Wilson Loops

Prerequisites: Gauge Models, Parallel Transport. Full path (18 pages)

• Why is it interesting?
• Intuitive
• Concrete
• Abstract

Why is it interesting?

Gauge fields like the photon field or gluon fields are usually described using gauge potentials. An alternative is to use Wilson lines. In contrast to gauge potentials, Wilson loops are completely gauge invariant. The set of all Wilson loops is sufficient to reconstruct the corresponding gauge potential.

The formulation of Yang-Mills theory which makes use of Wilson loops is called loop formulation.

Intuitive

A particular nice system to understand Wilson loops in a non-technical context are financial markets.

Say, we represent different countries by different points on a lattice. Each country has its own currency. However, the absolute values of the different currencies has no meaning of all on the global money market. Instead, all that counts are the relationships between the different currencies. The exchange rates between different currencies is encoded in something we call connection.

Now, how can we make money on the money market?

Well, let's say we are a banker in London. We have a budget of 100 pounds. Our goal is to trade our currency against other currencies in such a way that we have in the end more than 100 pounds. This means, we need to look for profitable loops through the global money lattice. We are interested in loops, because to actually determine if have earned money, we need to compare our final amount of money with the 100 pounds we started with. This is only possible if the final amount of money is also given in pounds.

Let's say the exchange rates are as follows:

• 1.5 dollars = 1 pound,
• 1 dollar = 10 pesos,
• 1 pound = 10 pesos.

How can we make money in this situation?

1. We start by trading our 100 pound against 150 dollars.
2. Then, we trade the 150 dollars against 1500 pesos.
3. Finally we trade the 1500 pesos against 150 pounds.

Thus, by trading in a loop we have gained 50 pounds.

The loops we considered here are exactly analogous to the Wilson loops used in quantum field theories. The gauge freedom corresponds here to the freedom to rescale the local currencies. For examples, England could introduce a new currency called "new-pound" and determine that 10 new-pounds are worth 1 pound. This wouldn't change the situation at the global money market at all, because all banks would simply adjust their exchange rates:

• 0.15 dollars = 1 new-pound,
• 1 dollar = 10 pesos,
• 1 new-pound = 1 peso.

Absolute value has no meaning and this is what we call gauge freedom. We can't make more money simply because we change the local value of our currency.

The thing we used in the situation above to earn money is called arbitrage. An arbitrage means a possibility to earn money without any risk. The arbitrage possibilities are completely encoded in closed loops.

In physics we usually don't use Wilson loops, but instead work with gauge field strengths (think electric or magnetic field strength). These gauge field strengths $F_{\mu\nu}$ correspond to infinitesimal Wilson loops around a given point $x$.

For more on this perspective see

• Physics of Finance by Kirill Ilinski
• The symmetry and simplicity of the laws of physics and the Higgs boson by Juan Maldacena
• Foreign exchange market as a lattice gauge theory by K. Young

The confining phase is usually defined by the behavior of the action of the Wilson loop, which is simply the path in spacetime traced out by a quark–antiquark pair created at one point and annihilated at another point. In a non-confining theory, the action of such a loop is proportional to its perimeter. However, in a confining theory, the action of the loop is instead proportional to its area. Since the area is proportional to the separation of the quark–antiquark pair, free quarks are suppressed. Mesons are allowed in such a picture, since a loop containing another loop with the opposite orientation has only a small area between the two loops. https://en.wikipedia.org/wiki/Color_confinement

Concrete

We take a gauge field $A_\mu = A_\mu^a T^a$ with Hermitian generators $T^a$ of the gauge group $G$, normalized as $\text{tr}(T^aT^b) = \frac12\delta^{ab}$ in the fundamental representation, and covariant derivative $D_\mu = \partial_\mu - igA_\mu$. Under a gauge transformation $\psi(x) \to U(x)\psi(x)$ the potential transforms as $A_\mu \to UA_\mu U^{-1} - \frac{i}{g}(\partial_\mu U)U^{-1}$.

Wilson lines. For a path $C$ from $y$ to $x$, the Wilson line is the path-ordered exponential

$$W_C(x,y) = \mathcal{P}\exp\left(ig\int_C dz^\mu A_\mu(z)\right),$$

where $\mathcal{P}$ puts fields at points closer to $x$ to the left. It is the solution of the parallel transport equation along $C$ and transforms as $W_C(x,y) \to U(x)\,W_C(x,y)\,U(y)^{-1}$. Therefore $\bar\psi(x)W_C(x,y)\psi(y)$ is gauge invariant. For a closed loop, $x = y$, the trace

$$W(C) = \text{tr}\,\mathcal{P}\exp\left(ig\oint_C dx^\mu A_\mu\right)$$

is gauge invariant. This is the Wilson loop.

Abelian case. For electromagnetism the path ordering is trivial and, by Stokes' theorem, a spatial loop gives, in signature $(+,-,-,-)$ where $A_\mu dx^\mu = A_0\,dt - \vec A\cdot d\vec x$, $W(C) = \exp\left(-ie\oint_C \vec A\cdot d\vec x\right) = \exp(-ie\Phi_B)$, where $\Phi_B$ is the magnetic flux through $C$ (units with $\hbar = c = 1$). This phase is what the Aharonov-Bohm effect measures.

Small loops and the lattice. On a hypercubic lattice with spacing $a$, the gauge field is replaced by link variables $U_\mu(x) = \exp\left(iag A_\mu(x)\right) \in G$, the parallel transporter from $x$ to $x+a\hat\mu$, which transforms as $U_\mu(x) \to U(x+a\hat\mu)\,U_\mu(x)\,U(x)^{-1}$. The smallest Wilson loop is the plaquette, the transport around an elementary square $x \to x+a\hat\mu \to x+a\hat\mu+a\hat\nu \to x+a\hat\nu \to x$:

$$U_{\mu\nu}(x) = U_\nu(x)^\dagger\,U_\mu(x+a\hat\nu)^\dagger\,U_\nu(x+a\hat\mu)\,U_\mu(x) = \exp\left(ia^2 g F_{\mu\nu}(x) + \mathcal{O}(a^3)\right),$$

with $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - ig[A_\mu, A_\nu]$.

For $G = SU(N)$ the Wilson action

$$S = \beta\sum_{\text{plaquettes}}\left(1 - \frac{1}{N}\text{Re}\,\text{tr}\,U_{\mu\nu}\right), \qquad \beta = \frac{2N}{g^2},$$

reduces to the Euclidean Yang-Mills action $\int d^4x\, \frac14 F^a_{\mu\nu}F^a_{\mu\nu}$ as $a \to 0$. This is the starting point of lattice QCD.

Static quark potential. Take a rectangular loop with spatial extent $R$ and Euclidean time extent $T$. The loop describes a heavy quark and antiquark created at separation $R$, held fixed for time $T$, and annihilated. For large $T$

$$\langle W(R,T)\rangle \sim e^{-V(R)\,T},$$

where $V(R)$ is the energy of the static pair. If $\langle W\rangle$ falls off with the area, $\langle W\rangle \sim e^{-\sigma RT}$, then $V(R) = \sigma R$ grows linearly and the quarks are confined, with string tension $\sigma$. A perimeter law, $\langle W\rangle \sim e^{-\mu\cdot 2(R+T)}$, corresponds to a potential that approaches a constant at large $R$, and the quarks are not confined.

Abstract

Let $P \to M$ be a principal $G$-bundle with connection $A$. For a loop $\gamma$ based at $x\in M$, parallel transport around $\gamma$ defines the holonomy $\text{Hol}_\gamma(A)$, an element of $G$ once a point in the fiber over $x$ is chosen; a different choice of point conjugates it. For a representation $R$ of $G$, the Wilson loop is the class function

$$W_R(\gamma) = \text{tr}_R\,\text{Hol}_\gamma(A),$$

which is invariant under gauge transformations, the vertical automorphisms of $P$. Holonomy defines a group homomorphism from the group of based loops (modulo reparametrization and retracing) to $G$. Conversely, the traced holonomies of all loops determine the connection up to gauge equivalence, for $G = U(N)$ or $SU(N)$ (Giles' reconstruction theorem). This is the basis of loop formulations of gauge theory. For $SU(N)$ the Wilson loops are not independent but satisfy algebraic relations, the Mandelstam identities; for $SU(2)$, for example, $W(\alpha)W(\beta) = W(\alpha\beta) + W(\alpha\beta^{-1})$ in the fundamental representation.

The curvature is the infinitesimal version of the holonomy: for the small loop $x \to x+\epsilon u \to x+\epsilon u+\epsilon v \to x+\epsilon v \to x$, $\text{Hol} = 1 + i g\epsilon^2 F_{\mu\nu}u^\mu v^\nu + \mathcal{O}(\epsilon^3)$ in the conventions of the concrete layer. In quantum Yang-Mills theory the expectation values $\langle W(\gamma_1)\cdots W(\gamma_k)\rangle$ satisfy the Makeenko-Migdal loop equations, which follow from the Schwinger-Dyson equations and close at large $N$ on single loops.

Confinement and center symmetry. In pure $SU(N)$ Yang-Mills theory the area law for fundamental Wilson loops is the standard criterion for confinement. On a thermal circle the Wilson loop that winds around the Euclidean time direction, the Polyakov loop, transforms under the center $\mathbb{Z}_N$ of $SU(N)$ by multiplication with a phase. Its expectation value vanishes in the confined, center-symmetric phase and is nonzero in the deconfined phase. In modern language, the Wilson lines are the charged operators of a $\mathbb{Z}_N$ one-form symmetry, and confinement is the statement that this one-form symmetry is unbroken. Adding dynamical fundamental quarks breaks the center symmetry explicitly, and large Wilson loops then obey a perimeter law because the string breaks into two mesons.

Topological field theory. In Chern-Simons theory on a three-manifold, Witten showed that expectation values of Wilson loops along knots and links are topological invariants. For $G = SU(2)$ and the fundamental representation, they reproduce the Jones polynomial.

Holography. In the AdS/CFT correspondence the expectation value of a suitable Wilson loop in $\mathcal{N}=4$ super Yang-Mills theory is computed at strong coupling by the area of a minimal string worldsheet in $AdS_5$ that ends on the loop at the boundary (Maldacena; Rey and Yee).

Path to this page

1. Calculus
2. Linear Algebra
3. Functional
4. Group Theory
5. Manifold
6. Newton's Second Law
7. Vector Calculus
8. Coulomb's law
9. Fiber Bundles
10. Newtonian Mechanics
11. Parallel Transport
12. Symmetry
13. Variational Calculus
14. Euler-Lagrange Equations
15. Maxwell Equations
16. Lagrangian Formalism
17. Gauge Symmetry
18. Gauge Models