  Surface operators are nonlocal probes of gauge theories capable of
distinguishing phases that are not discernible by the classic Wilson-'t Hooft
criterion. We prove that the correlation function of a surface operator with a
chiral primary operator in ${\cal N}=4$ super Yang-Mills is a finite polynomial
in the Yang-Mills coupling constant. Surprisingly, in spite of these
observables receiving nontrivial quantum corrections, we find that these
correlation functions are exactly captured in the 't Hooft limit by
supergravity in asymptotically $AdS_5\times S^5$ [1]! We also calculate exactly
the surface operator vacuum expectation value and the correlator of a surface
operator with 1/8-BPS Wilson loops using supersymmetric localization. We
demonstrate that these correlation functions in ${\cal N}=4$ SYM realize in a
nontrivial fashion the conjectured action of $S$-duality. Finally, we
perturbatively quantize ${\cal N}=4$ SYM around the surface operator
singularity and identify the Feynman diagrams that when summed over reproduce
the exact result obtained by localization.
