nLab ’t Hooft operator




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Chern-Weil theory

Differential cohomology



In gauge theory, where a Wilson line is a curve in ambient spacetime with a gauge field holonomy around the curve, dually a ‘t Hooft operator is a curve with Dirac monopole-like singularity of the ambinent gauge field along it (hence they may be thought of as 1-dimensional distributions of magnetic charge).

review includes (Kapustin-Witten 06, section 6.2)

At least in the additional presence of Higgs bundle fields the singularity makes the field strength curvature that of a differential form with logarithmic singularities along the specified curve

(Kapustin-Witten 06, (6.8), (6.9))


Relation to S-duality and geometric Langlands correspondence

Under the identification of the geometric Langlands correspondence with aspects of S-duality in super Yang-Mills theory, the t Hooft operators correspond to Hecke operator? (Kapustin-Witten 06, section 9).

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)


The original definition is due to

  • Gerard 't Hooft, On the phase transition towards permanent quark confinement, Nuclear Physics : B, volume: 138, issue: 1 (1978), pp. 1 - 25 (igitur)

Review includes

Discussion in the context of S-duality is in

and further discussion of this relating to the geometric Langlands correspondence is in

Last revised on August 11, 2021 at 11:03:33. See the history of this page for a list of all contributions to it.