Wilson loop

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.

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.

transforming in the fundamental representation of the gauge group, where the Wilson line is an operator that makes the combination

[4] It allows for the comparison of the matter field at different points in a gauge invariant way.

Alternatively, the Wilson lines can also be introduced by adding an infinitely heavy test particle charged under the gauge group.

Its charge forms a quantized internal Hilbert space, which can be integrated out, yielding the Wilson line as the world-line of the test particle.

However, the swampland conjecture known as the completeness conjecture claims that in a consistent theory of quantum gravity, every Wilson line and 't Hooft line of a particular charge consistent with the Dirac quantization condition must have a corresponding particle of that charge be present in the theory.

[6] Decoupling these particles by taking the infinite mass limit no longer works since this would form black holes.

Mathematically the term within the trace is known as the holonomy, which describes a mapping of the fiber into itself upon horizontal lift along a closed loop.

[7] Formally the set of all Wilson loops forms an overcomplete basis of solutions to the Gauss' law constraint.

The set of all Wilson lines is in one-to-one correspondence with the representations of the gauge group.

This can be reformulated in terms of Lie algebra language using the weight lattice of the gauge group

[8] An alternative view of Wilson loops is to consider them as operators acting on the Hilbert space of states in Minkowski signature.

Over large times the vacuum expectation value of the Wilson loop projects out the state with the minimum energy, which is the potential

are exponentially suppressed with time and so the expectation value goes as making the Wilson loop useful for calculating the potential between quark pairs.

This potential must necessarily be a monotonically increasing and concave function of the quark separation.

[11][12] Since spacelike Wilson loops are not fundamentally different from the temporal ones, the quark potential is really directly related to the pure Yang–Mills theory structure and is a phenomenon independent of the matter content.

[13] Elitzur's theorem ensures that local non-gauge invariant operators cannot have a non-zero expectation values.

This is motivated from the potential between infinitely heavy test quarks which in the confinement phase is expected to grow linearly

Four links around a single square are known as a plaquette, with their trace forming the smallest Wilson loop.

, denoted by[17] These Wilson loops are used to study confinement and quark potentials numerically.

Linear combinations of Wilson loops are also used as interpolating operators that give rise to glueball states.

gauge theory gives rise to an area law with a string tension of the form[21][22] where

gauge theories at zero temperature, instead they exhibit confinement at all values of the coupling constant.

holonomies with the delta functions yields a set of identities between Wilson loops.

supersymmetric Yang–Mills theory maximally helicity violating amplitudes factorize into a tree-level component and a loop level correction.

[39] Wilson lines also play a role in orbifold compactifications where their presence leads to greater control of gauge symmetry breaking, giving a better handle on the final unbroken gauge group and also providing a mechanism for controlling the number of matter multiplets left after compactification.

[40] These properties make Wilson lines important in compactifications of superstring theories.

[43] For this reason, Wilson loops are key observables on in these theories and are used to calculate global properties of the manifold.