Large black hole entropy from the giant brane expansion

Abstract

We show that the Bekenstein-Hawking entropy of large supersymmetric black holes in AdS5 × S5 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 → ∞ at fixed ratio $$\frac{{\text{Entropy}}}{{N}^{2}}$$, it recovers the 1/16 BPS black hole entropy by a saddle-point approximation of the giant graviton expansion.

Giant graviton expansion of Schur index and quasimodular forms

“…where s a,b are Stirling number of the first kind and P k are twisted Weierstrass functions defined in (A. 16). It can be shown that the index expression (2.6) does not depend on the choice of the auxiliary fugacity w and special choices of w may simplify expressions.…”

“…In the simplest example of four dimensional N = 4 U(N ) super Yang-Mills (SYM) theory dual to type IIB superstring in AdS 5 × S 5 , finite N effects originate from wrapped D3 brane states maximally stretched in S 5 called giant gravitons [13] and having charge of order JHEP05(2024)282 N . 1 From the corresponding giant-graviton expansion of the index, it is possible to reproduce the entropy of dual black holes in AdS 5 × S 5 as first shown in [15] for small black holes with charge Q ≪ N 2 and later generalized to black holes with charge Q ∼ N 2 , first in [16], using a large-charge expansion, and later in [17], using a large N expansion. Recently, the giant graviton expansion has also been described by enumerating BPS geometries studying bubbling solutions in supergravity [18,19].…”

“…24q 5 − 16q 6 + 24q 8 (5.5) + 88q 9 + 48q 10 + 144q 11 + 36q 12 + 192q 13 − 240q 14 + 280q 15 − 504q16 …”