nLab
Wilson loop

Idea

A Wilson loop or Wilson line is an observable in (both classical and quantum) gauge theory obtained from the holonomy of the gauge connection.

Hence if the gauge connection is given by a globally defined 1-form $A$, then the Wilson loop along a closed loop $C$ is the trace of the time-ordered exponential?

$$W_C=\mathrm{Tr}(𝒫\exp (i{\oint }_C{A}_{\mu }d{x}^{\mu }))$$W_C = Tr(\mathcal{P}exp(i\oint_C A_\mu d x^\mu))

where $𝒫$ is the “time-ordering operator” and $A_{\mu }$ are the components of the connection.

Properties

Relation to 1d Chern-Simons theory

For $G$ a suitable Lie group (compact, semi-simple and simply connected) the Wilson loops of $G$-principal connections are equivalently the partition functions of a 1-dimensional Chern-Simons theory.

This appears famously in the formulation of Chern-Simons theory with Wilson lines. More detailes are at orbit method.

Relation to defects

Wilson loop insertions may be thought of or at least related to defects in the sense of QFT with defects.

Examples

In Chern-Simons theory

In $\mathrm{SU}(3)$-Chern-Simons theory the Wilson line observables compute the Jones polynomial of the given curve. See there for more details.

References

General

• Kenneth Wilson, Confinement of quarks, Physical Review D 10 (8): 2445. doi (1974)

• Yuri Makeenko, Methods of contemporary gauge theory, Cambridge Monographs on Math. Physics, gBooks

• wikipedia Wilson loop

• R. Giles, Reconstruction of gauge potentials from Wilson loops, Physical Review D 24 (8): 2160, doi

• A. Andrasi, J. C. Taylor, Renormalization of Wilson operators in Minkowski space, Nucl. Phys. B516 (1998) 417, hep-th/9601122

• Amit Sever, Pedro Vieira, Luis F. Alday, Juan Maldacena, Davide Gaiotto, An Operator product expansion for polygonal null Wilson loops, arxiv.org/abs/1006.2788

In QFT with defects

Relation to QFT with defects is discussed in

slide 17 of

• Constantin Bachas?, Conformal defects in/and String theory (pdf)

slide 5 of

• Constantin Bachas?, Conformal defects in gauged WZW models (pdf)

As partition functions of 1d Chern-Simons theory

Expression of Wilson loops as partition functions of 1-dimensional Chern-Simons theories by the orbit method (as used notably in Chern-Simons theory) is in section 4 of

• Chris Beasley, Localization for Wilson Loops in Chern-Simons Theory, in J. Andersen, H. Boden, A. Hahn, and B. Himpel (eds.) Chern-Simons Gauge Theory: 20 Years After, AMS/IP Studies in Adv. Math., Vol. 50, AMS, Providence, RI, 2011. (arXiv:0911.2687)

referring to

• S. Elitzur, Greg Moore, A. Schwimmer, and Nathan Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B 326 (1989) 108–134.

in the context of Chern-Simons theory and in more general gauge theory to

• A. P. Balachandran, S. Borchardt, and A. Stern, Lagrangian And Hamiltonian Descriptions of Yang-Mills Particles, Phys. Rev. D 17 (1978) 3247–3256

for more general discussion