Finite- <i>N</i> holography and the giant graviton expansion

The Giant Graviton Expansion for superconformal indices provides a direct route to finite-N holography. We explain its different appearances and review its bulk derivation for the index counting states annihilated by half of the supersymmetries of [Formula: see text] supersymmetric Yang–Mills for different gauge groups. From the bulk perspective, the expansion is obtained via supersymmetric localization of the path integral computing the index, which localizes to giant gravitons with maximum size. This construction also explains the analytic continuation in the literature as an instance of wall-crossing.