Wilson Loop Variables Research Articles

SO(N) Lattice Gauge Theory, planar and beyond

AbstractLattice gauge theories have been studied in the physics literature as discrete approximations to quantum Yang‐Mills theory for a long time. Primary statistics of interest in these models are expectations of the so‐called Wilson loop variables. In this article we continue the program initiated by Chatterjee [3] to understand Wilson loop expectations in lattice gauge theories in a certain limit through gauge‐string duality. The objective in this paper is to better understand the underlying combinatorics in the strong coupling regime by giving a more geometric picture of string trajectories involving correspondence to objects such as decorated trees and noncrossing partitions. Using connections with free probability theory, we provide an elaborate description of loop expectations in the planar setting, which provides certain insights into structures of higher‐dimensional trajectories as well. Exploiting this, we construct an example showing that in any dimension, the Wilson loop area law lower bound does not hold in full generality. © 2018 Wiley Periodicals, Inc.

FROM EUCLIDEAN FIELD THEORY TO QUANTUM FIELD THEORY

In order to construct examples for interacting quantum field theory models, the methods of Euclidean field theory turned out to be powerful tools since they make use of the techniques of classical statistical mechanics. Starting from an appropriate set of Euclidean n-point functions (Schwinger distributions), a Wightman theory can be reconstructed by an application of the famous Osterwalder–Schrader reconstruction theorem. This procedure (Wick rotation), which relates classical statistical mechanics and quantum field theory, is, however, somewhat subtle. It relies on the analytic properties of the Euclidean n-point functions. We shall present here a C*-algebraic version of the Osterwalder–Schrader reconstruction theorem. We shall see that, via our reconstruction scheme, a Haag–Kastler net of bounded operators can directly be reconstructed. Our considerations also include objects, like Wilson loop variables, which are not point-like localized objects like distributions. This point of view may also be helpful for constructing gauge theories.

Non-Abelian Stokes Theorem for Wilson Loops Associated with General Gauge Groups

A formula constituting the non-Abelian Stokes theorem for general semi-simple compact gauge groups is presented. The formula involves a path integral over a group space and is applicable to Wilson loop variables irrespective of the topology of loops. Some simple expressions analogous to the 't Hooft tensor of a magnetic monopole are given for the 2-form of interest. A special property in the case of the fundamental representation of the group SU(N) is pointed out.

THE SEGAL–BARGMANN TRANSFORM FOR TWO-DIMENSIONAL EUCLIDEAN QUANTUM YANG–MILLS

The Segal–Bargmann transform of certain Wilson loop variables for quantum Yang–Mills theory on the plane is determined explicitly.

USp(2k) Matrix Model: -- Schwinger-Dyson Equations and Closed-Open String Interactions --

We derive the Schwinger-Dyson/loop equations for the USp(2k) matrix model which close among the closed and open Wilson loop variables. These loop equations exhibit a complete set of the joining and splitting interactions required for the nonorientable Type I superstrings. The open loops realize the SO(2n_f) Chan-Paton factor and their linearized loop equations derive the mixed Dirichlet/Neumann boundary conditions.

A RIGOROUS CONSTRUCTION OF ABELIAN CHERN-SIMONS PATH INTEGRALS USING WHITE NOISE ANALYSIS

Using recent results in white noise analysis we rigorously construct Feynman path integrals for abelian Chern-Simons theory. We define Wilson loop variables and show that their expectation is a topological invariant, namely computable in terms of pairwise linking numbers as conjectured by E. Witten.

Complexification of gauge theories

For the case of a first-class constrained system with equivariant momentum map, we study the conditions under which the double process of reducing to the constraint surface and dividing out by the group of gauge transformations G is equivalent to the single process of dividing out the initial phase space by the complexification G C of G. For the particular case of a phase space action that is the lift of a configuration space action, conditions are found under which, in finite dimensions, the physical phase space of a gauge system with first-class constraints is diffeomorphic to a manifold imbedded in the physical configuration space of the complexified gauge system. Similar conditions are shown to hold for the infinite-dimensional example of Yang-Mills theories. As a physical application we discuss the adequateness of using holomorphic Wilson loop variables as (generalized) global coordinates on the physical phase space of Yang-Mills theory.

Abelian Chern–Simons theory and linking numbers via oscillatory integrals

A rigorous mathematical model of Abelian Chern–Simons theory based on the theory of infinite-dimensional oscillatory integrals developed by Albeverio and Ho/egh-Krohn is introduced. A gauge-fixed Chern–Simons path integral is constructed as a Fresnel integral in a certain Hilbert space. Wilson loop variables are defined as Fresnel integrable functions and it is shown in this context that the expectation value of products of Wilson loops with respect to the Chern–Simons path integral is a topological invariant which can be computed in terms of pairwise linking numbers of the loops, as conjectured by Witten. Furthermore, a lattice Chern–Simons action is proposed which converges to the continuum limit.

Generalized coordinates on the phase space of Yang-Mills theory

We study the suitability of complex Wilson loop variables as (generalized) coordinates on the physical phase space of SU(2) Yang-Mills theory. To this end, we construct a natural one-to-one map from the physical phase space of the Yang-Mills theory with compact gauge group G to a subspace of the physical configuration space of the complex -Yang-Mills theory. Together with a recent result by Ashtekar and Lewandowski this implies that the complex Wilson loop variables form a complete set of generalized coordinates on the physical phase space of SU(2) Yang-Mills theory. They also form a generalized canonical loop algebra. Implications for both general relativity and gauge theory are discussed.

Solution of the SU(2) Mandelstam constraints

It is shown how the Mandelstam constraints for an SU(2) pure lattice gauge theory with 3 N physical degrees of freedom may be solved completely in terms of 3 N Wilson and Polyakov loop variables and N − 1 gauge invariant discrete ±1 variables, thus enabling a manifestly gauge invariant formulation of the theory.

Gauge invariant functions of connections

It is shown that, for certain gauge groups, central functions of the holonomy variables do not determine connections up to gauge equivalence. It is also shown that, for a large class of compact groups, such functions do determine connections up to gauge equivalence. It is then shown that, for the latter type of gauge groups, the Euclidean quantum gauge field measure is determined by the expectation values of the Wilson loop variables (products of characters evaluated on holonomies).

TWO-DIMENSIONAL QUANTUM CHROMODYNAMICS ON A CYLINDER

We study two-dimensional quantum chromodynamics with massive quarks on a cylinder in a light-cone formalism. We eliminate the non-dynamical degrees of freedom and express the theory in terms of the quark and Wilson loop variables. It is possible to perform this reduction without gauge fixing. The fermionic Fock space can be defined independent of the gauge field in this light-cone formalism.

Loop space Hamiltonians and numerical methods for large- N gauge theories

We consider large- N gauge theories in the hamiltonian, collective field approach. We derive an alternative collective representation which leads to significant reduction when translation invariance is invoked. It allows for a simplified computer simulation of loop rearrangements and the development of numerical techniques in the hamiltonian, loop space formalism. We proceed to give numerical evidence for validity of our representation and outline a general numerical approach for solving large- N QCD in terms of gauge-invariant Wilson loop variables.

Reconstruction of gauge potentials from Wilson loops

The algebraic relations between open and closed path-ordered phases and Wilson loop variables in gauge theories are discussed. It is shown that the generalized Mandelstam constraints of order $N$ constitute sufficient algebraic conditions on Wilson loop variables to allow reconstruction of a corresponding gauge potential.

Collective field approach to the large- N limit: Euclidean field theories

We apply the collective field method to study the large-N behavior of euclidean field theories. We show how this method leads to an effective action whose stationary points determine the N → ∞ limit. This approach applies to Yang-Mills gauge theories and the effective action is given in terms of gauge-invariant Wilson loop variables.