Alexander A. Migdal

Critical Properties of Randomly Triangulated Planar Random Surfaces

A discrete version of the Polyakov string is studied by analytical and numerical methods. The role of the intrinsic metric is played by random triangulation. The results only qualitatively agree with the Liouville perturbation theory. In particular, the critical exponents for the solvable cases D = 0 and D = −2 are shown to be larger than those calculated perturbatively. Our numerical simulations for D = 3 indicate a large but finite Hausdorff dimension dH = 10.0 ±0.2.

A nonperturbative treatment of two-dimensional quantum gravity

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.

Exact equation for the loop average in multicolor QCD

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.

Analytical and numerical study of a model of dynamically triangulated random surfaces

We report the results of our analytical and numerical investigation of a model of randomly triangulated random surfaces which is a discrete analog of the quantized string model suggested by Polyakov. A physical quantity of major interest was the mean square extent ~X 2 ) of the surface embedded in a D-dimensional space as a function of its area A. In the exactly solvable case, D = -2, the law (X 2 ) - In(A) is found. Some exact properties of the phase diagram of the model in the limiting cases D ~ _+ oo are discussed. The numerical results for D > 0 suggest the existence of a power law (X 2 ) - A 2~ with u * 0 in a rather large area of the phase plane. The distribution of the internal curvature is calculated analytically for D = 0 and measured for some other dimensions. Snapshots of typical triangulations are presented.

Quantum Chromodynamics as Dynamics of Loops

Abstract QCD is entirely reformulated in terms of white composite fields—the traces of the loop products. The 1/ N expansion turns out to be the WKB (Hartree-Fock) approximation for these fields. The “classical” equation describing the N = ∞ case is reduced to a bootstrap form. New, manifestly gauge-invariant perturbation theory in the loop space, reproducing asymptotic freedom, is developed by iterations of this equation. The area law appears to be a self-consistent solution at large loops.

Recent progress in the theory of noncritical strings

Abstract 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 are 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.

Induced gauge theory at large N

Abstract We propose and study at large N a new lattice gauge model, in which the Yang-Mills interaction is induced in the naive one-loop approximation by the heavy scalar field in the adjoint representation. At any dimension of space and any N the gauge fields can be integrated out yielding an effective field theory for the gauge-invariant scalar field, corresponding to eigenvalues of the initial matrix field. This field develops a vacuum average, the fluctuations of which describe elementary excitations of our gauge theory. At N = ∞ we find two phases of the model: the strong-coupling phase which may be relevant to QCD (if there are no phase transitions at some critical N ), and the weak-coupling phase. We have derived exact nonlinear integral equations for the vacuum average and for the scalar excitation spectrum.We propose and study at large N a new lattice gauge model , in which the Yang-Mills interaction is induced by the heavy scalar field in adjoint representation. At any dimension of space and any

Dilaton effective lagrangian in gluodynamics

N

Instantons in the Burgers equation.

the gauge fields can be integrated out yielding an effective field theory for the gauge invariant scalar field, corresponding to eigenvalues of the initial matrix field. This field develops the vacuum average, the fluctuations of which describe the elementary excitations of our gauge theory. At

Simulations of four-dimensional simplicial quantum gravity

N= \infty