M2-brane indices on Higgs vacua and black holes

TL;DR

This work analyzes the finite-$N$ superconformal index of the 3d ADHM quiver, a UV description of the $ ext{N}=8$ SCFT dual to M-theory on $AdS_4\times S^7$, to probe quantum black hole microstates beyond the large-$N$ limit. Using the factorized index, the authors compute high-order contributions and separate graviton versus black hole sectors by comparing microcanonical degeneracies and canonical complex-$\beta$ phase diagrams; they identify signatures of black hole states in finite-$N$ data and relate them to the large-$N$ entropy. They further develop the factorization for multi-flavor ($F$) ADHM theories, connect Higgs-vacuum resummations to the Hilbert series on the Higgs branch, and reveal additional saddles tied to Higgs vacua and Bethe Ansatz solutions. The results demonstrate that finite-$N$ indices encode rich quantum gravitational information and provide new tools for resolving microstates and phase structure in holographic M-theory setups.

Abstract

As an exact count of protected states, the superconformal index provides a powerful probe into holography and quantum aspects of gravity, reproducing the Bekenstein--Hawking entropy of supersymmetric AdS black holes in the large-$N$ limit. As a step toward understanding quantum black hole microstates, we study the finite-$N$ index of the 3d ADHM quiver gauge theory, a UV description of the 3d $\mathcal N=8$ SCFT dual to M-theory on AdS$_4 \times S^7$. In this note, we analyze both microcanonical and canonical features of the superconformal index. By computing the index to sufficiently high orders using the factorization formula, we identify signatures of quantum black hole states in the finite-$N$ spectrum of the ADHM quiver, which align with the leading large-$N$ contribution reflecting the holographic dual black hole entropy. Furthermore, we introduce the complex-$β$ phase diagram of the index, which exhibits distinct peaks potentially associated with different gravitational saddles. We also examine the Hilbert series limit of the factorized index. Our results demonstrate that the finite-$N$ index encodes rich information about black hole microstates and their quantum gravitational interpretation.

Figures (13)

• Figure 1: $\log I_{N=1}(Q)$ (blue & orange dots) vs $S(Q)$ (green line) up to $x^{100}$ on the left and up to $x^{300}$ on the right. Blue and orange dots represent coefficients with positive and negative signs, respectively. In the left plot, the small red dots indicate the graviton spectrum derived from the gravity side. While $\log I_{N=1}$ fits relatively well with $S$ for $Q < 100$, it deviates from $S$ for higher $Q$.
• Figure 2: $\log I_{N=2, \, 3}(Q)$ (blue & orange dots) vs $S(Q)$ (green line) on the left and $\log I_{N=2, \, 3}/I_{N=1}(Q)$ (blue & orange dots) vs $S_\text{int}(Q)$ (green line) on the right, both up to $x^{100}$. Blue and orange dots represent coefficients with positive and negative signs, respectively, while the small red dots indicate the graviton spectrum derived from the gravity side. The left and right plots in each line differ by the decoupled hypermultiplets, whose index contribution is the same as $I_{N=1}$.
• Figure 3: $\log I_{N=2}/I_{N=1}(Q)$ (blue & orange dots) vs $\tilde{S}(Q;\gamma = 0.53, \, \delta = -0.4)$ (green & red crosses) up to $x^{100}$. Blue dots and green crosses correspond to coefficients with the positive sign, whereas orange dots and red crosses correspond to coefficients with the negative sign. There is good agreement between the dots and the crosses in the oscillating region for large $Q$.
• Figure 5: $|\mathcal{I}_{N=2}/\mathcal{I}_{N=1}(\beta)|$ as a function of $\theta = \mathrm{Im} \,\beta/2$ when $\mathrm{Re} \,\beta=1$ (left), 0.453 (center), 0.1 (right), respectively. For $N=2$, the periodicity of $\theta$ is $\pi$, half of the other cases because only even powers appear in the series expansion. The transition occurs when $\beta = 0.453$ (center), where there are two peaks, up to sign, one at $\theta = 0$ and the other at $\theta = \pm1.34$.
• Figure 7: The left two are the black hole indices at $\beta = 1$ and $\beta = 0.504$ estimated from the large-$N$ entropy $\tilde{S}$, given by \ref{['eq:Stilde']}. The right two are the graviton indices $\mathcal{I}^\text{graviton}_{N=\infty}/\mathcal{I}_{N=1}$ at $\beta = 1$ and $\beta = 0.504$.