I will discuss a new approach to extracting the physical consequences of S-duality — that is, invariance under an SL(2,Z) action on the complexified gauge coupling — of four-dimensional maximally-supersymmetric super Yang-Mills theory. The main mathematical tool is the theory of harmonic analysis on the modular surface, i.e. the SL(2,Z) fundamental domain. Applying this technology leads to strong consequences for the analytic structure of observables in N=4 super Yang-Mills theory, such as the structure of instantons and the existence of non-perturbative effects. At large N, there are ramifications for holography: in particular, spectral decomposition reveals an equivalence between the strongly coupled planar theory, dual to type IIB supergravity, and the ensemble average over the conformal manifold of N=4 theories. Some wider implications for the AdS/CFT correspondence and connections to spacetime wormholes will (hopefully) be discussed. [The talk will also be streamed online, please ask the organizers for the link.]

IPhT