Lecture 1 on "Modular Structures in N=4 supersymmetric Yang-Mills theory"

In this lecture series we present recent results in the study of exact correlation functions of half-BPS operators in N=4 supersymmetric Yang-Mills theory (SYM) averaged over the space-time insertion points. After presenting some basic properties of 1/2-BPS operators in N=4 SYM, we review how these integrated correlation functions can be obtained from a matrix model formulation of the N=4 path-integral. We then move to present two different integrated correlation functions of four superconformal primary operators of the stress-tensor multiplet which are holographically related to scattering amplitudes of 4-gravitons in type IIB superstring theory on an AdS_5 x S^5 background. We derive exact expressions both in the number of colours N, as well as in the complexified Yang-Mills coupling \tau. A key player in our discussion is electro-magnetic duality of N=4 SYM which provides strong constraints on the coupling dependence of such observables which, in particular, have to be real-analytic modular invariant functions of \tau. We then discuss the large-N fixed-\tau limit to show how these results can be interpreted on the dual stringy side. We also present some details on how these integrated correlator can be used to supplement the standard bootstrap approach leading to exciting coupling dependent bounds on the anomalous dimensions of non-protected operators in N=4 SYM, such as the Konishi operator. Lastly, we discuss an integrated correlation function involving two superconformal primary operators in the stress tensor multiplet in the presence of a half-BPS line defect operator, such as a Wilson line. Electro-magnetic duality is again fundamental in understanding the exact dependence from the coupling constant \tau. [OPTIONAL: If time and energy permit, I can also present some new results regarding integrated correlation functions of two light operators, dual to gravitons on the holographic side, and heavy giant graviton operators, dual to D3 branes extended on the background geometry]