Hilbert space operators

An alternative view of Wilson loops is to consider them as operators acting on the Hilbert space of states in Minkowski signature.^[5] Since the Hilbert space lives on a single time slice, the only Wilson loops that can act as operators on this space are ones formed using spacelike loops. Such operators ${\displaystyle W[\gamma ]}$ create a closed loop of electric flux, which can be seen by noting that the electric field operator ${\displaystyle E^{i}}$ is nonzero on the loop ${\displaystyle E^{i}W[\gamma ]|0\rangle \neq 0}$ but it vanishes everywhere else. Using Stokes theorem it follows that the spatial loop measures the magnetic flux through the loop.^[9]

Makeenko–Migdal loop equation

Similarly to the functional derivative which acts on functions of functions, functions of loops admit two types of derivatives called the area derivative and the perimeter derivative. To define the former, consider a contour ${\displaystyle \gamma }$ and another contour ${\displaystyle \gamma _{\delta \sigma _{\mu \nu }}}$ which is the same contour but with an extra small loop at ${\displaystyle x}$ in the ${\displaystyle \mu }$-${\displaystyle \nu }$ plane with area ${\displaystyle \delta \sigma _{\mu \nu }=dx_{\mu }\wedge dx_{\nu }}$. Then the area derivative of the loop functional ${\displaystyle F[\gamma ]}$ is defined through the same idea as the usual derivative, as the normalized difference between the functional of the two loops^[25]

${\displaystyle {\frac {\delta F[\gamma ]}{\delta \sigma _{\mu \nu }(x)}}={\frac {1}{\delta \sigma _{\mu \nu }(x)}}[F[\gamma _{\delta \sigma _{\mu \nu }}]-F[\gamma ]].}$

The perimeter derivative is similarly defined whereby now ${\displaystyle \gamma _{\delta x_{\mu }}}$ is a slight deformation of the contour ${\displaystyle \gamma }$ which at position ${\displaystyle x}$ has a small extruding loop of length ${\displaystyle \delta x_{\mu }}$ in the ${\displaystyle \mu }$ direction and of zero area. The perimeter derivative ${\displaystyle \partial _{\mu }^{x}}$ of the loop functional is then defined as

${\displaystyle \partial _{\mu }^{x}F[\gamma ]={\frac {1}{\delta x_{\mu }}}[F[\gamma _{\delta x_{\mu }}]-F[\gamma ]].}$

In the large N-limit, the Wilson loop vacuum expectation value satisfies a closed functional form equation called the Makeenko–Migdal equation^[26]

${\displaystyle \partial _{\mu }^{x}{\frac {\delta }{\delta \sigma _{\mu \nu }(x)}}\langle W[\gamma ]\rangle =g^{2}N\oint _{\gamma }dy_{\nu }\delta ^{(D)}(x-y)\langle W[\gamma _{yx}]\rangle \langle W[\gamma _{xy}]\rangle .}$

Here ${\displaystyle \gamma =\gamma _{xy}\cup \gamma _{yx}}$ with ${\displaystyle \gamma _{xy}}$ being a line that does not close from ${\displaystyle x}$ to ${\displaystyle y}$, with the two points however close to each other. The equation can also be written for finite ${\displaystyle N}$, but in this case it does not factorize and instead leads to expectation values of products of Wilson loops, rather than the product of their expectation values.^[27] This gives rise to an infinite chain of coupled equations for different Wilson loop expectation values, analogous to the Schwinger–Dyson equations. The Makeenko–Migdal equation has been solved exactly in two dimensional ${\displaystyle {\text{U}}(\infty )}$ theory.^[28]

Mandelstam identities

Gauge groups that admit fundamental representations in terms of ${\displaystyle N\times N}$ matrices have Wilson loops that satisfy a set of identities called the Mandelstam identities, with these identities reflecting the particular properties of the underlying gauge group.^[29] The identities apply to loops formed from two or more subloops, with ${\displaystyle \gamma =\gamma _{2}\circ \gamma _{1}}$ being a loop formed by first going around ${\displaystyle \gamma _{1}}$ and then going around ${\displaystyle \gamma _{2}}$.

The Mandelstam identity of the first kind states that ${\displaystyle W[\gamma _{1}\circ \gamma _{2}]=W[\gamma _{2}\circ \gamma _{1}]}$, with this holding for any gauge group in any dimension. Mandelstam identities of the second kind are acquired by noting that in ${\displaystyle N}$ dimensions, any object with ${\displaystyle N+1}$ totally antisymmetric indices vanishes, meaning that ${\displaystyle \delta _{[b_{1}}^{a_{1}}\delta _{b_{2}}^{a_{2}}\cdots \delta _{b_{N+1}]}^{a_{N+1}}=0}$. In the fundamental representation, the holonomies used to form the Wilson loops are ${\displaystyle N\times N}$ matrix representations of the gauge groups. Contracting ${\displaystyle N+1}$ holonomies with the delta functions yields a set of identities between Wilson loops. These can be written in terms the objects ${\displaystyle M_{K}}$ defined iteratively so that ${\displaystyle M_{1}[\gamma ]=W[\gamma ]}$ and

${\displaystyle (K+1)M_{K+1}[\gamma _{1},\dots ,\gamma _{K+1}]=W[\gamma _{K+1}]M_{K}[\gamma _{1},\dots ,\gamma _{K}]-M_{K}[\gamma _{1}\circ \gamma _{K+1},\gamma _{2},\dots ,\gamma _{K}]-\cdots -M_{K}[\gamma _{1},\gamma _{2},\dots ,\gamma _{K}\circ \gamma _{K+1}].}$

In this notation the Mandelstam identities of the second kind are^[30]

${\displaystyle M_{N+1}[\gamma _{1},\dots ,\gamma _{N+1}]=0.}$

For example, for a ${\displaystyle {\text{U}}(1)}$ gauge group this gives ${\displaystyle W[\gamma _{1}]W[\gamma _{2}]=W[\gamma _{1}\circ \gamma _{2}]}$.

If the fundamental representation are matrices of unit determinant, then it also holds that ${\displaystyle M_{N}(\gamma ,\dots ,\gamma )=1}$. For example, applying this identity to ${\displaystyle {\text{SU}}(2)}$ gives

${\displaystyle W[\gamma _{1}]W[\gamma _{2}]=W[\gamma _{1}\circ \gamma _{2}^{-1}]+W[\gamma _{1}\circ \gamma _{2}].}$

Fundamental representations consisting of unitary matrices satisfy ${\displaystyle W[\gamma ]=W^{*}[\gamma ^{-1}]}$. Furthermore, while the equality ${\displaystyle W[I]=N}$ holds for all gauge groups in the fundamental representations, for unitary groups it moreover holds that ${\displaystyle |W[\gamma ]|\leq N}$.

Renormalization

Since Wilson loops are operators of the gauge fields, the regularization and renormalization of the underlying Yang–Mills theory fields and couplings does not prevent the Wilson loops from requiring additional renormalization corrections. In a renormalized Yang–Mills theory, the particular way that the Wilson loops get renormalized depends on the geometry of the loop under consideration. The main features are^{[31][32][33][34]}

• Smooth non-intersecting curve: This can only have linear divergences proportional to the contour which can be removed through multiplicative renormalization.
• Non-intersecting curve with cusps: Each cusp results in an additional local multiplicative renormalization factor ${\displaystyle Z[\phi ]}$ that depends on the cusp angle ${\displaystyle \phi }$.
• Self-intersections: This leads to operator mixing between the Wilson loops associated with the full loop and the subloops.
• Lightlike segments: These give rise to additional logarithmic divergences.

Scattering amplitudes

Wilson loops play a role in the theory of scattering amplitudes where a set of dualities between them and special types of scattering amplitudes has been found.^[35] These have first been suggested at strong coupling using the AdS/CFT correspondence.^[36] For example, in ${\displaystyle {\mathcal {N}}=4}$ supersymmetric Yang–Mills theory maximally helicity violating amplitudes factorize into a tree-level component and a loop level correction.^[37] This loop level correction does not depend on the helicities of the particles, but it was found to be dual to certain polygonal Wilson loops in the large ${\displaystyle N}$ limit, up to finite terms. While this duality was initially only suggested in the maximum helicity violating case, there are arguments that it can be extended to all helicity configurations by defining appropriate supersymmetric generalizations of the Wilson loop.^[38]

String theory compactifications

In compactified theories, zero mode gauge field states that are locally pure gauge configurations but are globally inequivalent to the vacuum are parameterized by closed Wilson lines in the compact direction. The presence of these on a compactified open string theory is equivalent under T-duality to a theory with non-coincident D-branes, whose separations are determined by the Wilson lines.^[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.^[41][42]

Topological field theory

In a topological field theory, the expectation value of Wilson loops does not change under smooth deformations of the loop since the field theory does not depend on the metric.^[43] For this reason, Wilson loops are key observables on in these theories and are used to calculate global properties of the manifold. In ${\displaystyle 2+1}$ dimensions they are closely related to knot theory with the expectation value of a product of loops depending only on the manifold structure and on how the loops are tied together. This led to the famous connection made by Edward Witten where he used Wilson loops in Chern–Simons theory to relate their partition function to Jones polynomials of knot theory.^[44]