S-duality transformation of $\mathcal{N}$ $=4$ SYM theory at the operator level

We consider the S-duality transformation of gauge invariant operators and states in $\mathcal{N}$ $=4$ SYM theory. The transformation is realized through an operator $S$ which is the $SL(2,Z)$ canonical transformation in loop space with the gauge invariant electric and the magnetic flux operators composing the canonical variables. Based on $S$, S-duals for all of the physical operators and states can be defined. The criterion for the theory to be S-duality invariant is that the superconformal charges and their S-duals differ by a $U(1{)}_Y$ phase. The verification can be done by checking the S transformation for supersymmetry and special supersymmetry variations of the loop operators. The fact that supercharges preserved by BPS Wilson operators and the S-dual BPS 't Hooft operators differ by a $4d$ chiral rotation could in some sense serve as a proof.