Fortuitous Chaos, BPS Black Holes, and Random Matrices

TL;DR

This work identifies a universal $\Gamma\times\Gamma$ random-matrix model that captures the chaotic fortuitous BPS sector in large $\bar{N}$ gauge theories and arises as a low-energy sector of double-scaled matrix models describing JT supergravity with extended supersymmetry. The model produces a leading spectral density with a gap $E_0$ and a BPS peak at $E=0$, plus a non-BPS continuum peaking near $2E_0$, and exhibits an interpolation between Bessel and Airy universality controlled by $\mu=\widetilde{\Gamma}/\sqrt{E_0}$ and $\tilde{\hbar}=\hbar/\mu$, yielding a topological expansion in $1/\Gamma$. The full definition uses a Wishart-type construction with $H=M^\dagger M$, governed by a string equation and an auxiliary Hamiltonian, and the universal low-energy limit yields $u_0(x)=\widetilde{\Gamma}^2/x^2$ with exact density given by a Bessel form in terms of $\xi=\mu\sqrt{E}/\hbar$, linking to intersection theory via deformed $\Theta$-classes and topological recursion relations. The results illuminate the chaotic nature and universality of fortuitous BPS chaos, connect to extended JT supergravity and Weil-Petersson volumes, and point to non-perturbative implications and broader applicability in holographic BPS sectors.

Abstract

The ``fortuitous'' Bogomol'nyi-Prasad-Sommerfield (BPS) sector states in gauge theory have been argued to furnish a description, through holography, of generic BPS black hole microstates. They are expected to be strongly chaotic, a necessary feature to capture the black hole dynamics. This dovetails nicely with the existence of various random matrix models of JT supergravity with extended supersymmetry, within which the BPS chaos must be contained as a subsector. This paper identifies and studies a simple random matrix model that underlies all known random matrix models of JT supergravity. It is argued that it captures many essential universal features of fortuitous BPS chaos. The model is topological, naturally interpolating between the Bessel and Airy models, where the gap energy $E_0$ controls the interpolation, and seems to have a simple intersection theory interpretation.

Figures (5)

• Figure 1: The triangle of connections.
• Figure 2: The leading density (\ref{['eq:leading']}) showing (red line) the BPS states at $E{=}0$ and the accompanying continuum that starts after a gap of $E_0$ (equal to 4 here). There is a peak at $2E_0$ of height set by the ratio $\Gamma/E_0$. The dashed asymptote is the leading part of the ${\cal N}{=}1$ result $\rho_0(E){\sim} 1/\sqrt{E}$.
• Figure 3: An example of $u_0(x)$. This is for ${\cal N}{=}2$ JT supergravity, with $E_0{=}0.2$. The BPS model curve $u_0{=}{\widetilde{\Gamma}}^2/x^2$ is shown as a red dashed line asymptote.
• Figure 4: Histogrammed statistical data, across $10^5$ samples, from a Wishart-type model with $\Gamma{=}10$, $N{=}100$, scaled into the endpoint. The Marchenko-Pastur leading distribution (\ref{['eq:marchenko-pastur']}) emerges (smooth curve). The inset shows the area close to the origin where the gap in the leading distribution is visible, with the continuum starting at $\lambda_-{\simeq}0.001191$.
• Figure 5: The histogrammed data from figure \ref{['fig:BPS-numerics-1']}, zoomed into the edge region, by scaling the energies as described in the text. Now the continuum begins at $E_0{=}{\widetilde{\Gamma}}^2=100$, with $\hbar$ effectively set to unity. Shown in red are the BPS states, and distributions of the first four non-zero energy levels, across $10^5$ samples. Curves showing the analytic expression for both the leading ($\rho_0(E)$, dotted) and exact ($\rho(E)$, solid) model are superimposed on top.