Wilson-’t Hooft operators in four-dimensional gauge theories andS-duality

Abstract

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 wellknown 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 AdS 2 E S 2 . 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 -angle by 2. We show that, depending on the gauge group, either SL2; 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.

Twist operators in higher dimensions

“…However, we must remind the reader that for the free conformally coupled scalar in d ≥ 3, the heat kernel calculations in appendix B produced a result for h n,2 which was not in agreement with eq. (3.26), i.e., our 22 We note that this expansion was recently extended to include a chemical potential in discussing a new class of 'charged' Rényi entropies [51].…”

“…However, we were able to use this expression to construct an expansion (2.35) of the conformal dimension in power series around n = 1 (where n is the order of the twist operator). 22 Further, h n,k = ∂ k n h n | n=1 , i.e., the coefficient of the term proportional to (n−1) k in eq. (2.35), is completely detemined by the (k + 1)-and k-point correlation functions of the stress tensor in flat space.…”

“…As noted in the introduction, in a higher dimensional CFT, the twist operators may be assigned a generalized notion of conformal dimension [22]. As in d = 2, the latter is defined by the leading singularity in the correlator T µν σ n .…”

The algebra of Wilson–'t Hooft operators

“…The 't Hooft loop operator is a disorder operator defined by the requirement that near a curve γ the gauge field has a singularity of a Dirac-monopole kind. Such singularities are labeled by conjugacy classes of homomorphisms from U(1) to G, which is equivalent to saying that they are labeled by orbits of the Weyl group in the coweight lattice Λ cw of G. More generally, Wilson-'t Hooft operators are labeled by Weyl orbits in the product Λ w (G) × Λ cw (G) [3]. In the N = 4 SYM theory there are more possibilities for loop operators, since one can construct them not only from gauge fields, but also from other fields.…”

“…3 Indeed, if γ is given by the equation x 1 = Rew = 0 and we require the gauge field to have a Dirac-like singularity in the x 1 , x 2 , x 3 plane:…”