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 path-ordered exponential?
where $\mathcal{P}$ is the “path-ordering operator” and $A_\mu$ are the components of the connection.
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.
The un-traced Wilson loop observable of perturbative Chern-Simons theory is the universal Vassiliev invariant (see there for more):
from Sati-Schreiber 19c
Wilson loop insertions may be thought of or at least related to defects in the sense of QFT with defects.
S-duality of 4d super Yang-Mills theory may exchange Wilson loop operators with 't Hooft operators, in an incarnation of the geometric Langlands correspondence (Kapustin-Witten 06)
In $SU(2)$-Chern-Simons theory the Wilson line observables compute the Jones polynomial of the given curve. See there for more details.
(Rozansky-Witten Wilson loop of unknot is square root of A-hat genus)
For $\mathcal{M}^{4n}$ a hyperkähler manifold (or just a holomorphic symplectic manifold) the Rozansky-Witten invariant Wilson loop observable associated with the unknot in the 3-sphere is the square root $\sqrt{{\widehat A}(\mathcal{M}^{4n})}$ of the A-hat genus of $\mathcal{M}^{4n}$.
This is Roberts-Willerton 10, Lemma 8.6, using the Wheels theorem and the Hitchin-Sawon theorem.
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
On super-Wilson lines via integration over supermanifolds:
See at perturbative quantization of 3d Chern-Simons theory/Vassiliev knot invariants.
Relation between Dehn surgery and Wilson loop observables in Chern-Simons theory:
E. Guadagnini, Surgery rules in quantum Chern-Simons field theory, Nuclear Physics B Volume 375, Issue 2, 18 May 1992, Pages 381-398 (doi:10.1016/0550-3213(92)90037-C)
Boguslaw Broda, Chern-Simons theory on an arbitrary manifold via surgery (arXiv:hep-th/9305051)
The Poisson bracket of Wilson line observables in 3d Chern-Simons theory was obtained in
For more see
In Chern-Simons theory as a topological string theory:
The identification of the Jones polynomial with Wilson loop observables in Chern-Simons theory is due to
see also
Volume 126, Number 1 (1989), 167-199 (euclid.cmp/1104179728)
Lecture notes:
The categorification of this relation to an identification of Khovanov homology with observables in D=4 super Yang-Mills theory:
Edward Witten, Khovanov homology and gauge theory, arxiv/1108.3103
Edward Witten, Fivebranes and Knots (arXiv:1101.3216)
Lecture notes:
See also
Discussion of BTZ black hole entropy and more generally of holographic entanglement entropy in 3d quantum gravity/AdS3/CFT2 via Wilson line observables in Chern-Simons theory:
Martin Ammon, Alejandra Castro, Nabil Iqbal, Wilson Lines and Entanglement Entropy in Higher Spin Gravity, JHEP 10 (2013) 110 (arXiv:1306.4338)
Jan de Boer, Juan I. Jottar, Entanglement Entropy and Higher Spin Holography in $AdS_3$, JHEP 1404:089, 2014 (arXiv:1306.4347)
Alejandra Castro, Stephane Detournay, Nabil Iqbal, Eric Perlmutter, Holographic entanglement entropy and gravitational anomalies, JHEP 07 (2014) 114 (arXiv:1405.2792)
Mert Besken, Ashwin Hegde, Eliot Hijano, Per Kraus, Holographic conformal blocks from interacting Wilson lines, JHEP 08 (2016) 099 (arXiv:1603.07317)
Andreas Blommaert, Thomas G. Mertens, Henri Verschelde, The Schwarzian Theory - A Wilson Line Perspective, JHEP 1812 (2018) 022 (arXiv:1806.07765)
Ashwin Dushyantha Hegde, Role of Wilson Lines in 3D Quantum Gravity, 2019 (spire:1763572)
Xing Huang, Chen-Te Ma, Hongfei Shu, Quantum Correction of the Wilson Line and Entanglement Entropy in the $AdS_3$ Chern-Simons Gravity Theory (arXiv:1911.03841)
Eric D'Hoker, Per Kraus, Gravitational Wilson lines in $AdS_3$ (arXiv:1912.02750)
Marc Henneaux, Wout Merbis, Arash Ranjbar, Asymptotic dynamics of $AdS_3$ gravity with two asymptotic regions (arXiv:1912.09465)
and similarly for 3d flat-space holography:
Arjun Bagchi, Rudranil Basu, Daniel Grumiller, Max Riegler, Entanglement entropy in Galilean conformal field theories and flat holography, Phys. Rev. Lett. 114, 111602 (2015) (arXiv 1410.4089)
Rudranil Basu, Max Riegler, Wilson Lines and Holographic Entanglement Entropy in Galilean Conformal Field Theories, Phys. Rev. D 93, 045003 (2016) (arXiv:1511.08662)
Wout Merbis, Max Riegler, Geometric actions and flat space holography (arXiv:1912.08207)
Discussion for 3d de Sitter spacetime:
Relation to QFT with defects is discussed in
slide 17 of
slide 5 of
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
referring to
in the context of Chern-Simons theory and in more general gauge theory to