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

TL;DR

This paper investigates Wilson-'t Hooft operators in 4D gauge theories, classifies them by electric and magnetic weights, and demonstrates their transformation properties under S-duality, providing evidence for S-duality in N=4 super-Yang-Mills.

Contribution

It introduces a classification of Wilson-'t Hooft operators via electric and magnetic weights and explores their transformation under S-duality in gauge theories.

Findings

Wilson-'t Hooft operators are classified by electric and magnetic weights.

The spectrum of operators transforms under SL(2,Z) or its subgroups.

The paper computes stress-energy tensor one-point functions at weak coupling.

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 2\pi. 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.