My understanding of Wilson loops

Let's work with classical electromagnetism. The 4-potential $A_\mu$ determines the electric and magnetic fields, which are the physical entities responsible for the electromagnetic force. The 4-potential has a gauge symmetry $A_\mu \rightarrow A_\mu +\partial_\mu \lambda$ with $\lambda$ any real function. Notably, the boundary conditions of the system matter for the gauge symmetry. If $\lambda$ is nonzero at the boundary, the symmetry becomes physical and we get conserved quantities from Noether's theorem, localized at the boundary.

Now, as I understand it, we can only ever observe gauge invariant quantities, such as the electric and magnetic fields. Another observable is given by Wilson loops

\begin{equation} W[\gamma]=e^{\oint_\gamma A_\mu dl^\mu} \end{equation}

The reason why this is an observable is simply Stokes' theorem: The integral over a closed loop becomes

\begin{equation} \oint_\gamma A_\mu dl^\mu = \int_R \vec{B} \cdot \vec{dS} \end{equation}

where $\vec{B}$ is the magnetic field, $R$ is the region enclosed by the loop $\gamma$ and $\vec{dS}$ is the surface element with normal perpendicular to $R$. Spatial Wilson loops are therefore just the magnetic flux through the loop. If we used a time-like loop, we would get a similar expression involving the integral of the electric field.

I get that physical observables need to be gauge invariant and therefore Wilson loops are good observables. Also, in Yang-Mills theories (i.e. not in the abelian case) the electric and magnetic fields are not even gauge invariant so the Wilson loops are trully some of the only observables (There's still the trace of the strength tensor $F_{\mu\nu}$ squared and other things like that too).

Things I don't get

In papers like this: Local subsystems in gauge theory and gravity, they talk about building a Hilbert space $\mathcal{H}$ for a Yang-Mills theory as the product of the Hilbert spaces of two subregions $\Sigma$ and $\Sigma'$ separated by some boundary $S$. Here's a direct quote:

in order for $\mathcal{H}$ to contain gauge-invariant operators such as a Wilson loop in a representation $R$, the space $\mathcal{H}_\Sigma$ has to contain a state that looks like a Wilson loop in the interior of $\Sigma$, but which terminates on $S$. The endpoint of the Wilson line acts as a surface charge transforming in the representation $R$. The generators of this gauge transformation are the nonabelian electric fields along links that cross the boundary.

I understand that gauge transformations that don't go to zero at the boundary are physical symmetries of the theory. That's why global phase transformations give electric charge conservation in Maxwell theory: They transform even at the boundary. I also understand that having a boundary means that there will be a Wilson loop that was a true degree of freedom in the full manifold but it will become a Wilson line with endpoints at the boundary in the bounded subregion. However, I don't understand what these endpoints represent, or why they transform under a gauge representation.

I'd like to know

I'd like to get some clarification on what's the physical interpretation for Wilson lines that end on boundaries (For example: Do they represent charge at the boundary? Do they represent a degree of freedom that captures what the unknown exterior is doing?) and, hopefully, a reference that explains all this without assuming that I already know what a Wilson loop/line is and how boundaries affect gauge theories.

In short: What's the specific relation between boundaries and gauge theories? What's the physical meaning of the ends of wilson lines at the boundaries of the theory? Is there some every day example of this? Like, Wilson lines on a conductor or something like that.

  • $\begingroup$ Short answer: $\lambda$ has no physical meaning $\endgroup$
    – basics
    Commented Jan 18 at 22:41

1 Answer 1


I give a qualitative overview addressing your questions below:

  1. Relation Between Boundaries and Gauge Theories:

    • In the presence of a boundary, the gauge transformations that do not vanish at the boundary become physical. This alteration affects the symmetry properties and, consequently, the observables of the theory.
    • Boundaries introduce localized physical degrees of freedom. For instance, in electromagnetism, boundary conditions can lead to the manifestation of surface charges or currents.
    • In Yang-Mills theories, the presence of boundaries can give rise to additional gauge-invariant observables, which are not present in the bulk theory.
  2. Physical Meaning of the Ends of Wilson Lines at Boundaries:

    • The endpoints of a Wilson line on a boundary represent the interaction of the gauge field with a 'charge' localized at the boundary. This 'charge' can be thought of as the endpoint of a non-abelian gauge field line.
    • These endpoints can be seen as sources or sinks of the gauge field, analogous to how electric charges are sources or sinks of electromagnetic fields in electromagnetism.
    • In the context of quantum field theory, these endpoints transform under a gauge representation, indicating that they carry quantum numbers associated with the gauge symmetry. For example, in QCD (a non-abelian gauge theory), these endpoints would represent color charges.
  3. An Everyday Example:

    • An everyday analogy might be how electric field lines terminate on electric charges. Imagine a conductor with a positive charge on one side and a negative charge on the other. The electric field lines (analogous to Wilson lines in this example) start from the positive charge and end on the negative charge.
    • In this analogy, the conductor's surface plays the role of the 'boundary.' The charges at the boundary (endpoints of electric field lines) alter the field configuration inside the conductor, similar to how endpoints of Wilson lines affect the gauge field configuration in a bounded region in gauge theories.
  • $\begingroup$ In the everyday example, what is the analogue of the gauge transformation of the endpoints? $\endgroup$ Commented Jan 19 at 21:15
  • $\begingroup$ A close analogue in-line with the above explanation would be akin to a redistribution of the charges without changing the observable electric field configuration outside a certain region. For example, redistributing charges on the surface of a conductor, while keeping the net charge constant, will change the local field lines near the surface but not the overall electric field pattern at larger distances (which is the observable effect we care about). Admittedly, this is a rather rough, but still workable analogy. $\endgroup$ Commented Jan 20 at 23:40

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