't Hooft operator
In theoretical physics, a 't Hooft operator, introduced by Gerard 't Hooft in the 1978 paper "On the phase transition towards permanent quark confinement", is a dual version of the Wilson loop in which the electromagnetic potential A is replaced by its electromagnetic dual Amag, where the exterior derivative of A is equal to the Hodge dual of the exterior derivative of Amag. In d spacetime dimensions, Amag is a (d-3)-form and so the 't Hooft operator is the integral of Amag over a (d-3)-dimensional surface.
While the Wilson loop is an order operator, the 't Hooft operator is an example of a disorder operator because it creates a singularity or a discontinuity in the fundamental fields such as the electromagnetic potential A. For example, in an SU(N) Yang Mills gauge theory a 't Hooft operator creates a Dirac magnetic monopole with respect to the center of SU(N). If a condensate is present which transforms in a representation of SU(N) which is invariant under the action of the center, such as the adjoint representation, then the magnetic monopole will be confined by a vortex lying along a Dirac string from the monopole to either an antimonopole or to infinity. This vortex is similar to a Nielsen-Olesen vortex, but it carries a charge under the center of SU(N), and so N such vortices may annihilate.
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