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Concept
Wilson loop
Summary
In quantum field theory, Wilson loops are gauge invariant operators arising from the parallel transport of gauge variables around closed loops. They encode all gauge information of the theory, allowing for the construction of loop representations which fully describe gauge theories in terms of these loops. In pure gauge theory they play the role of order operators for confinement, where they satisfy what is known as the area law. Originally formulated by Kenneth G. Wilson in 1974, they were used to construct links and plaquettes which are the fundamental parameters in lattice gauge theory. Wilson loops fall into the broader class of loop operators, with some other notable examples being 't Hooft loops, which are magnetic duals to Wilson loops, and Polyakov loops, which are the thermal version of Wilson loops.
To properly define Wilson loops in gauge theory requires considering the fiber bundle formulation of gauge theories. Here for each point in the -dimensional spacetime there is a copy of the gauge group forming what's known as a fiber of the fibre bundle. These fiber bundles are called principal bundles. Locally the resulting space looks like although globally it can have some twisted structure depending on how different fibers are glued together.
The issue that Wilson lines resolve is how to compare points on fibers at two different spacetime points. This is analogous to parallel transport in general relativity which compares tangent vectors that live in the tangent spaces at different points. For principal bundles there is a natural way to compare different fiber points through the introduction of a connection, which is equivalent to introducing a gauge field. This is because a connection is a way to separate out the tangent space of the principal bundle into two subspaces known as the vertical and horizontal subspaces. The former consists of all vectors pointing along the fiber while the latter consists of vectors that are perpendicular to the fiber.
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