Wilson loop


\infty-Chern-Weil theory

Differential cohomology



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 AA, then the Wilson loop along a closed loop CC is the trace of the path-ordered exponential?

W C=Tr(𝒫exp(i CA μdx μ)) W_C = Tr(\mathcal{P}exp(i\oint_C A_\mu d x^\mu))

where 𝒫\mathcal{P} is the “path-ordering operator” and A μA_\mu are the components of the connection.


Relation to 1d Chern-Simons theory

For GG a suitable Lie group (compact, semi-simple and simply connected) the Wilson loops of GG-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.

As the universal Vassiliev invariant

The un-traced Wilson loop observable of perturbative Chern-Simons theory is the universal Vassiliev invariant (see there for more):

from Sati-Schreiber 19c

Relation to defects

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

Duality with ‘t Hooft operators under S-duality and geometric Langlands

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 correspondenceS-duality in N=4 D=4 super Yang-Mills theory
Hecke transformation't Hooft operator
local system/flat connectionelectric eigenbrane (eigenbrane of Wilson operator)
Hecke eigensheafmagnetic eigenbrane (eigenbrane of 't Hooft operator )

(Kapustin-Witten 06)


In Chern-Simons theory

In SU(2)SU(2)-Chern-Simons theory the Wilson line observables compute the Jones polynomial of the given curve. See there for more details.

In Rozansky-Witten theory


(Rozansky-Witten Wilson loop of unknot is square root of A-hat genus)

For 4n\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 A^( 4n)\sqrt{{\widehat A}(\mathcal{M}^{4n})} of the A-hat genus of 4n\mathcal{M}^{4n}.

This is Roberts-Willerton 10, Lemma 8.6, using the Wheels theorem and the Hitchin-Sawon theorem.



On super-Wilson lines via integration over supermanifolds:

In Chern-Simons theory

See at perturbative quantization of 3d Chern-Simons theory/Vassiliev knot invariants.

Relation between Dehn surgery and Wilson loop observables in Chern-Simons theory:

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:

Jones polynomial as Wilson loop observables

The identification of the Jones polynomial with Wilson loop observables in Chern-Simons theory is due to

see also

Lecture notes:

  • Edward Witten, A New Look At The Jones Polynomial of a Knot, Clay Conference, Oxford, October 1, 2013 (pdf)

The categorification of this relation to an identification of Khovanov homology with observables in D=4 super Yang-Mills theory:

Lecture notes:

  • Edward Witten, Khovanov Homology And Gauge Theory, Clay Conference, Oxford, October 2, 2013 (pdf)

See also

  • Vaughan Jones, Index for subfactors, Invent. Math. 72, I (I983); A polynomial invariant for links via yon Neumann algebras, Bull. AMS 12, 103 (1985); Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126, 335 (1987)

Wilson lines computing holographic entropy in AdS 3/CFT 2AdS_3/CFT_2

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 3AdS_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 3AdS_3 Chern-Simons Gravity Theory (arXiv:1911.03841)

  • Eric D'Hoker, Per Kraus, Gravitational Wilson lines in AdS 3AdS_3 (arXiv:1912.02750)

  • Marc Henneaux, Wout Merbis, Arash Ranjbar, Asymptotic dynamics of AdS 3AdS_3 gravity with two asymptotic regions (arXiv:1912.09465)

and similarly for 3d flat-space holography:

Discussion for 3d de Sitter spacetime:

  • Alejandra Castro, Philippe Sabella-Garnier, Claire Zukowski, Gravitational Wilson Lines in 3D de Sitter (arXiv:2001.09998)

In QFT with defects

Relation to QFT with defects is discussed in

slide 17 of

slide 5 of

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

referring to

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

In S-duality and relation to t Hoofts operators