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

TL;DR

Wilson-'t Hooft operators are line defects that create electric and magnetic flux in 4d gauge theories. The authors develop a UV conformal framework where these operators correspond to maximally symmetric boundary conditions on AdS2 x S2, and they classify them by pairs of electric and magnetic weights modulo Weyl transformations. They analyze how S-duality acts on these operators, providing a lattice-like structure that supports S-duality in N=4 SYM and giving explicit expressions for leading scaling weights in both abelian and nonabelian theories. The work also clarifies the nonabelian generalization of Wilson and 't Hooft operators, outlines the role of GNO monopoles and Langlands duality, and points to future extensions to theories with less supersymmetry and other dualities.

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 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 AdS^2\times S^2. We show that Wilson-'t Hooft operators are classifed 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 theta-angle by 2π. We show that, depending on the gauge group, either SL(2,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.