Witten index of BMN matrix quantum mechanics

TL;DR

This work analyzes the Witten index of the BMN matrix quantum mechanics, a mass-deformed BFSS model with ${\rm SU}(2|4)$ supersymmetry, to count BPS ground states and protected nonzero-spin sectors. By exploiting non-renormalization, the index is computed in the weak-coupling regime, revealing sector-dependent behavior: the trivial vacuum sector yields an $N^2$-scaled entropy signaling BPS black holes in the plane-wave M-theory dual, while the irreducible vacuum sector matches the ABJM superconformal index in the large-$N$ limit up to a divergent factor and a careful mapping of fugacities. The analysis connects the BMN index to the ABJM index and motivates further exploration of BPS black hole phases, localization derivations, and M2-brane dynamics in the plane-wave background. These results advance the holographic understanding of BPS microstates and offer concrete handles on phase structure in supersymmetric matrix models.

Abstract

We compute the Witten index of the Berenstein-Maldacena-Nastase matrix quantum mechanics, which counts the number of ground states as well as the difference between the numbers of bosonic and fermionic BPS states with nonzero spins. The Witten index sets a lower bound on the entropy, which exhibits an $N^2$ growth that predicts the existence of BPS black holes in M-theory, asymptotic to the plane wave geometry. We also discuss a relation between the Witten index in the infinite $N$ limit and the superconformal index of the Aharony-Bergman-Jafferis-Maldacena theory.

Figures (2)

• Figure 1: The $\log(|d_j^{\rm BMN}|)$, $s_{\rm BMN}$ (blue) and $\log(|d_j^{{\cal N}=4}|)$, $s_{{\cal N}=4}$ (orange) at $j=N^2$ with increasing $N$. The green line is the asymptotic value for $s_{{\cal N}=4}$.
• Figure 2: The blue and orange regions are given by \ref{['eqn:phase_bdry']} and $C>0$ for $C$ given in \ref{['eqn:specific_heat']}. The red curve is given by the function $\beta(Q)$ for $Q$ varying from 0 to $\infty$.