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Universal 3d Cardy Block and Black Hole Entropy

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.