On $\frac18$-BPS black holes and the chiral algebra of $\mathcal{N}=4$ SYM

TL;DR

This paper explores the absence of certain BPS black hole microstates in $ ext{AdS}_5 imes ext{S}^5$, providing rigorous proofs and algebraic evidence supporting the conjecture that the chiral algebra only contains graviton operators, thus no $rac{1}{8}$-BPS black holes.

Contribution

It offers a rigorous proof that flavored Schur and MacDonald indices do not grow with black hole entropy and analyzes the algebraic structure of the chiral algebra, supporting the conjecture of no $rac{1}{8}$-BPS black holes.

Findings

Flavored Schur index cannot exhibit black hole entropy growth.

Numerical evidence suggests flavored MacDonald index also does not grow.

Constructed cohomologies recover only graviton operators.

Abstract

We investigate the existence of $\frac18$-BPS black hole microstates in the $\mathfrak{su}(1,1|2)$ sector of Type IIB string theory on $\mathrm{AdS}_5 \times \mathrm{S}^5$. As will be explained, these states are in one-to-one correspondence with the Schur operators comprising the chiral algebra of $\mathcal{N}=4$ super-Yang-Mills, and a conjecture of Beem et al. implies that the Schur sector only contains graviton operators and hence $\frac18$-BPS black holes do not exist. We scrutinize this conjecture from multiple angles. Concerning the macroscopic counting, we rigorously prove that the flavored Schur index cannot exhibit black hole entropy growth, and provide numerical evidence that the flavored MacDonald index also does not exhibit such growth. Next, we go beyond counting to examine the algebraic structure, beginning by presenting evidence for the well-definedness of the super-$\mathcal{W}$ algebra of Beem et al., then using modular differential equations to argue for an upper bound on the lightest non-graviton operator if existent, and finally performing a systematic construction of cohomologies to recover only gravitons. Along the way, we clarify key aspects of the 4d/2d correspondence using the formalism of the holomorphic topological twist.