Large black hole entropy from the giant brane expansion

Abstract: We show that the Bekenstein-Hawking entropy of large supersymmetric black holes in AdS$_5\times S<sup>5$ emerges from remarkable cancellations in the giant graviton expansions recently proposed by Imamura, and Gaiotto and Lee, independently. A similar cancellation mechanism is shown to happen in the exact expansion in terms of free fermions recently put-forward by Murthy. These two representations can be understood as sums over independent systems of giant D3-branes and free fermions, respectively. At large charges, the free energy of each independent system localizes to its asymptotic expansion near the leading singularity. The sum over the independent systems maps their localized free energy to the localized free energy of the superconformal index of $U(N)$ $\mathcal{N}=4$ SYM. This result constitutes a non-perturbative test of the giant graviton expansion valid at any value of $N$. Moreover, in the holographic scaling limit $N\to\infty$ at fixed ratio $\frac{\text{Entropy}}{N<sup>2}\,$, it recovers the 1/16 BPS black hole entropy by a saddle-point approximation of the giant graviton expansion.