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?
where $\mathcal{P}$ is the “time-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.
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)
geometric Langlands correspondence | S-duality in N=4 D=4 super Yang-Mills theory |
---|---|
Hecke transformation | 't Hooft operator? |
local system/flat connection | electric eigenbrane (eigenbrane of Wilson operator) |
Hecke eigensheaf | magnetic eigenbrane (eigenbrane of 't Hooft operator? ) |
In $SU(2)$-Chern-Simons theory the Wilson line observables compute the Jones polynomial of the given curve. See there for more details.
't Hooft operator?
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
The Poisson bracket of Wilson line observables in 3d Chern-Simons theory was obtained in
For more see
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