# Microstate Counting of $AdS_4$ Hyperbolic Black Hole Entropy via the Topologically Twisted Index

## TL;DR

This paper calculates the topologically twisted index for certain supersymmetric theories on hyperbolic spaces and demonstrates its precise agreement with the entropy of corresponding $AdS_4$ hyperbolic black holes, providing a microscopic understanding of their entropy.

## Contribution

It extends the computation of the topologically twisted index to $ ext{AdS}_4$ black holes with hyperbolic horizons and confirms the index's role in counting black hole microstates.

## Key findings

- Exact match between black hole entropy and topologically twisted index for $ABJM$ theories.
- Generalization of the index computation to $ ext{AdS}_4$ hyperbolic black holes.
- Supports the holographic microstate counting for hyperbolic black holes.

## Abstract

We compute the topologically twisted index for general $\mathcal{N} = 2$ supersymmetric field theories on $\mathbb{H}_2\times S^1$. We also discuss asymptotically $AdS_4$ magnetically charged black holes with hyperbolic horizon, in four-dimensional $\mathcal{N} = 2$ gauged supergravity. With certain assumptions, put forward by Benini, Hristov and Zaffaroni, we find precise agreement between the black hole entropy and the topologically twisted index, for $ABJM$ theories.