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We discuss the impact of quadratic quantum fluctuations on the Wilson loop extracted from classical string theory. We show that a large class of models, which includes the near horizon limit of D_p branes with 16 supersymmetries, admits a L\"{u}scher type correction to the classical potential. We confirm that the quantum determinant associated with a BPS configuration of a single quark in the AdS_5 \times S^5 model is free from divergences. We find that for the Wilson loop in that model, unlike the situation in flat space-time, the fermionic determinant does not cancel the bosonic one. For string models that correspond to gauge theories in the confining phase, we show that the correction to the potential is of a L\"{u}scher type and is attractive., Comment: 1+33 pages, 3 figures, a minor correction, references added

At high temperatures, some SU (N) gauge theories with fermions have metastable states related to the ZN symmetry of the theory. It has recently been argued that the possible existence of domains of these metastable states may lead to some interesting cosmological consequences. We show that these states, generically, have unaacceptable thermodynamic behavior. Their free energy F∝T4 with a positive proportionality constant. This leads not only to negative pressure but also to negative specific heat and, more seriously, to negative entropy. We argue that although such domains are important in the euclidean theory, they cannot be interpreted as physical domains in Minkowski space. This conclusion remains valid even if sufficient color singlet particles (such as leptons and weak gauge bosons) are added to the system to make the total free energy negative.

Particles in two spatial dimensions with fractional statistics known, generically, as anyons, have been of interest to particle physicists for nearly ten years. A major change in the direction of research occurred when it was discovered that anyons could play a role as quasiparticles in condensed-matter systems. This was originally discovered to be the case in systems exhibiting the Fractional Quantum Hall Effect. The application of anyons to condensed-matter systems received yet another boost when it was discovered by Laughlin that even an ideal gas of anyons was a superfluid and, as a result, a gas of charged anyons would be a superconductor. This led immediately to attempts to explain the superconductivity of high-T{sub c} materials which are layered ceramics in terms of anyons. The main challenge was to find a reasonable model for these materials which has quasiparticles obeying anyonic statistics. The goal of this article is to review the theory of anyonic superconductivity and its possible relation to high-T{sub c} materials. The emphasis in this review is on field-theoretical methods. In this paper the authors explain what an anyon is and how it can be modeled mathematically. The authors discuss the possible relationship between anyons and high-T{sub c}more » materials. The authors review several of the attempts to obtain anyonic quasiparticles from the Hubbard model which is commonly used to describe these materials. The authors describe the mathematical modeling of anyons in terms of their interaction with an Abelian gauge field with a Chern-Simons term. This description of anyons is used extensively in this article. The authors discuss the possible criteria for superconductivity in anyonic systems with particular emphasis on criteria which would be useful in the Chern-Simons description.« less

We attempt the numerical construction of an effective action in three dimensions for Ising spins which represent the Wilson lines in the four-dimensional SU(3) gauge theory at finite temperature. For each configuration of the gauge theory, each spin is determined by averaging the Wilson lines over a small neighborhood and then projecting the average to +/-1 according to whether the neighborhood is ordered or disordered. The effective Ising action, determined via the lattice Schwinger-Dyson equations, contains even (two-spin) and odd (one- and three-spin) terms with short range. We find that the truncation to Ising degrees of freedom produces an effective action which is discontinuous across the gauge theory's phase transition. This discontinuity may disappear if the effective action is made more elaborate., Comment: 12 pages, revtex, including 2 Postscript figures; conclusions have changed slightly to conform to the theorem of Van Enter, Fernandez, and Sokal on the continuity of RG maps

We study the Quantum Field Theory of nonrelativistic bosons coupled to a Chern--Simons gauge field at nonzero particle density. This field theory is relevant to the study of anyon superconductors in which the anyons are described as {\bf bosons} with a statistical interaction. We show that it is possible to find a mean field solution to the equations of motion for this system which has some of the features of bose condensation. The mean field solution consists of a lattice of vortices each carrying a single quantum of statistical magnetic flux. We speculate on the effects of the quantum corrections to this mean field solution. We argue that the mean field solution is only stable under quantum corrections if the Chern--Simons coefficient $N=2\pi\theta/g^2$ is an integer. Consequences for anyon superconductivity are presented. A simple explanation for the Meissner effect in this system is discussed., Comment: 10 Pages

The existence and uniqueness of a Yang-Mills potential for a given field strength is discussed. Previous results are reviewed. In two space-time dimensions every nonzero field strength can be derived from any element of a very large class of potentials which are not related by a gauge transformation. We exhibit these potentials explicitly. In four dimensions we show that both a constant electric field and a Coulomb field can be described by a variety of nongauge-related potentials which describe very different physical situations.

SU(n) gauge theories at finite temperature are examined in terms of the spontaneous breaking of a Z(n) symmetry. It is suggested that the phase transition at finite temperature results from the formation of "bubbles" of an alternate Z(n) vacuum. These Z(n) bubbles are analyzed for lattice gauge theories in the dilute gas approximation and the results are compared with the Monte-Carlo calculations of McLerran and Svetitsky.

The phase structure of gauge theories with fermions as a function of imaginary chemical potential is related to the confining properties of the theory. It is controlled by a remnant of the ZN which is present in the absence of fermions. At high temperature the theory has a first-order phase transition as a function of imaginary chemical potential . This transition is expected to be absent in the low-temperature phase. Monte Carlo simulations are proposed which would check these ideas. It is shown that properties of the theory at nonzero fermion density can be deduced from its behavior at finite imaginary chemical potential.

$\mathrm{SU}(n)$ gauge theories at finite temperature $T={\ensuremath{\beta}}^{\ensuremath{-}1}$ are analyzed in terms of the spontaneous breakdown of a $Z(n)$ symmetry corresponding to the order parameter $L(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})=(\frac{1}{n})\mathrm{Tr}P\mathrm{exp}[ig\ensuremath{\int}{0}^{\ensuremath{\beta}}{A}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}},t)dt]$. An "effective potential" for $L$ is evaluated in the one-loop approximation for both the continuum and the lattice gauge theories. It is shown that the $Z(n)$ symmetry is broken, so that the continuum theory does not confine for high temperatures. Similarly the lattice theory does not confine for sufficiently weak coupling if the number of time sites ${N}_{t}$ is finite. It is argued that as ${N}_{t}\ensuremath{\rightarrow}\ensuremath{\infty}$ the $Z(n)$ symmetry is restored and the theory will confine for all values of the coupling.

A relativistic quantum field theory at finite temperature T = β−1 is equivalent to the same field theory at zero temperature but with one spatial dimension of finite length β. This equivalence is discussed for scalars, for fermions, and for gauge theories. The relationship is checked for free field theory. The translation of correlation functions between the two formulations is described with special emphasis on the nonlocal order parameters of gauge theories. Possible applications are mentioned.

The phases of $\mathrm{SU}(N)$ gauge theories at finite temperature can be analyzed in terms of the Wilson line $L(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})=\mathrm{Tr}P\mathrm{exp}[ig \ensuremath{\int}{0}^{\ensuremath{\beta}}{A}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}},\ensuremath{\tau})d\ensuremath{\tau}]$. The question of confinement is related to the spontaneous breakdown of a $Z(N)$ symmetry: $〈L〉\ensuremath{\ne}0$ corresponds to symmetry breaking and deconfinement, $〈L〉=0$ implies symmetry restoration and confinement. A previous calculation of the one-loop effective potential for $L$ is discussed in $\mathrm{SU}(N)$ theories, both with and without fermions. A possible mechanism for the confining-deconfining transition in terms of the formation of "$Z(N)$ bubbles" is discussed. For SU(2) theories, the $Z(2)$ bubbles are simply finite-temperature instantons. However, $Z(N)$ bubbles are not finite-temperature instantons for $\mathrm{SU}(N)$. It is shown that a bag picture of hadronic structure similar to that of Callan, Dashen, and Gross may emerge from this picture.

We study in detail the classical Yang-Mills field equations in the presence of static external sources. Their formulation as an initial-value problem in the ${A}_{0}=0$ gauge provides us with a powerful tool for determining the existence of new solutions. In the case of point sources, the only static solutions known so far are the various Coulomb solutions which we classify according to their total energy and isospin. In the case of a localized but extended source, there are, besides the well-known Coulomb solution, two new types of solutions: the "magnetic dipole" solution which has the long-range behavior of a magnetic dipole field and which has lower energy than the Coulomb solution when the total external charge is large enough, and the "total screening" solution which has no long-range field strengths at all and which can have an arbitrarily low energy. We present a detailed study of these new solutions.

We present two new solutions to the classical Yang-Mills field equations in the presence of a localized external source. These solutions totally screen the charge of the source. They have lower energy than the corresponding Coulomb solution.