Giant Gravitons and Volume Minimisation

We establish a precise correspondence between the giant graviton expansion of the superconformal index of field theories in $D\leq 4$, and the master volume formalism of Gauntlett, Martelli and Sparks (GMS) which determines the near horizon geometries of certain BPS black holes and black strings in supergravity. We focus on 4d $\mathcal{N}=1$ superconformal field theories arising on the world volume of $N$ D3 branes placed at the tip of a cone over a toric Sasaki-Einstein manifold SE$_{5}$, the simplest example of which is $S^5$, corresponding to $\mathcal{N}=4$ super-Yang-Mills. The giant graviton expansion realises the superconformal index as the sum of contributions from wrapped D3 branes in the dual AdS$_{5}\times \text{SE}_{5}$. We argue that, for large wrapping numbers, the asymptotics of each such contribution is governed by the master volume of a particular metric deformation of $\text{SE}_5$ (suitably fibred over $S^{3}$). In particular, the wrapping numbers of a generic giant graviton configuration are identified with K\"{a}hler moduli of the corresponding metric. We further show that at large $N$ the entropy function of the relevant AdS$_5\times \text{SE}_5$ BPS rotating black hole is recovered by extremising over these moduli. Our results suggest that the complex Euclidean geometries corresponding to rotating BPS black holes in AdS$_{5}$ are determined by a close analogue of GMS volume minimisation, and that conversely, the off-shell geometries considered in such minimisation procedures should be understood as the near-horizon geometries of back-reacted giant gravitons. We present analogous results for 3d $\mathcal{N}=2$ theories holographically dual to M-theory on AdS$_4\times \text{SE}_7$.