• modularity
• N=4 super-Yang-Mills
• supersymmetric localization
• Wilson loop

In the context of N = 4 Supersymmetric Yang-Mills (SYM) theory in 4 dimensions, we want to study the defect operator sector and explore its properties under S-duality.
We introduce Wilson-'t Hooft line operators that represent the response of the gauge fields to a dyonic probe labelled by an index (q,p) with q and p respectively the magnetic and electric charge. Wilson-'t Hooft operators are exchanged under electromagnetic S-duality, in particular (q,p) transforms in the antifundamental representation of SL(2,Z).
Studying the defect operator sector is therefore the most direct way to test the conjectured SL(2, Z) duality.
We consider the integral of the correlator of a Wilson line with the insertion of two local operators against a supersymmetry preserving measure.
We propose a class of functions with an index (q,p) that changes under a modularity transformation as the integrated correlator and check that the basis is sufficient in the large N expansion, at fixed g, of the observable and in the finite N case. All our results include the instanton sector of the theory and are exact in g.
Some drawbacks of the basis of functions proposed are discussed together with the implications of these drawbacks on our understanding of the modularity properties of the integrated correlator.