Localized states of BFSS super quantum mechanics

TL;DR

This paper shows that BFSS matrix quantum mechanics exhibits localized and non-uniform phases arising from Gregory-Laflamme instabilities in the 11D gravity dual. It constructs a Carrollian map from Kaluza-Klein black strings/holes to pp-wave BFSS states, enabling exact translation of gravitational thermodynamics into BFSS thermodynamics and detailed phase diagrams. The authors derive analytic BFSS thermodynamics for uniform and perturbative localized phases and numerically construct non-uniform and localized gravity solutions, locating precise first- and second-order transitions and the corresponding phase boundaries. The results illuminate the strongly coupled BFSS regime, providing quantitative predictions for phase dominance in microcanonical and canonical ensembles and offering concrete targets for lattice, bootstrap, and machine-learning tests of holography. Overall, the work deepens the holographic understanding of BFSS and expands the landscape of black-hole/black-string phase structure in higher dimensions with explicit BFSS dual data.

Abstract

We analyze the recently discovered localized and non-uniform phases of the Banks-Fischler-Shenker-Susskind (BFSS) matrix quantum mechanics. Building on [1], we provide first-principles derivations of their properties and extend the results with new analytic and numerical insights. We show that strongly coupled BFSS dynamics emerge from a specific Carrollian transformation of 11-dimensional supergravity, which we justify in detail. In this framework, the uniform BFSS phase corresponds to a black string in a $pp$-wave background. We demonstrate that this background is unstable to a Gregory-Laflamme instability and, for the first time, compute the associated growth rate. The instability gives rise to non-uniform and localized phases that dominate the microcanonical ensemble in certain low-energy regimes, with the localized phase also prevailing in the canonical ensemble at low temperatures. We identify the corresponding first- and second-order phase transitions and derive analytic formulas for the thermodynamics of the localized phase, accurate to better than $0.3\%$ against numerical results.

Figures (11)

• Figure 1: A schematic drawing of the transverse horizon radius for different black hole phases with standard Kaluza-Klein asymptotics: (a) the localized black hole phase, (b) the non-uniform string phase, and (c) the uniform string phase. The boundaries at the horizontal axis describe the two periodically identified boundaries of the circle $S^1_{\hbox{\tiny $L$}}$.
• Figure 2: Schematic regime of validity for matrix black holes (BHs), $d=11$ and IIA supergravities and perturbative super quantum mechanics (SQM). Bright colors describe the region where the given theory is valid and the increasingly faded colors represent the regions where it becomes an increasingly poor approximation. Note that when type IIA supergravity is valid, so is $d=11$ supergravity (they both have the same upper cut-off temperature where $\alpha'$ corrections start being non-negligible and both supergravities stop being a good approximation to type IIA string theory and M-theory, respectively). However, the 11-dimensional supergravity description is valid for lower temperatures $\tau$ than type IIA supergravity. Here, we take $N\gg 1$, which is required for the Gregory-Laflamme (GL) transition at $\mathcal{O}(N^{-5/9})$ to be visible within 11-dimensional supergravity.
• Figure 3: Quasinormal mode frequency $\varpi$ as a function of $\varepsilon / N^{4/9}$. The real part is shown by blue disks and the imaginary part by orange squares. The instability sets in at $\varepsilon / N^{4/9}=1.7133(8)$, coinciding with the onset of the non-uniform phase. The behaviour reflects a second-order transition characterised by an exchange of stability: the uniform phase becomes unstable to the Gregory-Laflamme mode, while the non-uniform phase acquires stability.
• Figure 4: Integration domain for the non-uniform strings with six patches and displaying regions where higher resolution is required.
• Figure 5: The gauge-invariant metric component $\lvert g_{\tilde{T}\tilde{T}} \rvert / L^2$ is shown in the $\{x,y\}$ coordinates for $x_+ \approx 0.411377$. The steep gradients near $x=0$ illustrate why six coordinate patches are necessary. In the near-horizon region, where $\lvert g_{\tilde{T}\tilde{T}} \rvert / L^2$ is nearly constant, a different set of variables is employed, allowing for an accurate extraction of the energy and tension of the non-uniform string (see Eq. \ref{['nonU:widehatQ']}).