Quantum Gravity Microstates from Fredholm Determinants

TL;DR

The statistics of the first several energy levels of a nonperturbative definition of JT gravity are constructed explicitly using numerical methods, and the full quenched free energy F_{Q}(T) of the system is computed for the first time.

Abstract

A large class of two dimensional quantum gravity theories of Jackiw-Teitelboim form have a description in terms of random matrix models. Such models, treated fully non-perturbatively, can give an explicit and tractable description of the underlying ``microstate'' degrees of freedom. They play a prominent role in regimes where the smooth geometrical picture of the physics is inadequate. This is shown using a natural tool for extracting the detailed microstate physics, a Fredholm determinant ${\rm det}(\mathbf{1}{-}\mathbf{ K})$. Its associated kernel $K(E,E^\prime)$ can be defined explicitly for a wide variety of JT gravity theories. To illustrate the methods, the statistics of the first several energy levels of a non-perturbative definition of JT gravity are constructed explicitly using numerical methods, and the full quenched free energy $F_Q(T)$ of the system is computed for the first time. These results are also of relevance to quantum properties of black holes in higher dimensions.

Figures (2)

• Figure 1: Spectral density $\rho(E)$ (solid black), $\rho_{_0}\!(E)$ (blue dashed), and probability densities (also cumulative probabilities, dashed) of the first 6 states of the JT gravity microstate spectrum. Inset: Close-up of $\rho(E)$ and distributions for the ground state, with $\langle E_0\rangle{\simeq}0.66$. Note that $\hbar{=}{\rm e}^{-S_0}{=}1$ here.
• Figure 2: The quenched and annealed free energies of JT gravity, computed using direct sampling (see text). As $T{\to}0$, $F_Q(T)$ lands at $\langle E_0\rangle{\simeq}0.66$. Here, $\hbar{=}{\rm e}^{-S_0}{=}1$.