nLab ’t Hooft operator

Idea

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))

Properties

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 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 )

(Kapustin-Witten 06)

References

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

• Wikipedia, ‘t Hooft operator

Discussion in the context of S-duality is in

• Anton Kapustin, Wilson-‘t Hooft operators in four-dimensional gauge theories and S-duality, Phys.Rev. D74 (2006) 025005 (arXiv:hep-th/0501015)

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

• Anton Kapustin, Edward Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Communications in Number Theory and Physics Volume 1 (2007) Number 1 (arXiv:hep-th/0604151)