Universal Spinning Black Holes and Theories of Class $\mathcal R$

TL;DR

The paper identifies a universal spinning black hole in AdS$_4$ within minimal gauged supergravity and proves a universal Cardy-like relation between the 3d ${\\cal N}=2$ SCFT superconformal index ${\\cal I}_{S^2}$ and the $S^3$ free energy $F_{S^3}$ in the large $N$ limit, $\\log {\\cal I}_{S^2} \\approx { m i}\\frac{F_{S^{3}}}{\\pi} \\frac{\\varphi^{2}}{\\omega}$. This universality holds across uplifts to string/M-theory (M5, M2, D2, wrapped D4 and $(p,q)$-fivebranes) and in theories of class $\\cal R$ arising from M5-branes on hyperbolic 3-manifolds, where the index reproduces the black hole entropy at large $N$ via the 3d-3d correspondence. For class $\\cal R$, the index is computed through complex Chern-Simons theory on the hyperbolic manifold and matches the gravity result, confirming a microscopic origin for the entropy. The work points to a field-theory Cardy-like formula governing the index in a universal regime and outlines multiple future directions to derive it from first principles and extend the framework to broader holographic contexts.

Abstract

We study a supersymmetric, rotating, electrically charged black hole in AdS$_{4}$ which is a solution of four-dimensional minimal gauged supergravity. Using holography we show that the free energy on $S^3$ and the superconformal index of the dual three-dimensional $\mathcal N=2$ SCFT, in the planar limit, are related in a simple universal way. This result applies to large classes of SCFTs constructed from branes in string and M-theory which we discuss in some detail. For theories of class $\mathcal R$, which arise from $N$ M5-branes wrapped on hyperbolic three-manifolds, we show that the superconformal index agrees with the black hole entropy in the large $N$ limit.