Giant Gravitons and Volume Minimisation

TL;DR

The paper uncovers a precise bridge between two developments in AdS/CFT: the giant graviton expansion of the superconformal index in D ≤ 4 and the Gauntlett–Martelli–Sparks master-volume formalism for near-horizon geometries. Focusing on 4d N=1 SCFTs from N D3-branes at toric SE5 tips, the authors show that the large-wrapping (giant graviton) contributions are controlled by the master volume of a metric deformation of SE5 fibred over S^3, with the wrapping numbers identifying with Kähler moduli via λ_I = n_I and the Reeb data b = ∑ Δ_I v_I. In AdS5 × SE5 this yields a concrete saddle-point relation for the giant-graviton index: log I_{(n_1,...,n_D)} ∼ - (ω1 ω2)/(2π)^3 × V(b, λ_I). At large N this reproduces the black-hole entropy function after extremising over the moduli, suggesting a geometrical, complexified version of GMS volume minimisation governs rotating BPS black holes and their near-horizon geometries sourced by back-reacted giant gravitons; analogous results are derived for 3d N=2 theories dual to M-theory on AdS4 × SE7. The work thus weaves a coherent holographic narrative in which giant gravitons and variational master volumes encode the microstates and near-horizon data of BPS black holes and strings, offering new avenues to interpret complex Euclidean geometries in holography and to extend the variational paradigm beyond static, magnetically charged solutions.

Abstract

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 Kähler 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$_4\times \text{SE}_7$.

Figures (2)

• Figure 1: The toric diagram of Kähler cone $X_6$, with toric fan vectors $\{\mathbf{v}_I\}$.
• Figure 2: A diagrammatic visualisation of the solution space to the four inequalities (\ref{['eq: inequalities']}) with $I=1$.