Wilson loop - Imported Knowlpedia Content

Definition

Let $P\to M$ be a principal $G$ -bundle with connection $A$ , let $\rho:G\to\operatorname{GL}(V)$ be a finite-dimensional representation, and let $\gamma$ be a piecewise smooth closed loop based at $x$ . Choosing $p\in P_x$ , parallel transport around $\gamma$ gives a holonomy element $h_{A,\gamma,p}\in G$ . The Wilson loop in the representation $\rho$ is

W_{\rho,\gamma}(A) := \operatorname{tr}\!\bigl(\rho(h_{A,\gamma,p})\bigr).

Changing $p$ conjugates the holonomy element, so invariance of the trace under conjugation makes $W_{\rho,\gamma}(A)$ independent of this choice.

Gauge invariance

Under a gauge transformation, based holonomy is conjugated by the value of the transformation at the basepoint. Therefore

W_{\rho,\gamma}(u\cdot A)=W_{\rho,\gamma}(A).

Thus the Wilson loop is a function on gauge-equivalence classes of connections, even though the holonomy element itself depends on a point in the fiber. This is the basic mechanism by which a geometric transport operator produces a gauge-invariant observable.

Geometric meaning

The Wilson loop packages the conjugacy class of the holonomy along $\gamma$ through the character of $\rho$ . For a flat connection, it depends only on the based homotopy class of $\gamma$ . For a curved connection, smoothly homotopic loops can have different Wilson values; infinitesimal loops detect the curvature to leading order.

If $\gamma^{-1}$ is the oppositely oriented loop, then

W_{\rho,\gamma^{-1}}(A) = \operatorname{tr}\!\bigl(\rho(h_{A,\gamma,p}^{-1})\bigr),

which need not equal $W_{\rho,\gamma}(A)$ for an arbitrary representation.

Conventions

Some authors normalize the trace by $\dim V$ , insert a sign or coupling constant in the exponential convention for holonomy, or reserve “Wilson loop” for its expectation value in a quantum gauge theory. The representation and normalization must therefore be specified. Before taking the trace, the parallel-transport operator is often called a Wilson line; along an open path it transforms at both endpoints and is not by itself gauge invariant.

References

Kenneth G. Wilson, “Confinement of quarks,” Physical Review D 10 (1974), 2445–2459. DOI record. Relevant: the original lattice-gauge Wilson observable.
John C. Baez and Javier P. Muniain, Gauge Fields, Knots and Gravity, World Scientific, 1994. DOI record. Relevant: holonomy and gauge-invariant loop observables.