Narrow Results

The equations of motion of quantum Yang–Mills theory (in the planar "large-N" limit), when formulated in loop-space are shown to have an anomalous term, which makes them analogous to the equations of motion of WZW models. The anomaly is the Jacobian of the change of variables from the usual ones, i.e. the connection one-form A, to the holonomy U. An infinite-dimensional Lie algebra related to this change of variables (the Lie algebra of loop substitutions) is developed, and the anomaly is interpreted as an element of the first cohomology of this Lie algebra. The Migdal–Makeenko equations are shown to be the condition for the invariance of the Yang–Mills generating functional Z under the action of the generators of this Lie algebra. Connections of this formalism to the collective field approach of Jevicki and Sakita are also discussed.

We give observations about dualities where one of the dual theories is geometric. These are illustrated with a duality between the simple harmonic oscillator and a topological field theory. We then discuss the Wilson loop in the context of the AdS/CFT duality. We show that the Wilson loop calculation for certain asymptotically AdS scalar field space–times with naked singularities gives results qualitatively similar to that for the AdS black hole. In particular, it is apparent that (dimensional) metric parameters in the singular space–times permit a "thermal screening" interpretation for the uark potential in the boundary theory, just like black hole mass. This suggests that the Wilson loop calculation merely captures metric parameter information rather than true horizon information.

We derive an exact equation for simple self non-intersecting Wilson loops in non-Abelian gauge theories with gauge fields interacting with fermions in two-dimensional Euclidean space.

We consider general supersymmetric Wilson loops in ABJM model, which is Chern–Simons-matter theory in (2+1) dimensions with 𝒩 = 6 supersymmetry. The Wilson loops of our interest are so-called Zarembo-type: they have generic contours in spacetime, but the scalar field coupling is arranged accordingly so that there are unbroken supersymmetries. Following the supermatrix construction of Wilson loops by Drukker and Trancanelli and the generalization by Griguolo et al., we study 1/6-BPS Wilson loops and check that their expectation value is protected using perturbation up to two loops. We also study the dual string configuration in AdS₄×ℂℙ³ background and check the supersymmetry.

A novel approach to evaluate the Wilson loops associated with a SU(∞) gauge theory in terms of pure string degrees of freedom is presented. It is based on the Guendelman–Nissimov–Pacheva formulation of composite antisymmetric tensor field theories of area (volume) preserving diffeomorphisms which admit p-brane solutions and which provide a new route to scale-symmetry breaking and confinement in Yang–Mills theory. The quantum effects are discussed and we evaluate the vacuum expectation values (VEV) of the Wilson loops in the large N limit of the quenched reduced SU(N) Yang–Mills theory in terms of a path integral involving pure string degrees of freedom. The quenched approximation is necessary to avoid a crumpling of the string worldsheet giving rise to very large Hausdorff dimensions as pointed out by Olesen. The approach is also consistent with the recent results based on the AdS/CFT correspondence and dual QCD models (dual Higgs model with dual Dirac strings). More general Loop wave equations in C-spaces (Clifford manifolds) are proposed in terms of generalized holographic variables that contain the dynamics of an aggregate of closed branes (p-loops) of various dimensionalities. This allows us to construct the higher-dimensional version of Wilson loops in terms of antisymmetric tensor fields of arbitrary rank which couple to p-branes of different dimensionality.

We discuss a new class of non-renormalization theorems in and Super-Yang-Mills theory, obtained by using a superspace which makes a lower dimensional subgroup of the full supersymmetry manifest. Certain Wilson loops (and Wilson lines) belong to the chiral ring of the lower dimensional supersymmetry algebra, and their expectation values can be computed exactly.

We consider a two-parameter family of ℤ₂ gauge theories on a lattice discretization of a three-manifold and its relation to topological field theories. Familiar models such as the spin-gauge model are curves on a parameter space Γ. We show that there is a region Γ₀ ⊂ Γ where the partition function and the expectation value 〈W_R(γ)〉 of the Wilson loop can be exactly computed. Depending on the point of Γ₀, the model behaves as topological or quasi-topological. The partition function is, up to a scaling factor, a topological number of . The Wilson loop on the other hand, does not depend on the topology of γ. However, for a subset of Γ₀, 〈W_R(γ)〉 depends on the size of γ and follows a discrete version of an area law. At the zero temperature limit, the spin-gauge model approaches the topological and the quasi-topological regions depending on the sign of the coupling constant.

We review quantum field theory approach to the knot theory. Using holomorphic gauge, we obtain the Kontsevich integral. It is explained how to calculate Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial way which can be programmed on a computer. We discuss experimental results and temporal gauge considerations which lead to representation of Vassiliev invariants in terms of arrow diagrams. Explicit examples and computational results are presented.