Black hole microstates in AdS$_4$ from supersymmetric localization

TL;DR

This work presents a microscopic derivation of the entropy of supersymmetric AdS$_4$ black holes by computing the topologically twisted index of ABJM theory at large $N$ and extremizing it over trial R-symmetries. The index is evaluated via localization, reduced to a contour integral over gauge holonomies, and solved in the large-$N$ limit through a Bethe Ansatz framework governed by a Bethe potential, yielding an $N^{3/2}$ scaling. The resulting extremized index exactly reproduces the Bekenstein-Hawking entropy of static AdS$_4$ black holes, with the critical chemical potentials corresponding to horizon values of the bulk scalar fields and thus encoding the exact R-symmetry of the IR superconformal quantum mechanics. On the gravity side, the analysis connects the asymptotic magnetic AdS$_4$ vacuum to the near-horizon AdS$_2 imes S^2$ geometry through an attractor mechanism, and shows how the microscopic count aligns with the macroscopic entropy, supporting an I-extremization principle in 1D systems and opening avenues for generalized dualities and quantum corrections.

Abstract

This paper addresses a long standing problem, the counting of the microstates of supersymmetric asymptotically AdS black holes in terms of a holographically dual field theory. We focus on a class of asymptotically AdS$_4$ static black holes preserving two real supercharges which are dual to a topologically twisted deformation of the ABJM theory. We evaluate in the large $N$ limit the topologically twisted index of the ABJM theory and we show that it correctly reproduces the entropy of the AdS$_4$ black holes. An extremization of the index with respect to a set of chemical potentials is required. We interpret it as the selection of the exact R-symmetry of the superconformal quantum mechanics describing the horizon of the black hole.

Figures (4)

• Figure 1: Analytic structure of $\mathop{\mathrm{Li}}\nolimits_n(e^{iu})$.
• Figure 2: Plots of the eigenvalues $u_i$, $\tilde{u}_j$ for $N=25$ (in orange) and $N=101$ (in blue) for $y_a=i$ and $k=1$. When $N\rightarrow 4N$, the imaginary parts of $u_i$, $\tilde{u}_j$ are approximately doubled---consistently with a scaling $N^\frac{1}{2}$---while the real parts remain constant. For comparison, we also plot the analytical result.
• Figure 3: Plots of $u_i$, $\tilde{u}_j$ for $N=50$ (on the left) and $N=75$ (on the right), $\Delta_1=0.3$, $\Delta_2=0.4$, $\Delta_3=0.5$ with $\sum_a\Delta_a=2\pi$ and $k=1$.
• Figure 4: Plots of the density of eigenvalues $\rho(t)$ and the function $\delta v(t)$ for $N=75$, $\Delta_1=0.3$, $\Delta_2=0.4$, $\Delta_3=0.5$ with $\sum_a\Delta_a=2\pi$ and $k=1$. The blue dots represent the numerical simulation, while the solid grey line is the analytical result.