Магниттік контур операторы және Вильсон контурының дуалы

Magnetic loop operator dual to the Wilson loop

In quantum field theory, the 't Hooft loop is a magnetic analogue of the Wilson loop for which spatial loops give rise to thin loops of magnetic flux associated with magnetic vortices. 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.

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. For a gauge group where is the universal covering group with a Lie algebra and is a subgroup of the center, then the set of allowed Wilson lines is in one to one correspondence with the representations of This can be formulated more precisely by introducing the weights of the Lie algebra, which span the weight lattice Denoting as the lattice spanned by the weights associated with the algebra of rather than , the Wilson lines are in one to one correspondence with the lattice points lattice where is the Weyl group. The Lie algebra valued charge of the 't Hooft line can always be written in terms of the rank Cartan subalgebra as , where is an dimensional charge vector. Due to Wilson lines, the 't Hooft charge must satisfy the generalized Dirac quantization condition , which must hold for all representations of the Lie algebra. The generalized quantization condition is equivalent to the demand that holds for all weight vectors. To get the set of vectors that satisfy this condition, one must consider roots which are adjoint representation weight vectors. Co roots, defined using roots by , span the co root lattice These vectors have the useful property that 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 of with a weight lattice , in which case the 't Hooft lines are described by
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 up to the identification that for all it holds that

Line operators play a role in indicating differences in gauge theories of the form that differ by the center subgroup 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. Despite this, the theories do differ in their global properties, such as having different sets of allowed line operators. For example, in gauge theories, Wilson loops are labelled by while 't Hooft lines by However in the lattices are reversed where now the Wilson lines are determined by while the 't Hooft lines are determined by .

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. For a gauge group where is the universal covering group with a Lie algebra and is a subgroup of the center, then the set of allowed Wilson lines is in one to one correspondence with the representations of This can be formulated more precisely by introducing the weights of the Lie algebra, which span the weight lattice Denoting as the lattice spanned by the weights associated with the algebra of rather than , the Wilson lines are in one to one correspondence with the lattice points lattice where is the Weyl group. The Lie algebra valued charge of the 't Hooft line can always be written in terms of the rank Cartan subalgebra as , where is an dimensional charge vector. Due to Wilson lines, the 't Hooft charge must satisfy the generalized Dirac quantization condition , which must hold for all representations of the Lie algebra. The generalized quantization condition is equivalent to the demand that holds for all weight vectors. To get the set of vectors that satisfy this condition, one must consider roots which are adjoint representation weight vectors. Co roots, defined using roots by , span the co root lattice These vectors have the useful property that 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 of with a weight lattice , in which case the 't Hooft lines are described by
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 up to the identification that for all it holds that

Line operators play a role in indicating differences in gauge theories of the form that differ by the center subgroup 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. Despite this, the theories do differ in their global properties, such as having different sets of allowed line operators. For example, in gauge theories, Wilson loops are labelled by while 't Hooft lines by However in the lattices are reversed where now the Wilson lines are determined by while the 't Hooft lines are determined by .