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

TL;DR

This work develops a localization-based framework for the topologically twisted index of general ${\\cal N}=2$ theories on $\\mathbb{H}_2\\times S^1$ and applies it to a deformed ABJM theory, revealing a discrete spectrum of normalizable Landau-like modes that control the one-loop determinants. By carefully regularizing the determinants and handling non-compact boundary conditions, the authors derive the leading large-$N$ behavior of the ABJM index and show that it reproduces the Bekenstein–Hawking entropy of magnetically charged hyperbolic AdS$_4$ black holes in a dual ${\\cal N}=2$ gauged supergravity, under a precise dictionary between field-theory fluxes/holonomies and bulk charges. The key technical advance is the explicit treatment of the flux-Laplacian spectrum on $\\mathbb{H}_2$ and the resulting cohomological cancellations that yield a tractable 1-loop determinant. The results provide a concrete holographic check of AdS$_4$/CFT$_3$ via microstate counting on non-compact horizons and point to a deep connection between boundary extremization and bulk attractor-like mechanisms, with potential extensions to dyons and other compactifications.

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.