, also has a connection with the string model in that its basic degrees of freedom are “string bits”, and the Wilson loop operator creates a string-like excitation. Nambu [2], Polyakov [3] and Gervais and Neveu [4] proposed to write equations of motion for the Wilson loop operators in order to establish a direct connection with the string model. Similar ideas were subsequently explored in the context of lattice gauge theories by Eguchi and Wadia [5], Foerster [6], Eguchi [7], and Weingarten [8]…

A closed equation for the loop average is obtained in QCD with an infinite number of colors. It is shown how this equation generates the planar graphs. The lattice regularization of this equation is considered.

Simple model systems like the O(N)σ model, the Gross-Neveu model, and the random matrix model are solved at N → ∞ using Dyson-Schwinger equations and the fact that the Hartree-Fock approximation is exact at N → ∞. The complete string equations of the U(∞) lattice gauge theory are presented. These must include both string rearrangement and splitting. Comparison is made with the “continuum” equations of Makeenko and Migdal which are structurally different. The difference is ascribed to inequivalent regularization procedures in the treatment of string splitting or rearrangement at intersections.

It is shown how to calculate the Wilson loop average for self-intersecting contours in two-dimensional lattice U(∞) gauge theory by means of the lattice contour equations. Some examples are given and the structure of the general solution is discussed. The free lattice string is not recovered even in the strong-coupling limit. The Gross–Witten phase transition becomes a first-order one when massless fermions are included.

It is pointed out that at N → ∞, for finite temperature, the Schwinger-Dyson equations imply that below the deconfining phase transition the Wilson loops are independent of the temperature. This suggests a first-order deconfinement phase transition.

We now turn to the second theme in the subject, where 1/N acts as a small expansion parameter. This is especially relevant in 4-dim. QCD where the coupling constant grows with the distance one is probing. In an influential paper Witten [1] has argued that all the qualitative features of QCD can be understood in the framework of the 1/N expansion. In particular, if one assumes confinement, QCD is a weakly interacting theory of an infinite number of mesons with couplings of the order of 1/N and baryons are the solitons with masses proportional to N, their size and shape being N-independent…

In this paper the existing results concerning mesons and glue states in the large-N limit of QCD arc reviewed, and it is shown how to fit baryons into this picture.

It is shown that ordinary baryons can be understood as solitons in current algebra effective lagrangians. The formation of color flux tubes can also be seen in current algebra, under certain conditions.

The collective coordinate method of Gervais and Sakita is used to quantise the large-N topological soliton in the SU(3) chiral model. The leading-order hamiltonian is that of a symmetrical top. The Wess-Zumino term plays a role in determining the flavour and spin quantum numbers of the baryon multiplets.

Starting from the large-N power counting which suggests that the baryons are QCD solitons, the authors derive an exact large-N equation identical to the so-called bootstrap condition of static strong-coupling theory. This equation determines the group structure of the baryon multiplets at N → ∞. One solution is the standard nonrelativistic quark model.

We prove that the group theoretic structure of the chiral soliton model is identical to that of the naive quark model in the large- N_C limit. This implies that all group theoretic calculations such as the F/D ratios, g_πNN/g_πNΔ, etc. are identical in the soliton and quark models in large N. This result is true for an arbitrary number of flavors. We also compare the two models for finite N_C.

Chromodynamics with n flavors of massless quarks is invariant under chiral U(n)⊗ U(n). It is shown that in the limit of a large number of colors, under reasonable assumptions, this symmetry group must spontaneously break down to diagonal U(n).

A fully interacting effective chiral Lagrangian obeying the anomalous axial-baryon-current conservation law is constructed. This Lagrangian is a generalization of one implied by the 1/N approximation. In a certain sense, the old σ model is recovered. Our Lagrangian displays the dependence of amplitudes on the quantum-chromodynamic vacuum angle θ, gives soft η′ theorems, and hints at a possible complementarity between the instanton and 1/N approaches. We can rewrite our model in terms of a gauge-invariant gluon field.

We present a qualitative derivation of the chiral model from QCD. This is based on using a Nambu–Jona-Lasinio–type effective Lagrangian as an intermediate step. A detailed derivation of the anomalous low-energy Wess-Zumino term is presented. This includes vector, axial-vector, and pseudoscalar particles. The low-energy scale is set by . We also present the low-energy nonlinear chiral model which generalizes the Skyrme model. The possibility of soliton solutions is indicated. There is a possible application of these ideas to electroweak theory.

We consider baryons in a two-flavor Nambu–Jona-Lasinio–type model for self-interacting quarks in the large-N limit using mean-field-theory techniques. We show that for slowly varying mean fields, all baryon properties are exactly those obtained from collective-coordinate quantization of the soliton in a nonlinear a model of mesons. We also indicate how the higher Fermi couplings can stabilize the soliton.

The following sections are included:

• Eguchi-Kawai Model

• Collective Fields

• Numerical Methods
  • Eguchi-Kawai models
  • Loop equations
  • Variational methods

It is pointed out that the factorization of disconnected Wilson loop amplitudes implies a major reduction in the dynamical degrees of freedom in the large-N limit of lattice gauge theory; the original model may be replaced by a much simpler one (d is the space-time dimensionality),

Thus the field theory may be reduced to an integration over a finite number of matrices in large-N limit.

It is argued that spontaneous symmetry breaking occurs in the recently proposed Eguchi–Kawai model. The argument is based on an analytic investigation and on Monte Carlo simulations. A quenched version of the model is proposed which gives good behavior at weak couplings.

We argue that in a U(N) invariant theory in presence of a random background gauge field, the quenched expectation values of U(N) invariant objects are volume independent in the limit N → ∞.

We study the large-N reduced model recently proposed by the present authors. This model is a modified version of the Eguchi-Kawai model incorporating twisted boundary conditions. It is shown that the Schwinger-Dyson equations of our model are the same as in the infinite-lattice theory provided [U(1)]⁴ symmetry is not spontaneously broken. We study the model at strong coupling, weak coupling, and intermediate coupling using analytical and Monte Carlo techniques. At weak coupling, it is shown that for a particular choice of twist, [U(1)]⁴ symmetry is not broken and we prove how one recovers usual planar perturbation theory. Monte Carlo data for χ ratios show striking agreement with Wilson-theory results.

We formulate a general method of collective fields in quantum theory, which represents a direct generalization of the Bohm–Pines treatment of plasma oscillations. The present method provides a complete procedure for reformulating a given quantum system in terms of a most general (overcomplete) set of commuting operators. We point out and exemplify how this formalism offers a new powerful method for studying the large-N limit. For illustration we discuss the collective motions of N identical harmonic oscillators. As a much more important application, we show how, based on the present formalism, one solves the planar limit of a non-trivial SU(N) symmetric quantum theory.

We study SU(∞) gauge theory on an asymmetric lattice using a generalisation of the twisted Eguchi–Kawai model. We show that it is possible to remove the large-N bulk transition by choosing a sufficiently asymmetric lattice and hence to study the physical deconfinement transition without being affected by the former. Results for N = 16 indicate a strong first order deconfinement transition.

A new approach to QCD-momentum loop dynamics – is proposed. The basic quantity is the local momentum p_μ(s) of quarks propagating along the closed loop in a gluonic vacuum. Using the equations of QCD at N = ∞, we derive the stochastic motion of p_μ(s) in extra proper time H. We propose an explicit method for computer simulation, preserving all the symmetries of the continuum theory. An advantage of this approach is the genuine reduction of degrees of freedom. The problems are related with cancellations in numerical simulations.

The coherent state variational algorithm provides a method for solving the large-N limit of non-abelian gauge theories. An implementation of this algorithm, capable of minimizing the large-N effective action and computing meson and glueball spectra, has recently been completed. Hamiltonian or euclidean formulations of lattice gauge theories, in any dimension, may be studied. Bose or Fermi fundamental representation matter fields may be included. This paper discusses the design and testing of this implementation. The method involves explicit manipulation of expectation values of physical operators and may be applied directly in infinite volume. The error introduced by the truncation of the set of physical obseivables (necessary to obtain a finite procedure) is studied by applying the algorithm to a variety of exactly soluble model theories. These include φ⁴ scalar field theories, fermion theories, 2-dimensional euclidean pure gauge theory, and 1 + 1 dimensional QCD. Modest size calculations are shown to yield accurate results, even in theories possessing asymptotic freedom, spontaneous symmetry breaking, or large-N phase transitions.

Random matrix ensembles were originally introduced to study systems with a large number of energy levels like complex nuclei [1],[2]. Their introduction to study the large N limit in quantum field theory was done by Brézin, Itzykson, Parisi and Zuber (BPIZ) [3], with the hope of performing a sum over planar diagrams. The zero-dimensional model where the matrix does not depend on space-time was used to count the number of planar diagrams. The result is in agreement with combinatorial methods [4]. The method of BPIZ essentially involved expressing the classical equations (as N = ∞) in terms of the density of eigenvalues.

We investigate the planar approximation to field theory through the limit of a large internal symmetry group. This yields an alternative and powerful method to count planar diagrams. Results are presented for cubic and quartic vertices, some of which appear to be new. Quantum mechanics treated in this approximation is shown to be equivalent to a free Fermi gas system.

The large-N limit of the two-dimensional U(N) (Wilson) lattice gauge theory is explicitly evaluated for all fixed λ = g²N by steepest-descent methods. The λ dependence is discussed and a third-order phase transition, at λ = 2, is discovered. The possible existence of such a weak- to strong-coupling third-order phase transition in the large-N four-dimensional lattice gauge theory is suggested, and its meaning and implications are discussed.

An exactly soluble class of model U(N) lattice gauge theories is considered. The ground state is discussed as a separable N-fermion problem solved by Mathieu functions. Some exact correlation functions are presented. The N = ∞ limit exhibits a third order phase transition demarcating the strong and weak phases at (g²N)⁻¹ ≈ 0.55.

Several problems in lattice gauge theories such as mean field theory or the few plaquette problem lead to the evaluation of the properties of one link in an external matrix source. This problem is solved here in the large N limit. There are two phases characterized by a single parameter, the average value of the inverse of the modulus of the eigenvalues of the external source. The third derivative of the free energy is discontinuous at the transition point.

The planar approximation is reconsidered. It is shown that a saddle point method is ineffective, due to the large number of degrees of freedom. The problem of eliminating angular variables is illustrated on a simple model coupling two N × N matrices.

The integral over two n × n hermitan matrices is evaluated in the limit of large n. For this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non-local weight. The above integral arises in the study of the planar approximation to quantum field theory.

A method for evaluating the integral

In his original contribution to the 1/N expansion of gauge theories 't Hooft [1] had observed that the Feynman diagrams, written in the double line notation, are in one-to-one correspondence with two-dimensional surfaces characterised by their Euler number χ, and can be arranged as a topological expansion in powers of N_χ. This is the precise basis of the conjectured relation (as yet without proof) between nonabelian gauge theories and the string model. Several authors [3],[4],[5] proposed that the 't Hooft construction can provide a lattice formulation of 2-dim. surfaces or equivalently 2-dim. gravity, because the Feynman diagrams actually provide a “fishnet” or a dual lattice on the 2-dim. surface. David [3] and Kazakov [4] went beyond the proposal and implemented it as a viable computational tool…

Some discrete lattice models for quantum two-dimensional euclidean gravity are shown to be equivalent to zero-dimensional planar field theories. Explicit expressions are given for partition functions. A universal continuum limit exists for open surfaces, but not for closed ones, and is argued to describe a space with negative average curvature. Extensions of those models to higher dimensions and to surface models are briefly discussed.

A new formulation of a Weingarten-type model in terms of bilocal (no-matrix) variables in continuous space is given. This model is equivalent to the model of triangular planar random surfaces. It generalizes in a reparametrization invariant way the model recently suggested by Billoire, Gross and Marmari. The number-of-surfaces critical index is calculaledd in the zero-dimensional case of this model.

We compare the results of analytical and numerical studies of lattice 2D quantum gravity, where the internal quantum metric is described by random (dynamical) triangulation, with the recent results of conformal approach developed by Knizhnik, Polyakov and Zamolodchikov. The remarkable agreement is underlined for the interactions of gravity with matter fields: Potts spins, D-dimensional Gaussian fields (bosonic string). Some new results arc presented for D = 1 discretized bosonic strings satisfying the predictions of conformal theory for the critical exponents: γ_str = 0, ν_str = 0, but with unusual logarithmic corrections.

It is established that various critical regimes may occur for a model of two-dimensional pure quantum gravity. These regimes correspond to the presence of effective fields with scaling dimensions Δ_k = − γ_str · k/2, k = 1, 2, 3 …, where γ_str = −1/m, m = 2, 3, 4 … is the critical exponent of “string susceptibility” (with respect to the cosmological constant). This behaviour is typical for unitary conformal fields with the central charge c = 1 − 6/m(m + 1) in the presence of 2D-quantum gravity. We use the framework of loop equations for the invariant boundary functional, which are exactly solvable in this case.

We reconsider a recently solved Ising model on a random planar graph. The Yang–Lee edge singularity, familiar from the ordinary Ising model, is exposed. It is shown to correspond to an exactly solvable critical dimer counting problem on the random surface in the infinite temperature limit. This suggests an interesting interpretation of a recently proposed phenomenological model exhibiting multicritical behavior. The critical exponents are found to be γ = − ⅓ (string susceptibility) and σ = ½ (edge singularity). The result is at odds with the Knizhnik–Polyakov–Zamolodchikov formula in conjunction with the Yang–Lee edge singularity's central charge C = – 22/5. Possible explanations are discussed. The result σ = – coincides with the corresponding exponent of the ordinary three-dimensional spherical model, as does the set of exponents of the random Ising critical point found previously.

Field theories of closed strings are shown to be exactly solvable for a central charge of matter fields c = 1−6/m(m + 1), m = 1, 2, 3,… The two-point function χ(λ, N), in which λ is the cosmological constant and N⁻¹ is the string coupling constant, obeys a scaling law χ(λ, N) = N^{−(m + 1/2)}f((λ_c − λ)N^{m/(m + 1/2)}) in the limit in which N⁻¹ goes to zero and λ goes to a critical value λ_c; we have determined the universal non-linear differential equation satisfied by the function f. From this equation it is found that a phase transition takes place for some finite value of the scaling parameter (λ_c - λ)N^{m/(m + 1/2)}; this transition is a “condensation of handles” on the world sheet, characterized by a divergence of the averaged genus of the world sheets. The cases m = 2, 3 are elaborated in more details, and the case m = 1, which corresponds to the embedding of a bosonic string in −2 dimensions, is reduced to explicit quadratures.

Starting from the random triangulation definition of two-dimensional euclidean quantum gravity, we define the continuum limit and compute the partition function for closed surfaces of any genus. We discuss the appropriate way to define continuum string perturbation theory in these systems and show that the coefficients (as well as the critical exponents) are universal. The universality classes are just the multicritical points described by Kazakov. We show how the exact non-perturbative string theory is described by a non-linear ordinary differential equation whose properties we study. The behavior of the simplest theory, c = 0 pure gravity, is governed by the Painlevé transcendent of the first kind.

We propose a nonperturbative definition of two-dimensional quantum gravity, based on a double-scaling limit of the random-matrix model. We derive an exact differential equation for the partition function of two-dimensional gravity coupled to conformal matter as a function of the string coupling constant that governs the genus expansion of two-dimensional surfaces, and discuss its properties and consequences. We also construct and discuss the correlation functions of an infinite set of local operators for spherical topology.

We propose a nonperturbative definition of two-dimensional quantum gravity, based on a double scaling limit of the random matrix model. We develop an operator formalism for utilizing the method of orthogonal polynomials that allows us to solve the matrix models to all orders in the genus expansion. Using this formalism we derive an exact differential equation for the partition function of two-dimensional gravity as a function of the string coupling constant that governs the genus expansion of two-dimensional surfaces, and discuss its properties and consequences. We construct and discuss the correlation functions of an infinite set of pointlike and loop operators to all orders in the genus expansion.

We find the differential equation satisfied by the exact specific heat for a series of models of matter coupled to 2D gravity, indexed by positive integers p and q, and present evidence that they are the c < 1 minimal models with diagonal modular invariants.

We study the correlation functions of microscopic loops (local operators) and of macroscopic loops in non-perturbative two dimensional quantum gravity. They are easily calculated using a free fermion formalism. The microscopic loop correlation functions can be expressed in terms of the KdV flows. The specific heat as a function of the scaling fields obeys the generalized KdV equation. The physical interpretation of macroscopic loop correlation functions is discussed.

We consider a supersymmetric discretized string. The full string theory is defined as the sum over the triangulations of the surface, which is imbedded in the superspace. In the continuum limit such a string theory is described by an appropriate Wess-Zumino model. We present an explicit computation of the properties of the string in the 1D case: we find that supersymmetry is spontaneously broken.

We study the m = 3 multicritical string model at d = 0, and we find a solution free of singularities on the real axis. We discuss the consequences of such a finding on the behaviour of macroscopic loops.

We present the loop equations of motion which define the correlation functions for loop operators in two-dimensional quantum gravity. We show that non-perturbative correlation functions constructed from real solutions of the Painlevé equation of the first kind violate these equations by non-perturbative terms.

We argue that the leading weak coupling nonperturbative effects in closed string theories should be of order exp(−C/κ) where κ² is the closed string coupling constant. This is the case in the exactly soluble matrix models. These effects are in principle much larger than the exp(−C/κ²) effects typical of the low energy field theory. We argue that this behavior should be generic in string theory because string perturbation theory generically behaves like (2g)! at genus g.

We study the continuum Schwinger-Dyson equations for nonperturbative two-dimensional quantum gravity coupled to various matter fields. The continuum Schwinger-Dyson equations for the one-matrix model are explicitly derived and turn out to be a formal Virasoro condition on the square root of the partition function, which is conjectured to be the τ function of the KdV hierarchy. Furthermore, we argue that general multi-matrix models are related to the W algebras and suitable reductions of KP hierarchy and its generalizations.

We give a derivation of the loop equation for two-dimensional gravity from the KdV equations and the string equation of the one-matrix model. We find that the loop equation is equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the Virasoro algebra. We give an interpretation of these equations in topological gravity and discuss their extension to multi-matrix models. For the multi-critical models the loop equation naturally singles out the operators corresponding to the primary fields of the minimal models.

The continuum Schwinger–Dyson equations for Ihe two-matrix model, around Ihe Ising critical point, are derived for operators involving one of the two matrices. It is shown that, within the space of the corresponding couplings, the resulting constraints obey a W₃ algebra.

These are notes based on two lectures given at the Conference on Geometry and Topology (Harvard University, April 1990). The first was mainly devoted to explaining a conjecture according to which stable intersection theory on moduli space of Riemann surfaces is governed by the KdV hierarchy of integrable equations. The second lecture was primarily an introduction to the “hermitian matrix model” of two-dimensional gravity, which is a crucial part of the background for the conjecture. Analogous but more general theories also exist and are sketched in these notes. The generalization in the first lecture involves considering intersection theory on the moduli space of pairs consisting of a Riemann surface Σ and a holomorphic map of Σ to a fixed Kähler manifold K. The simplest analogous generalization in the second lecture involves a chain of hermitian matrices.

We prove the equivalence between the recent matrix model formulation of 2D gravity and lattice integrable models. For even potentials this system is the Volterra hierarchy, and many properties of the continuum matrix model like the Virasoro conditions on the partition function stem directly from the integrability properties of the lattice model and its hamiltonian properties.

The O(n) model on a two-dimensional dynamical random lattice is reformulated as a random matrix problem. The critical properties of the model are encoded in the spectral density of the random matrix which satisfies an integral equation with Cauchy kernel. The analysis of its singularities shows that the model can be critical for − 2 ≤ n ≤ 2 and allows the determination of the anomalous dimensions of an infinite series of magnetic operators. The results coincide with those found in Ref. 11 for 2d quantum gravity.

Unoriented surfaces generated by real symmetric one-matrix models are solved in the scaling limit in which the size of the matrix (related to the string coupling constant) goes to infinity and the cosmological constant approaches a multicritical point of a suitably chosen potential. The solution involves skew orthogonal polynomials, and in spite of the non-local character of the operations d/dx or multiplication by x acting on these polynomials, a local differential formalism is shown to be present in this problem as well. The Gel'fand–Dikii pseudo-differential operator ((∂² + f)^m−1/2)₊ appears here factorized as a product of two differential operators of degrees m and (m − 1) respectively. The relations with other ensembles of random matrices are examined and the difficulties associated with multi-matrix models are pointed out.

We consider a formulation of nonperturbalive two-dimensional quantum gravity coupled to a single bosonic field (d = 1 matter). Starling from a matrix realization of the discretized model, we express the continuum theory as a double scaling limit in which the 2D cosmological constant g tends towards a critical value g_c, and the string coupling 1/N → 0, with the scaling parameter α ∝ In{g − g_c)/(g − g_c)N held fixed. We find that in this formulation logarithmic corrections already present at tree level persist to all higher genus, suggesting a behavior different from the previously considered cases of d < 1 matter.

We describe a field theoretic formulation for one-dimensional string theory. It is given by the collective field representation of the matrix model and leads to a physical interpretation of the theory as that of a massless scalar field in two dimensions. The additional dimension, coming from the large-N color of the matrix model, has an extent which goes to infinity in the continuum limit. The interactions of the field theory are non-zero only at the boundaries of this additional dimension.

We discuss the singlet sector of the d = 1 matrix model in terms of a Dirac fermion formalism. The leading order two- and three-point functions of the density fluctuations are obtained by this method. This allows us to construct the effective action to that order and hence provide the equation of motion. This equation is compared with the one obtained from the continuum approach. We also compare continuum results for correlation functions with the matrix model ones and discuss the nature of gravitational dressing for this regularization. Finally, we address the question of boundary conditions within the framework of the d = 1 unitary matrix model, considered as a regularized version of the Hermitian model, and study the implications of a generalized action with an additional parameter (analogous to the θ parameter) which give rise to quasi-periodic wave functions.

We discuss two results: (i) we calculate the two-point function of the density fluctuations to in the fermonic formulation of the d = 1 string theory and compare with the result from the candidate collective field Hamiltonian. The latter result is divergent, showing the inequivalence of the two theories. We find out the corrections to the collective field Hamiltonian (both in the form of infinite counterterms and additional finite pieces) needed to match with the fermion theory, (ii) We study tree-level scattering processes between bosons due to the localized interaction near the boundary (in a region of order ). The reflection problem at the boundary is treated by an analytic continuation of the time-of-flight variable.

We investigate the double-scaled free fermion theory of the c = 1 matrix model. We compute correlation functions of the eigenvalue density field and compare with the predictions of a relativistic boson theory. The c = 1 theory behaves as a relativistic theory at long distances, but has softer behavior at short distances. The soft short-distance behavior is closely related to the breakdown of the topological expansion at high energies. We also compute macroscopic loop amplitudes at c = 1, finding an integral representation for n-loop amplitudes to all orders of perturbation theory. We evaluate the integrals explicitly for two, three, and four macroscopic loops. The small loop length asymptotic expansion then gives correlation functions of local operators in the theory. The two-macroscopic-loop formula gives information on the spectrum and wave functions in the theory. The three- and four-loop amplitudes give scattering amplitudes for tachyon operators to all orders of perturbation theory. Again, the topological expansion breaks down at high energies. We compare our amplitudes with predictions from the Liouville theory.

We find the general classical solution of the Das–Jevicki collective field theory, corresponding to a tachyon background in (1 + 1)-dimensional string theory. The solution has a simple interpretation in the equivalent free Fermi theory, as a state with a dynamical Fermi surface. In terms of the variables corresponding to the upper and lower Fermi momenta, the collective field hamiltonian separates into right- and left-moving pieces. As one application, we discuss the tree-level S-matrix. We also describe briefly a number of interesting particular solutions.

We discuss random matrix-model representations of D = 1 string theory, with particular emphasis on the case in which the target space is a circle of finite radius. The duality properties of discretized strings are analyzed and shown to depend on the dynamics of vortices. In the representation in terms of a continuous circle of matrices we find an exact expression for the free energy, neglecting non-singlet states, as a function of the string coupling and the radius which exhibits exact duality. In a second version, based on a discrete chain of matrices, we find that vortices induce, for a finite radius, a Kosterlitz–Thouless phase transition that takes us to a c = 0 theory.

The following sections are included:

• ADDENDUM: W_∞