Wilson-'t Hooft operators in four-dimensional gauge theories and S-duality

By A. Kapustin. 2005

394 citations recorded by the source corpus.

We study operators in four-dimensional gauge theories which are localized on a straight line, create electric and magnetic flux, and in the UV limit break the conformal invariance in the minimal possible way. We call them Wilson-'t Hooft operators, since in the purely electric case they reduce to the well-known Wilson loops, while in general they may carry 't Hooft magnetic flux. We show that to any such operator one can associate a maximally symmetric boundary condition for gauge fields on AdSE2×S2. We show that Wilson-'t Hooft operators are classified by a pair of weights (electric and magnetic) for the gauge group and its magnetic dual, modulo the action of the Weyl group. If the magnetic weight does not belong to the coroot lattice of the gauge group, the corresponding operator is topologically nontrivial (carries nonvanishing 't Hooft magnetic flux). We explain how the spectrum of Wilson-'t Hooft operators transforms under the shift of the θ-angle by 2π. We show that, depending on the gauge group, either SL(2,[openface Z]) or one of its congruence subgroups acts in a natural way on the set of Wilson-'t Hooft operators. This can be regarded as evidence for the S-duality of N=4 super-Yang-Mills theory. We also compute the one-point function of the stress-energy tensor in the presence of a Wilson-'t Hooft operator at weak coupling.

Subfields: Condensed Matter Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/7a1f1bb24ea78be7941ee60dff16207b92f187fa

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Condensed Matter Physics

1. F. Benini
2. D. Gaiotto
3. Cyril Closset
4. E. Witten
5. N. Seiberg
6. Sergei Gukov
7. Hee-Cheol Kim
8. O. Aharony

In Nuclear and High Energy Physics

1. D. Gaiotto
2. F. Benini
3. E. Witten
4. Sergei Gukov
5. J. Maldacena
6. N. Seiberg
7. N. Drukker
8. Yuji Tachikawa

Citations in the corpus

• Holographic Wilson loops (2006, background): “We study the modification on the low energy effective field theory on the N D3-branes that generate the AdS5×S(5) background due to the 11 This foliation structure and the relation with N = 4 SYM defined on the AdS2×S 2 boundary – which makes manifest the symmetries left unbroken by the insertion of”
• Electric-Magnetic Duality And The Geometric Langlands Program (2006, background): “The answer is to a large extent fixed by symmetries, but turns out to be rather nontrivial [38].”
• Quantum black holes, localization, and the topological string (2010, methodology): “Note that we are using localization techniques to evaluate a bulk functional integral of supergravity whereas in [14, 56, 57] it was used to evaluate a functional integral in the boundary gauge theory.”
• Gauge theory loop operators and Liouville theory (2009, result): “The study of Wilson [17, 18, 19, 20, 10] and ’t Hooft loop operators [21, 22, 23, 24] in N = 4 super YangMills has yielded quantitative evidence in favour of the S-duality conjecture, demonstrating that the correlation function of Wilson loops and ’t Hooft loops are exchanged under Sduality [23, 24]”
• Comments on twisted indices in 3d supersymmetric gauge theories (2016, background): “57) (7)Note that a Wilson loop is defined in terms of a representation R of the gauge group G instead of the algebra g, although we will not discuss any of the interesting subtleties associated to this fact — see for instance [51, 61].”
• BUBBLING GEOMETRIES FOR HALF-BPS WILSON LINES (2006, background): “The spacetime part of this symmetry SL(2,R)×SO(3) can be seen more clearly by Weyl transformation[10].”
• On the Defect Group of a 6D SCFT (2015, background): “For example, in the context of four-dimensional theories, the spectrum of line operators and surface operators must be specified [1–6].”
• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals (2008, background): “On the other hand, lifting the configuration (via T-duality) to M-theory will allow us to relate these theories (at low energies) to M2-branes probing a C4/Zk singularity, and thereby to justify the duality conjecture we made in the previous section.”
• The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects (2012, neutral): “This operator is described by boundary conditions for the gauge field along C corresponding to inserting a GNO monopole [42, 43] (a Dirac monopole embedded in the gauge group G).”
• Comments on 3d Seiberg-like dualities (2011, methodology): “The (20)Note that the lattice of allowed magnetic fluxes for an orthogonal group is finer than the coroot lattice of the associated Lie algebra [62], since orthogonal groups have nontrivial fundamental group.”
• Fivebranes and Knots (2011, background): “Just as our boundary condition was described by specifying the singularity that fields must have along the boundary of V , so an ’t Hooft operator is defined, as explained in [65], by describing the singular behavior that four-dimensional fields should have along a chosen one-manifold S , which usua”
• Defects in conformal field theory (2016, background): “the scalar Wilson line in four dimensions is one example [28].”
• On gravitational description of Wilson lines (2006, background): “Moreover, a generic 1/2 BPS state broke the R (2)The consequences of AdS2 × S(2) symmetry for the field theory were recently studied in [16, 14, 17].”
• Generalized Symmetries in Condensed Matter (2022, neutral): “In the free theory, this operator can be argued (45) to have an anomalous dimension ∆ k ( g ) = g 2 k 2 32 π 2 ; for large enough k , ∆ p ( g ) passes through 4 at some g c < √ 4 π , and we can interpret this as the location of a transition in the low-energy physics from a Coulomb phase to a phase…”
• Wormholes, emergent gauge fields, and the weak gravity conjecture (2015, background): “The classification of conformal transformations of finite line operators is more complicated, so far not too much seems to have been said about it (although see [43] for some remarks on the infinite case).”
• Framed BPS states (2010, neutral): “The Witten effect for line operators [18, 25] shows that the ’t Hooft operator cannot be a single-valued function of ζ.”