Non-planar diagrams in the large N limit of U(N) and SU(N) lattice gauge theories

It is shown that the limit as N → ∞ with g²N fixed of the strong coupling expansion for the vacuum expectation values of a U(N) or SU(N) lattice gauge theory is not given by a sum of planar diagrams. This contradicts a result claimed by De Wit and 't Hooft.     

Comparing O(N) and SU(N) × SU(N) spin systems in 1 + 1 dimensions to SU(N) gauge theories in 3 + 1 dimensions

We have computed the scale breaking Λ parameters of the euclidean and hamiltonian formulations of the lattice regulated O(N) and SU(N) × SU(N) spin systems in 1 + 1 dimensions in terms of the Λ_PV parameters of the Pauli-Villars regulated continuum models. Using lattice perturbation theory, the renormalized mass gap has been determined in terms of Λ_PV for each model. These results are compared to analogous calculations in SU(N) gauge theories.     

Gaussian fluctuations to U(N) and SU(N) lattice gauge theories in the mean field approximation

The mean field can be considered as a classical solution of an appropriately reformulated version of lattice gauge theories. Axial gauge fixing renders it stable. The quadratic forms for the fluctuations in the gaussian approximation are analyzed. The gaussian correction to the mean field free energy is expressed for all U(N) and SU(N) in terms of structure functions that are explicitly calculated for U(N), SU(∞, and SU(∞) numerical calculations are performed for the phase transition point, its latent heat, and some correlation lengths that are characteristic for this kind of mean field approach.     

SU(N) invariant non-linear σ models

Two series of SU(N) invariant non-linear σ models are constructed using exceptional orbits of the adjoint representation of SU(N). Each of these models possesses infinitely many local as well as non-local conserved charges.     

Nonconvergence of the 1/N expansion for SU(N) gauge fields on a lattice

We present specific examples that demonstrate the non-convergence of the 1/N expansion for the lattice theory of SU(N) gauge fields.     

The phase structure of SU(N)/Z(N) lattice gauge theories

The phase structure of pure SU(N)/Z(N) lattice gauge theories in four dimensions is discussed. The presence of ZN monopoles plausibly leads to a phase transition. A Monte Carlo simulation of SO(3) shows the presence of a very strong, may be first order, phase transition.     

Chiral symmetry,the 1/N expansion and the SU(N) thirring model

In the two-dimensional SU(N) Thirring model, the 1/N expansion seems to predict spontaneous breaking of the continuous chiral symmetry. This is impossible in two-dimensions. Reasoning along the lines of Berezinski, Kosterlitz and Thouless for the two-dimensional XY model, we argue that, in fact, rather than showing long-range order, $\text{〈}\text{ψ}\text{ψ(x)}\text{ψ}\text{ψ(0)〉}$ vanishes in this model as |x|^?1/N at large |x|. The 1/N expansion is, in fact, a rather good guide to the properties of this model.     

Gauge hierarchies of SU(N) grand unification

The structure of spontaneous breaking of SU(N) gauge symmetry for grand unification is investigated. The results obtained are applied to the analysis of SU(8) symmetry for which possible ways of breaking and intermediate symmetries are considered. It is assumed that the SU(8) group unifies the subgroups of colour, standard electroweak and horizontal symmetries. We find conditions which it is necessary to impose on the vacuum expectation values of Higgs multiplets to provide an arbitrary breaking pattern of SU(N) symmetry and conserve any intermediate symmetry. If in the SU(8) models considered fermions and mirror fermions do not violate the (V-A) and (V+A) structure of weak interactions, then their masses should not be greater than ～10² GeV. It is also shown that the contributions of fermion and Higgs multiplets to the renormalization group equation for the coupling constant of any subgroup of SU(N) are identical. Renormalization group identities for the case of arbitrary SU(N) breaking are given where the contribution of Higgs multiplets have been taken into account (but they cancel each other). Using these identities one can calculate the mass values for the breaking of the intermediate symmetries in the SU(8) models, and also exclude part of the possible breaking patterns.     

On the large N limit of conformal field theory

Following recent advances in large N matrix mechanics, I discuss here the free (Cuntz) algebraic formulation of the large N limit of two-dimensional conformal field theories of chiral adjoint fermions and bosons. One of the central results is a new affine free algebra which describes a large N limit of affine Lie algebra. Other results include the associated free-algebraic partition functions and characters, a free-algebraic coset construction, free-algebraic construction of , free-algebraic vertex operator constructions in the large N Bose systems, and a provocative new free-algebraic factorization of the ordinary Koba-Nielsen factor.     

Non-local continuity equations for self-dual SU(N) Yang-Mills fields

Using a manifestly gauge invariant formulation, an infinite set of classical non-local continuity equations is constructed for the self-dual SU(N) Yang-Mills fields.     

All-loop group-theory constraints for color-ordered SU(N) gauge-theory amplitudes

We derive constraints on the color-ordered amplitudes of the L-loop four-point function in SU(N) gauge theories that arise solely from the structure of the gauge group. These constraints generalize well-known group theory relations, such as U(1) decoupling identities, to all loop orders.     

Axially symmetric static solutions of four-dimensional SU(N) σ-models by the inverse scattering method

A systematic framework is derived for constructing a superpotential in static, axially symmetric four-dimensional SU(N) principal σ-models by applying an inverse scattering method.     

The quantum collective field method and its application to the planar limit

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.     

Improved mean field studies of SU(N) chiral models and comparison with numerical simulations

SU(N) × SU(N) matrix models are investigated using a systematic expansion around the classical mean field result. One-loop corrections for N = 2?5, and ∞ are computed for the two-, three-, and four-dimensional models. The results are in good agreement with numerical simulations of these models.     

Mean field and Monte Carlo studies of SU(N) chiral models in three dimensions

SU(N) chiral models defined on three-dimensional cubic lattices arestudied using mean field and Monte Carlo techniques. Mean field theory predicts first-order transitions for all finite N greater than two. The mean field estimates of the transition temperature and discontinuity of the order parameter are in good agreement with computer simulations for N = 3 and 4. The N → ∞ limit of mean field theory has a first-order phase transition.     

Kinematics of periodic structures in SU(N) gauge theories

We study periodicity by use of the holonomy group. For a general field configuration, which spans the whole SU(N) group, we show that periodicity of gauge-invariant quantities implies that either the flux is quantized or that there is no flux at all. Finally we show that in a special gauge, the quantized 't Hooft flux can be expressed as a line integral over one of the components of the vector potential. The other components then have “Higgs-like” zeros, with winding numbers related to the 't Hooft flux.     

Grand unified theories in the framework of the SU(N) group

Due to the problem of flavour, the general SU(N) gauge group is considered as a candidate for grand unification. Starting from a set of general requirements, we find restrictions and conditions for the construction of grand unified theories within the framework of such a group. The allowed groups get restricted to SU(6), SU(7) and SU(8).     

Inequalities for order and disorder parameters in SU(N) gauge theories

We discuss inequalities of the Mack-Fröhlich type in SU(N) gauge theories and their possible implications for the various phases of those theories.     

The external field problem in the large N limit of QCD

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.