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We consider the S-duality transformation of gauge invariant operators andstates in $\mathcal{N}$ $=4$ SYM theory. The transformation is realized throughan operator $ S $ which is the $ SL(2,Z) $ canonical transformation in loopspace with the gauge invariant electric and the magnetic flux operatorscomposing the canonical variables. Based on $ S $, S-duals for all of thephysical operators and states can be defined. The criterion for the theory tobe S-duality invariant is that the superconformal charges and their S-dualsdiffer by a $ U(1)_{Y} $ phase. The verification can be done by checking the Stransformation for supersymmetry and special supersymmetry variations of theloop operators. The fact that supercharges preserved by BPS Wilson operatorsand the S-dual BPS 't Hooft operators differ by a $4d $ chiral rotation couldin some sense serve as a proof.

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