Duality for Z(N) gauge theories

The duality properties of simple Z(N) gauge theories are discussed. For N ? 4 we find self duality in four dimensions and we give the transition points. For N > 4 these systems are not self dual. Also, the order parameter is discussed. The general Z(N) gauge theory is found to be self dual for all N.     

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.     

Transition from strong to weak coupling in SU(N) lattice gauge theories

A possible explanation is proposed for the crossover from strong to weak coupling region in SU(N) lattice gauge theories. We predict the pointswhere the crossover takes place for all SU(N)M: For example, g² ≈ 2.0 for SU(2), g² ≈ 1.0 for SU(3) and lim_N→∞Ng²(SU(N) ≈ 2.0.     

Monte Carlo studies of SU(N)/ZN lattice gauge theories in four dimensions

Using Monte Carlo techniques on a four-dimensional space-time lattice, we study SU(N)/Z_N gauge theories for N = 3, 4, 5 and 6. We find first-order phase transitions at critical inverse temperatures of β_c = 6.40, 12.0, 19.5 and 32.0 and SU(3)/Z₃,SU(4)/Z₄,SU(5)/Z₅and SU (6)Z₆, respectively.     

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.     

Dynamical generation of a Coulomb phase in the mean-field approach to Z(N) lattice gauge theories

The phase diagram of Z(N) lattice gauge theories is examined in the mean-field approach. It is shown how large non-perturbative fluctuations around the mean-field may naturally occur in these theories. When this happens, a massless phase which is absent in the mean-field approximation is dynamically generated. An order parameter which characterizes the Coulomb to Higgs phase transition is introduced. The predictions for the values of the critical coupling for this transition are in excellent agreement with the Monte Carlo data in four dimensions.     

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.     

Comparison of lattice gauge theories with gauge groups Z2 and SU(2)

We study a model of a pure Yang Mills theory with gauge group SU(2) on a lattice in Euclidean space. We compare it with the model obtained by restricting variables to $\text{Z}$₂. An inequality relating expectation values of the Wilson loop integral in the two theories is established. It shows that confinement of static quarks is true in our SU(2) model whenever it holds for the corresponding $\text{Z}$₂-model. The SU(2) model is shown to have high and low temperature phases that are distinguished by a qualitatively different behavior of the 't Hooft disorder parameter.     

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.     

Phase structure of SU(2) and SU(3) lattice gauge theories

Using a mean field we obtain the phase diagrams for SU(2) and SU(3) with a mixed fundamental-adjoint representation action.     

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.     

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 large-N phase transition in the U(N) lattice gauge theory

We present an outline for a proof (the precise details of which will be presented in a follow-up paper) of a large-N phase transition in dimensions greater than two. The critical couplings are calculated in d=3 and d=4 and are found to be β=0.44 and β=0.40, respectively.     

The phase structure of mixed lattice gauge theories

The mixed compact-non-compact U(1) model is shown to be equivalent to a compact U(1) Higgs model. It is argued that the mixed SO(3)-SU(2) model is dual to an SO(3) gauge theory coupled to a scalar field in the fundamental representation. The degrees of freedom are Z(2) monopoles and charges or, in a dual picture, monopoles and loops. This picture is supported by a Monte Carlo calculation. The implications for the SU(2) transition region are discussed.     

The deconfinement transition in SU(2) lattice gauge theories

Using Monte Carlo techniques we study the SU(2) Yang-Mills system at finite temperatures for two different forms of lattice action, proposed by Villain and Monton respectively. In both cases the energy density ε exhibits a similar behaviour to the case of the Wilson action, being an order of magnitude smaller at lower temperatures and approaching the Stefan-Boltzmann limit for a free gluon gas at higher temperatures. The transition between these two temperature regimes appears rather abrupt and the specific heat of gluon matter exhibits a sharp peak at the transition point. The transition temperature, expressed in MeV, is found to be consistent in both the cases with that obtained by using the Wilson action, although in the natural units of the corresponding Λ-parameters it differs substantially, being 10.7, 27.3 and 42.8 for Manton, Villain and Wilson actions, respectively.     

A boson representation for SU (N) lattice gauge theories

SU(N) lattice gauge theories are reformulated in terms of fields varying over non-compact spaces ^N , transforming asN dimensional representations of SU(N) and integrated with Gaussian measure. This reformulation is equivalent to a boson operator representation. Strong coupling expansions based on this formalism do not involve SU(N) vector coupling coefficients.     

The action of SU(N) lattice gauge theory in terms of the heat kernel on the group manifold

We consider the heat kernel on the group manifold as an alternative to the Wilson action in lattice gauge theory, and we exhibit its strict analogy with the well-known Berezinski-Villain action. With the heat kernel action, the Gross-Witten singularity is rigorously absent in two dimensions. The similarity of the heat kernel action to the hamiltonian approach should provide a better convergence of the lagrangian strong coupling expansion, while its behaviour at weak coupling should simplify the analysis of the weak coupling perturbative expansion.     

On the phase structure of five-dimensional SU(2) gauge theories with anisotropic couplings

The phase diagram of five-dimensional SU(2) gauge theories is explored using Monte Carlo simulations of the theory discretized on a Euclidean lattice using the Wilson plaquette action and periodic boundary conditions. We simulate anisotropic gauge couplings which correspond to different lattice spacings a₄ in four dimensions and a₅ along the extra dimension. In particular we study the case where a₅>a₄. We identify a line of first order phase transitions which separate the confined from the deconfined phase. We perform simulations in large volume at the bulk phase transition staying in the confined vacuum. The static potential measured in the hyperplanes orthogonal to the extra dimension hints at dimensional reduction. We also locate and analyze second order phase transitions related to breaking of the center along one direction.