They play the role of a disorder parameter for the Higgs phase in pure gauge theory.
Consistency conditions between electric and magnetic charges limit the possible 't Hooft loops that can be used, similarly to the way that the Dirac quantization condition limits the set of allowed magnetic monopoles.
They were first introduced by Gerard 't Hooft in 1978 in the context of possible phases that gauge theories admit.
[1] There are a number of ways to define 't Hooft lines and loops.
they are equivalent to the gauge configuration arising from the worldline traced out by a magnetic monopole.
[2] These are singular gauge field configurations on the line such that their spatial slice have a magnetic field whose form approaches that of a magnetic monopole where in Yang–Mills theory
is the generally Lie algebra valued object specifying the magnetic charge.
't Hooft lines can also be inserted in the path integral by requiring that the gauge field measure can only run over configurations whose magnetic field takes the above form.
More generally, the 't Hooft loop can be defined as the operator whose effect is equivalent to performing a modified gauge transformations
By constructing such gauge transformations, an explicit form for the 't Hooft loop can be derived by introducing the Yang–Mills conjugate momentum operator If the loop
, then an explicitly form of the 't Hooft loop operator is[4] Using Stokes' theorem this can be rewritten in a way which show that it measures the electric flux through
, analogous to how the Wilson loop measures the magnetic flux through the enclosed surface.
The 't Hooft loop is a disorder operator since it creates singularities in the gauge field, with their expectation value distinguishing the disordered phase of pure Yang–Mills theory from the ordered confining phase.
On the basis of the commutation relation between the 't Hooft and Wilson loops, four phases can be identified for
gauge theories that additionally contain scalars in representations invariant under the center
The mixed phase requires the gauge bosons to be massless particles and does not occur for
Since 't Hooft operators are creation operators for center vortices, they play an important role in the center vortex scenario for confinement.
[6] In this model, it is these vortices that lead to the area law of the Wilson loop through the random fluctuations in the number of topologically linked vortices.
In the presence of both 't Hooft lines and Wilson lines, a theory requires consistency conditions similar to the Dirac quantization condition which arises when both electric and magnetic monopoles are present.
is a subgroup of the center, then the set of allowed Wilson lines is in one-to-one correspondence with the representations of
, the Wilson lines are in one-to-one correspondence with the lattice points
The Lie algebra valued charge of the 't Hooft line can always be written in terms of the rank
Due to Wilson lines, the 't Hooft charge must satisfy the generalized Dirac quantization condition
meaning that the only magnetic charges allowed for the 't Hooft lines are ones that are in the co-root lattice This is sometimes written in terms of the Langlands dual algebra
More general classes of dyonic line operators, with both electric and magnetic charges, can also be constructed.
Sometimes called Wilson–'t Hooft line operators, they are defined by pairs of charges
it holds that Line operators play a role in indicating differences in gauge theories of the form
Unless they are compactified, these theories do not differ in local physics and no amount of local experiments can deduce the exact gauge group of the theory.
gauge theories, Wilson loops are labelled by
the lattices are reversed where now the Wilson lines are determined by