The Giant Graviton Expansion

We propose and test a novel conjectural relation satisfied by the superconformal index of maximally supersymmetric $U(N)$ gauge theory in four dimensions. Analogous relations appear to be also valid for the superconformal indices of a large collection of other gauge theories, as well as for a broad class of index-like generating functions. The relation expresses the finite $N$ index as a systematic series of corrections to a large $N$ answer. Individual corrections have an holographic interpretation as the analytic continuation of contributions from "giant graviton" branes fixed by a specific symmetry generator.