Nonanalyticity in scale in the planar limit of QCD

Using methods of numerical lattice gauge theory we show that, in the limit of a large number of colors, properly regularized Wilson loops have an eigenvalue distribution which changes nonanalytically as the overall size of the loop is increased. This establishes a large-N phase transition in continuum planar gauge theory, a fact whose precise implications remain to be worked out.     

Two dimensional lattice gauge theory based on a quantum group

In this article we analyse a two dimensional lattice gauge theory based on a quantum group. The algebra generated by gauge fields is the lattice algebra introduced recently by A.Yu. Alekseev, H. Grosse and V. Schomerus in [1]. We define and study Wilson loops. This theory is quasi-topological as in the classical case, which allows us to compute the correlation functions of this theory on an arbitrary surface.Laboratoire Propre du CNRS UPR 14     

Finite size Wilson loops

A study of Wilson loop averages for finite size loops is initiated. Within the framework of euclidean four-dimensional lattice SU(2) gauge theory with elementary Wilson action we compute the expectation values of all rectangular loops to 12th order in the strong coupling expansion. The leading term for weak coupling is evaluated for loops up to 4 × 4. A comparison to Monte Carlo data is presented. Other related issues are also discussed.     

Effective actions from block diagonalization in lattice gauge theories

It is shown that the first step in Wilson's real space renormalization program for lattice gauge theories — namely the integration of internal degrees of freedom within one block — can be performed for “block-diagonalized” actions, which possess the proper continuum limit, just like the Wilson action. As a result, we obtain, on a lattice with double lattice spacing, an effective action which contains in the weak-coupling limit, in addition to the familiar plaquette terms, planar Wilson loops with six and eight links.     

Numerical study of the Wilson loop-plaquette correlations for four-dimensional compactU (1) gauge theory

The Wilson loop-plaquette correlations for 3×3 Wilson loops are investigated by Monte Carlo simulations of four-dimensional compactU(1) gauge theory.     

Brief introduction to Wilson loops and large <Emphasis Type="Italic">N</Emphasis>

A brief pedagogical introduction to Wilson loops, lattice gauge theory, and 1/N expansion in QCD is presented.     

Three-dimensional lattice gauge theory and strings

We consider SU(2) lattice gauge theory in three dimensions. The Wilson loops are found to be well described by a simple string model in the approximate scaling region.     

Simplicial gauge theory and quantum gauge theory simulation

We propose a general formulation of simplicial lattice gauge theory inspired by the finite element method. Numerical tests of convergence towards continuum results are performed for several SU(2) gauge fields. Additionally, we perform simplicial Monte Carlo quantum gauge field simulations involving measurements of the action as well as differently sized Wilson loops as functions of β.     

Adjoint sources in lattice gauge theory

Variance reduction techniques for the evaluation of Wilson loops in lattice gauge theory are analysed. The method is extended to Wilson loops in the adjoint representation. Variational methods are also applied to adjoint sources. The combination of these techniques allows the potential V(R) between two static adjoint sources to be determined in SU(2) gauge theory. One isolated static adjoint source is also studied and the energy and distribution of the gluon field of this “glue-lump” is obtained. This is relevant to the saturation of the adjoint potential V(R) at large R.     

Some approximate calculations in SU(2) lattice mean field theory

Approximate calculations are performed for small Wilson loops of SU(2) lattice gauge theory in the mean field approximation. Reasonable agreement is found with Monte Carlo data. Ways of improving these calculations are discussed.     

Constraints on two-dimensional conformal field theories from knot invariants

The connection between Witten's topological three-dimensional gauge theory and RCFTs provides a natural setting to study the interplay between surface diffeomorphisms and intertwining of Wilson loops. These considerations lead directly to constraints on RFCTs including those previously derived by Vafa.     

Monte Carlo studies of wilson loops in four-dimensional U(2) gauge theory

Wilson loops are calculated using Monte Carlo simulations for pure U(2) gauge theory on a 6⁴ lattice. The loops appear to contain an area law piece in both the high and low temperature regions. The string tension is discontinuous at β = β_c, where β_c is the critical inverse temperature. This suggests that the first-order phase transition in U(2) gauge theory is not a deconfining phase transition. The determinant of the Wilson loop, however, extracts the U(1) part of the theory and appears to lose the area law at low temperature.     

Field theory on a non‐commutative plane: a non‐perturbative study

The 2d gauge theory on the lattice is equivalent to the twisted Eguchi–Kawai model, which we simulated at N ranging from 25 to 515. We observe a clear large N scaling for the 1‐ and 2‐point function of Wilson loops, as well as the 2‐point function of Polyakov lines. The 2‐point functions agree with a universal wave function renormalization. The large N double scaling limit corresponds to the continuum limit of non‐commutative gauge theory, so the observed large N scaling demonstrates the non‐perturbative renormalizability of this non‐commutative field theory. The area law for the Wilson loops holds at small physical area as in commutative 2d planar gauge theory, but at large areas we find an oscillating behavior instead. In that regime the phase of the Wilson loop grows linearly with the area. This agrees with the Aharonov‐Bohm effect in the presence of a constant magnetic field, identified with the inverse non‐commutativity parameter. Next we investigate the 3d λϕ⁴ model with two non‐commutative coordinates and explore its phase diagram. Our results agree with a conjecture by Gubser and Sondhi in d = 4, who predicted that the ordered regime splits into a uniform phase and a phase dominated by stripe patterns. We further present results for the correlators and the dispersion relation. In non‐commutative field theory the Lorentz invariance is explicitly broken, which leads to a deformation of the dispersion relation. In one loop perturbation theory this deformation involves an additional infrared divergent term. Our data agree with this perturbative result. We also confirm the recent observation by Ambjø rn and Catterall that stripes occur even in d = 2, although they imply the spontaneous breaking of the translation symmetry.     

“Loop the loop”

The applicability of the collective coordinate method (saddle-point approximation) for large-N planar models is discussed. Some unstated assumptions are clarified. Statements that Wilson loops form a complete set of gauge invariant operators are also examined and a set of generalized algebraic Mandelstam relations among Wilson loops is presented. The inclusion of loops that wind around themselves and cross many times, as independent variables, is stressed.     

Comments on the Morita equivalence

It is known that the noncommutative Yang-Mills (YM) theory with periodical boundary conditions on a torus at a rational noncommutativity parameter value is Morita equivalent to the ordinary YM theory with twisted boundary conditions on a dual torus. We give a simple derivation of this fact. We describe the one-to-one correspondence between these two theories and the corresponding gauge invariant observables. In particular, we show that under the Morita map, the Polyakov loops in the ordinary YM theory are converted to the open noncommutative Wilson loops discovered by Ishibashi, Iso, Kawai, and Kitazawa.     

U(3) four-dimensional lattice gauge theory and Monte Carlo calculations

Monte Carlo results for the pure U(3) lattice gauge theory on a 6⁴ lattice are reported. Wilson loops and the string tension are presented. The first-order phase transition in U(3) is reflected quite clearly in a discontinuity in the string tension at β = β_c. The U(1) factor of U(3) is extracted using the determinant of the Wilson loops. As expected, the U(1) component appears to deconfine at the phase transition..     

Wilson loops in the light of spin networks

If G is any finite product of compact orthogonal, unitary and symplectic matrix groups, then Wilson loops generate a dense subalgebra of continuous observables on the configuration space of lattice gauge theory with structure group G. If G is orthogonal, unitary or symplectic, then Wilson loops associated to the natural representation of G are enough.

This extends a result of Sengupta [Proc. Am. Math. Soc. 1221 (3) (1994) 897] and earlier work by Durhuus [Lett. Math. Phys. 4 (6) (1980) 515]. In particular, our approach includes the cases of even orthogonal and symplectic groups.     

Hedgehogs in Wilson loops and phase transition in SU(2) Yang–Mills theory

We suggest that the gauge-invariant hedgehog-like structures in the Wilson loops are physically interesting degrees of freedom in the Yang–Mills theory. The trajectories of these “hedgehog loops” are closed curves corresponding to center-valued (untraced) Wilson loops and are characterized by the center charge and winding number. We show numerically in the SU(2) Yang–Mills theory that the density of hedgehog structures in the thermal Wilson–Polyakov line is very sensitive to the finite-temperature phase transition. The (additively normalized) hedgehog line density behaves like an order parameter: The density is almost independent of the temperature in the confinement phase and changes substantially as the system enters the deconfinement phase. In particular, our results suggest that the (static) hedgehog lines may be relevant degrees of freedom around the deconfinement transition and thus affect evolution of the quark–gluon plasma in high-energy heavy-ion collisions.     

Glueball mass calculations on an array of computers

We have calculated the mass of the 0⁺ glueball in SU(2) pure gauge theory in 4 dimensions, with very high statistics. The computation was done on an array of microprocessors with nearest-neighbor connections which run concurrently. We discuss, in detail, the implementation of the pure gauge algorithm for SU(2) and SU(3) and also the algorithm for calculating arbitrarily shaped Wilson loops on the array. The extension of these algorithms to the inclusion of dynamical fermions is also discussed. Finally, we present the results of our variational calculation of glueball masses which are in agreement with published results.