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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))
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).
The original definition is due to
Review includes
Discussion in the context of S-duality is in
and further discussion of this relating to the geometric Langlands correspondence is in