Universal 3d Cardy Block and Black Hole Entropy

TL;DR

This work develops a universal Cardy-limit framework for 3d supersymmetric partition functions, showing that hemisphere (Cardy) blocks dominate in the Cardy limit and enable a factorization of the generalized superconformal index, refined topologically twisted index, and squashed S^3 partition function. It derives two index theorems linking these partition functions to one another and to the round sphere free energy, revealing deep connections between field theory entropies and dual AdS$_4$ gravity entropies in the large $N$ limit. The authors illustrate the framework with explicit large $N$ examples, notably ${\cal N}=4$ $U(N)$ theories and ABJM-type quivers, showing that the entropy functions computed from field theory reproduce the Bekenstein-Hawking entropies of rotating BPS black holes in AdS$_4$ and related holographic spacetimes. The results suggest the Cardy block as a unifying microscopic object that encodes black hole microstates across M-theory and massive IIA duals, and they point to potential exactness at finite $N$ in certain large-$N$ saddles. Altogether, the paper provides a cohesive framework for relating 3d SUSY partition functions to gravity entropies and offers explicit checks across diverse holographic theories.

Abstract

We discuss the Cardy limit of 3d supersymmetric partition functions which allow the factorization into the hemisphere indices: the generalized superconformal index, the refined topologically twisted index and the squashed sphere partition function. In the Cardy limit, the hemisphere index can be evaluated by the saddle point approximation where there exists a dominant saddle point contribution, which we call the Cardy block. The Cardy block turns out to be a simple but powerful object as it is a building block of other partition functions in the Cardy limit. The factorization to the Cardy block allows us to find universal relations among the partition functions, which we formulate as index theorems. Furthermore, if we consider a holographic 3d SCFT and its large $N$ limit, those partition functions relate to various entropic quantities of the dual gravity theory in AdS$_4$. As a result, our result provides the microscopic derivation of the universal relations among those entropic quantities of the gravity theory. We also discuss explicit examples, which confirm our general index theorems.