nLab ’t Hooft operator

Redirected from "'t Hooft operators".
Note: ’t Hooft operator and ’t Hooft operator both redirect for "'t Hooft operators".
Contents

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Chern-Weil theory

Differential cohomology

Contents

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 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)

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

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.