In quantum field theory, the 't Hooft loop is the magnetic analogue of the Wilson loop. When acting on a quantum state, spatial 't Hooft loop operators give rise to thin loops of magnetic flux associated with magnetic vortices. They can also be used to diagnose the phase of a pure gauge theory where they act as the disorder parameter for the Higgs phase. Consistency conditions between electric and magnetic charges limit the possible 't Hooft loops that exist in a theory, 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.

Definition

There are a number of ways to define 't Hooft lines and loops. For timelike they are equivalent to the gauge configuration arising from the worldline traced out by a magnetic monopole. 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 theories is generally a 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 fields take the above form.

with a winding number satisfies

These modified gauge transformations are not true gauge transformations as they do not leave the action invariant. For temporal loops they create the aforementioned field configurations while for spatial loops they instead create loops of color magnetic flux, referred to as center vortices. By constructing such gauge transformations, an explicit form for the 't Hooft loop can be derived by introducing the Yang–Mills conjugate momentum operator

, then an explicitly form of the 't Hooft loop operator is

Using Stokes' theorem this can be rewritten in a way which shows that it measures the electric flux, analogous to how the Wilson loop measures the magnetic flux through the enclosed surface.

satisfy

is an element of the center of the gauge group. This relation can be taken as a defining feature of 't Hooft loops. The commutation properties between these two loop operators is often utilized in topological field theory where these operators form an algebra.

Disorder operator

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. Similarly to the Wilson loop, the expectation value of the 't Hooft loop can follow either the area law

is a constant, or it can follow the perimeter law

is a constant.

gauge theories that additionally contain scalars symmetry. The four phases are

• Confinement: Wilson loops follow the area law while 't Hooft loops follow the perimeter law.

• Higgs phase: Wilson loops follow the perimeter law while 't Hooft loops follow the area law.

• Confinement together with a partially Higgsed phase: both follow the area law.

• Mixed phase: both follow the perimeter law.

In the third phase the gauge group is only partially broken down to a smaller non-abelian subgroup. The mixed phase requires the gauge bosons to be massless particles theories, but is similar to the Coulomb phase for abelian gauge theory.

Since 't Hooft operators are creation operators for center vortices, they play an important role in the center vortex scenario for confinement. 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.

Charge constraints

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 the universal covering group is a subgroup of the center, then the set of allowed Wilson lines is in one-to-one correspondence with the representations. This can be formulated more precisely by introducing the weights is the Weyl group.

, which must hold for all representations of the Lie algebra.

that satisfy this condition, one must consider roots which are adjoint representation 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 up to the identification it holds that

. Unless they are compactified, these theories do not differ in local physics and no amount of local experiments.

See also

• Polyakov loop

References