[Paper Review] Study of Wilson loop functionals in 2D Yang-Mills theories

J.M. Aroca, Yu. A. Kubyshin|ArXiv.org|Feb 8, 1998

Black Holes and Theoretical Physics8 references3 citations

TL;DR

This paper derives an explicit formula for the vacuum expectation value of the Wilson loop functional in 2D Yang-Mills theories on arbitrary orientable two-dimensional manifolds, both in the continuum and on the lattice, for any gauge group. It identifies a contribution from the space of invariant connections that resembles monopole effects, offering a unified framework for computing Wilson loop observables in 2D gauge theories.

ABSTRACT

The derivation of the explicit formula for the vacuum expectation value of the Wilson loop functional for an arbitrary gauge group on an arbitrary orientable two-dimensional manifold is considered both in the continuum case and on the lattice. A contribution to this quantity, coming from the space of invariant connections, is also analyzed and is shown to be similar to the contribution of monopoles.

Motivation & Objective

• To derive a general formula for the vacuum expectation value of the Wilson loop functional in 2D Yang-Mills theories on arbitrary orientable two-dimensional manifolds.
• To extend the computation of Wilson loop functionals to both continuum and lattice formulations of 2D Yang-Mills theory.
• To analyze the contribution of the space of invariant connections to the Wilson loop functional and relate it to monopole-like effects.
• To provide a unified framework applicable to arbitrary compact gauge groups in two-dimensional quantum field theories.

Proposed method

• The authors employ differential geometric and topological techniques to analyze the structure of connections on two-dimensional manifolds.
• They derive the Wilson loop functional using path integral methods in the continuum and lattice formulations.
• The space of invariant connections is isolated and its contribution to the Wilson loop is computed via group-theoretic and cohomological methods.
• The formalism incorporates gauge group invariance and topological invariants of the underlying manifold.
• The derivation is validated through consistency checks and comparison with known results in symmetric and abelian cases.
• The role of curvature and holonomy is systematically analyzed to extract the functional dependence of the Wilson loop.

Experimental results

Research questions

• RQ1 What is the explicit form of the vacuum expectation value of the Wilson loop functional in 2D Yang-Mills theory on an arbitrary orientable two-dimensional manifold?
• RQ2 How do the contributions from invariant connections in the gauge field configuration space affect the Wilson loop expectation value?
• RQ3 To what extent does the contribution from invariant connections resemble monopole contributions in the path integral?
• RQ4 How does the formula for the Wilson loop functional generalize across different gauge groups in 2D?
• RQ5 Can the continuum and lattice formulations of 2D Yang-Mills theory yield consistent expressions for the Wilson loop functional?

Key findings

• The paper derives a closed-form expression for the vacuum expectation value of the Wilson loop functional valid for any compact gauge group on an arbitrary orientable 2D manifold.
• The contribution from the space of invariant connections is shown to be non-trivial and structurally similar to monopole contributions in the path integral.
• The formula unifies results from both continuum and lattice formulations of 2D Yang-Mills theory.
• The derivation confirms consistency with known abelian and symmetric cases, validating the general approach.
• The method reveals that topological invariants of the manifold and group representation theory jointly determine the Wilson loop value.
• The revised version corrects typographical errors and updates references, ensuring technical accuracy and completeness.