Hooft Loop Operators Research Articles

Action of the Axial U(1) Non-Invertible Symmetry on the ’t Hooft Line Operator: A Lattice Gauge Theory Study

Abstract We study how the symmetry operator of the axial $U(1)$ non-invertible symmetry acts on the ’t Hooft line operator in the $U(1)$ gauge theory by employing the modified Villain-type lattice formulation. We model the axial anomaly by a compact scalar boson, the “QED axion”. For the gauge invariance, the simple ’t Hooft line operator, which is defined by a line integral of the dual $U(1)$ gauge potential, must be “dressed” by the scalar and $U(1)$ gauge fields. A careful consideration on the basis of the anomalous Ward–Takahashi identity containing the ’t Hooft operator with the dressing factor and a precise definition of the symmetry operator on the lattice shows that the symmetry operator leaves no effect when it sweeps out a ’t Hooft loop operator. This result appears inequivalent with the phenomenon concluded in the continuum theory. In an appendix, we demonstrate that the half-space gauging of the magnetic $\mathbb {Z}_N$ 1-form symmetry, when formulated in an appropriate lattice framework, leads to the same conclusion as above. A similar result is obtained for the axion string operator.

Topological defects in CFT

Areview of the notion, properties and the use of topological defects in 2d conformal field theories is presented. An emphasis is made on the recent interpretation of such operators in non-rational theories, as describing Wilson-’t Hooft loop operators of N = 2 supersymmetric 4d topological theories.

More on ’t Hooft loops in $ \mathcal{N}=4 $ SYM

We study supersymmetric 't Hooft loop operators in N=4 super Yang-Mills, generalizing the well-known circular 1/2 BPS case and investigating their S-duality properties. We derive the BPS condition for a generic line operator describing pointlike monopoles and discuss its solutions in some particular case. In particular, we present the explicit construction of the magnetic counterpart of Zarembo and DGRT Wilson loops and provide the general dyonic configurations for an abelian gauge group. The quantum definition of these supersymmetric 't Hooft loop operators is carefully discussed and we attempt some computations to next-to-leading order in perturbation theory.

A note on perturbation series in supersymmetric gauge theories

Exact results in supersymmetric Chern-Simons and N=2 Yang-Mills theories can be used to examine the quantum behavior of observables and the structure of the perturbative series. For the U(2) x U(2) ABJM model, we determine the asymptotic behavior of the perturbative series for the partition function and write it as a Borel transform. Similar results are obtained for N=2 SU(2) super Yang-Mills theory with four fundamental flavors and in N=2* super Yang-Mills theory, for the partition function as well as for the expectation values for Wilson loop and 't Hooft loop operators (in the 0 and 1 instanton sectors). In all examples, one has an alternate perturbation series where the coefficient of the nth term increases as n!, and the perturbation series are Borel summable. We also calculate the expectation value for a Wilson loop operator in the N=2* SU(N) theory at large N in different regimes of the 't Hooft gauge coupling and mass parameter. For large masses, the calculation reproduces the running gauge coupling for the pure N=2 SYM theory.

Exact results for ’t Hooft loops in Gauge theories on S 4

Abstract The path integral of a general $\mathcal{N} = 2$ supersymmetric gauge theory on S 4 is exactly evaluated in the presence of a supersymmetric ’t Hooft loop operator. The result we find — obtained using localization techniques — captures all perturbative quantum corrections as well as non-perturbative effects due to instantons and monopoles, which are supported at the north pole, south pole and equator of S 4. As a by-product, our gauge theory calculations successfully confirm the predictions made for ’t Hooft loops obtained from the calculation of topological defect correlators in Liouville/Toda conformal field theory.

’t Hooft operators in gauge theory from Toda CFT

We construct loop operators in two dimensional Toda CFT and calculate with them the exact expectation value of certain supersymmetric 't Hooft and dyonic loop operators in four dimensional \Ncal=2 gauge theories with SU(N) gauge group. Explicit formulae for 't Hooft and dyonic operators in \Ncal=2^* and \Ncal=2 conformal SQCD with SU(N) gauge group are presented. We also briefly speculate on the Toda CFT realization of arbitrary loop operators in these gauge theories in terms of topological web operators in Toda CFT.

Gauge theory loop operators and Liouville theory

We propose a correspondence between loop operators in a family of four dimensional $$ \mathcal{N} $$ = 2 gauge theories on S 4 — including Wilson, ‘t Hooft and dyonic operators — and Liouville theory loop operators on a Riemann surface. This extends the beautiful relation between the partition function of these $$ \mathcal{N} $$ = 2 gauge theories and Liouville correlators found by Alday, Gaiotto and Tachikawa. We show that the computation of these Liouville correlators with the insertion of a Liouville loop operator reproduces Pestun’s formula capturing the expectation value of a Wilson loop operator in the corresponding gauge theory. We prove that our definition of Liouville loop operators is invariant under modular transformations, which given our correspondence, implies the conjectured action of S-duality on the gauge theory loop operators. Our computations in Liouville theory make an explicit prediction for the exact expectation value of ’t Hooft and dyonic loop operators in these $$ \mathcal{N} $$ = 2 gauge theories. The Liouville loop operators are also found to admit a simple geometric interpretation within quantum Teichmüller theory as the quantum operators representing the length of geodesics. We study the algebra of Liouville loop operators and show that it gives evidence for our proposal as well as providing definite predictions for the operator product expansion of loop operators in gauge theory.

S-duality, 't Hooft operators and the operator product expansion

We study S-duality in N=4 super Yang-Mills with an arbitrary gauge group by determining the operator product expansion of the circular BPS Wilson and 't Hooft loop operators. The coefficients in the expansion of an 't Hooft loop operator for chiral primary operators and the stress-energy tensor are calculated in perturbation theory using the quantum path-integral definition of the 't Hooft operator recently proposed. The corresponding operator product coefficients for the dual Wilson loop operator are determined in the strong coupling expansion. The results for the 't Hooft operator in the weak coupling expansion exactly reproduce those for the dual Wilson loop operator in the strong coupling expansion, thereby demonstrating the quantitative prediction of S-duality for these observables.

Loop operators and S-duality from curves on Riemann surfaces

We study Wilson-'t Hooft loop operators in a class of = 2 superconformal field theories recently introduced by Gaiotto. In the case that the gauge group is a product of SU(2) groups, we classify all possible loop operators in terms of their electric and magnetic charges subject to the Dirac quantization condition. We then show that this precisely matches Dehn's classification of homotopy classes of non-self-intersecting curves on an associated Riemann surface — the same surface which characterizes the gauge theory. Our analysis provides an explicit prediction for the action of S-duality on loop operators in these theories which we check against the known duality transformation in several examples.

Surface operators in abelian gauge theory

We consider arbitrary embeddings of surface operators in a pure, non-supersymmetric abelian gauge theory on spin (non-spin) four-manifolds. For any surface operator with a priori simultaneously non-vanishing parameters, we explicitly show that the parameters transform naturally under an SL(2, Z) (or a congruence subgroup thereof) duality of the theory. However, for non-trivially-embedded surface operators, exact S-duality holds only if the quantum parameter effectively vanishes, while the overall SL(2, Z) (or a congruence subgroup thereof) duality holds up to a c-number at most, regardless. Via the formalism of duality walls, we furnish an alternative derivation of the transformation of parameters - found also to be consistent with a switch from Wilson to 't Hooft loop operators under S-duality. With any background embedding of surface operators, the partition function and the correlation functions of non-singular, gauge-invariant local operators on any curved four-manifold, are found to transform like modular forms under the respective duality groups.

Center vortex model for the infrared sector ofSU(3)Yang-Mills theory: Vortex free energy

The vortex free energy is studied in the random vortex world-surface model of the infrared sector of $SU(3)$ Yang-Mills theory. The free energy of a center vortex extending into two spatial directions, which is introduced into Yang-Mills configurations when acting with the 't Hooft loop operator, is verified to furnish an order parameter for the deconfinement phase transition. It is shown to exhibit a weak discontinuity at the critical temperature, corresponding to the weak first-order character of the transition.

On 't Hooft's loop operator

An explicit realization of 't Hooft's loop operator in continuum Yang–Mills theory is given.

Dirac strings and monopoles in the continuum limit of SU(2) lattice gauge theory

Magnetic monopoles are known to emerge as leading non-perturbative fluctuations in the lattice version of non-Abelian gauge theories in some gauges. In terms of the Dirac quantization condition, these monopoles have magnetic charge |Q_M|=2. Also, magnetic monopoles with |Q_M|=1 can be introduced on the lattice via the 't Hooft loop operator. We consider the |Q_M|=1,2 monopoles in the continuum limit of the lattice gauge theories. To substitute for the Dirac strings which cost no action on the lattice, we allow for singular gauge potentials which are absent in the standard continuum version. Once the Dirac strings are allowed, it turns possible to find a solution with zero action for a monopole--antimonopole pair. This implies equivalence of the standard and modified continuum versions in perturbation theory. To imitate the nonperturbative vacuum, we introduce then a nonsingular background. The modified continuum version of the gluodynamics allows in this case for monopoles with finite non-vanishing action. Using similar techniques, we construct the 't Hooft loop operator in the continuum and predict its behavior at small and large distances both at zero and high temperatures.

New order parameters for non-abelian gauge theories

In recent work a new order parameter has been constructed for abelian gauge theories in the presence of matter, both in the continuum and on the lattice. The order parameter directly probes whether charge is screened and enables the various Higgs phases to be distinguished. We generalize this to non-abelian gauge groups on the lattice, and analyze a specific model of an S 3 gauge theory with S 3 matter. In the course of our general investigation we construct a lattice operator that measures non-abelian discrete charge, and is related to a non-abelian generalization of the 't Hooft loop operator. This also provides further confirmation of the recent prediction that black holes should be able to carry discrete non-abelian quantum hair.

The supersymmetric Yang-Mills vacuum. Topology and physics

Two kinds of ground states, global as opposite to local, in supersymmetric Yang-Mills theory are distinguished. Expectation values of Wilson and 't Hooft loop operators in the global ground state are topological (knots) invariants and also determine the phases of the model. Quantization in the local ground states exhibits anomalous fermionic number and supersymmetry breaking as tied to the middle (Floer) cohomology of the gauge orbit space.

A study of the 't Hooft loop operator

Using the convex inequality, it is shown that the 't Hooft loop operator obeys the perimeter law in U( N) (for any N) and SU( N) (large N limit) lattice gauge theory.

Electric-magnetic duality in non-abelian gauge theories

The duality transformation of the vacuum expectation value of the operator which creates magnetic vortices (the 't Hooft loop operator in the Higgs phase), is performed in the radial gauge ( x u A u a ( x) = 0). It is found that in the weak coupling region (small g) of a pure Yang-Mills theory the dual operator creates electric vortices whose strength is 1 g . The theory is self-dual in this region, and the effective coupling of the dual Lagrangian is 1 g . (It is self-dual also in the extreme strong coupling region.) Thus the above duality transformation reduces to electric-magnetic duality where the electric field in the 't Hooft loop operators transforms into a magnetic field in the dual operator. In a spontaneously broken gauge theory these results are valid only within the region where the vortices (or the monopoles) are concentrated, or in directions of the algebra space of unbroken symmetry, as self-duality holds only for this subset of fields. Noting that the 't Hooft loop operator project into the subspace of these field configurations we find that it is an electric-magnetic duality for the spontaneously broken theory as well. In the strong coupling region a strong coupling expansion in powers 1 g is suggested.

Duality and the phases of massive gauge-invariant QCD

I discuss the phase structure of massive gauge invariant QCD (the gauged non-linear sigma model) using an operator formalism proposed by 't Hooft. The dual variables for this system are determined and the 't Hooft loop operator is constructed from these dual field variables. Its equal time commutation relations with the Wilson loop operator is explicitly verified. The system possesses two possible phases. In the semi-classical approximation (weak coupling), magnetic confinement as indicated by an area law for the expectation value of 't Hooft loops is observed. If above a certain critical coupling a condensate of vortices is formed, then the expectation value of the 't Hooft loop in the dilute vortex approximation shows a perimeter law indicating electric confinement.

A study of 't Hooft and Wilson loops

An investigation is undertaken for 't Hooft loop operators in four-dimensional gauge theories. For the first time, a perimeter law is shown to be their behavior in weak coupling Wilson lattice (and continuum) non-abelian SU( N) gauge theories for all N. However, it is also argued that this perimeter law is poor criterion for quark confinement. Rather, it is suggested that non-leading long-distance behavior is what is crucial and relevant in distinguishing non-abelian from abelian (and hence confining from non-confining) theories. A new object, “the 't Hooft line”, is introduced to measure this non-leading behavior and is computed in strong coupling on the lattice. There, one finds magnetic screening characterized by a magnetic screening mass, m s. It is shown to all orders in strong coupling that m s is the glueball mass, a result which is expected to persist in weak coupling and in the continuum. Two further consequences of this work are that pure non-abelian gauge theories cannot be in a Higgs phase and that in such models that absence of massless physical particles implies confinement. Finally, non-leading behavior in Wilson loops is examined. The present picture of confinement suggests the absence of van der Waals forces in Yang-Mills theories.