We provide a quantum path integral definition of an ’t Hooft loop operator, which inserts a point-like monopole in a four-dimensional gauge theory. We explicitly compute the expectation value of the circular ’t Hooft operators in $N = 4$ super Yang–Mills with arbitrary gauge group $G$ up to next to leading order in perturbation theory. We also compute in the strong coupling expansion the expectation value of the circular Wilson loop operators. The result of the computation of an ’t Hooft loop operator in the weak coupling expansion exactly reproduces the strong coupling result of the conjectured dual Wilson loop operator under the action of $S$-duality. This paper demonstrates — for the first time — that correlation functions in $N = 4$ super Yang–Mills admit the action of $S$-duality.
"Quantum 't Hooft operators and $S$-duality in $N=4$ super Yang-Mills." Adv. Theor. Math. Phys. 13 (6) 1941 - 1981, December 2009.