The Low Energy Limit of BFSS Quantum Mechanics

Abstract

We investigate the low-energy regime of BFSS quantum mechanics using its holographic dual. We identify three distinct thermodynamic phases (black holes) and analyze their thermodynamic properties extensively, including phase transitions amongst the several phases. While the properties of the canonical ensemble aligns with existing conjectures on BFSS thermodynamics, we uncover intriguing and unexpected behavior in the microcanonical ensemble. Specifically, for sufficiently low energies, we observe the dominance of the localized phase. Surprisingly, we also identify an energy range where the non-uniform phase becomes dominant. The transition between these phases is mediated by a Kol-type topology-changing phenomenon.

Figures (6)

• 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.
• Figure 2: $\Delta \mathfrak{f}$, the free energy difference between a given thermal phase and the uniform phase at the same $\tau$, as a function of $\tau$. The black disks represent exact numerical data for the localized BFSS phase, while the red squares denote the non-uniform BFSS phase.
• Figure 3: $\Delta \sigma$, the entropy density difference between a given thermodynamic phase and the uniform phase at the same energy $\varepsilon$, as a function of $\varepsilon$. The black disks represent exact numerical data for the localized BFSS phase, while the red squares denote the non-uniform BFSS phase.
• Figure 4: Sketch of integration domain in $\{r,Z\}$ coordinates. The $\{\rho,\xi\}$ coordinates we use in the far region are related to these coordinates through $r=\frac{\pi}{L}\rho\sqrt{2-\rho^2}/(1-\rho^2)$ and $Z=2 \pi \arcsin(\xi/\sqrt 2)$. The grid lines are lines of constant $x$ and constant $y$ where $\{x,y\}$ are the coordinates used near the horizon.
• Figure 5: Integration domain with two patches $I$ and $II$. Chebyshev-Gauss-Lobatto grids with $50\times 50$ points are placed using transfinite interpolation. Left panel: Patch $I$ (in the far region) uses $\{\rho,\xi\}$ coordinates and patch $II$ (near the horizon in the quarter circle in the lower left) is mapped from $\{x,y\}$ coordinates using \ref{['coordmap']}. Right panel: Patch II (near the horizon) uses $\{x,y\}$ coordinates.