Alpha states demystified: Towards microscopic models of AdS$_2$ holography

TL;DR

This work presents a concrete construction of alpha-states in JT gravity and shows that factorizing dilaton gravities with a universal bilocal term have discrete spectra, linked to local potentials. It demonstrates that distinct bulk descriptions can be nonperturbatively equivalent, with null states arising from redundancies and quotienting required to define the physical baby universe Hilbert space. The authors also explore a canonical JT theory that, despite a disk-only geometric expansion, admits a dual quantum-mechanical system with a discrete spectrum, offering a new holographic pairing in AdS2. Furthermore, the Coleman-Giddings-Strominger mechanism is clarified in this JT context, connecting ensemble averaging to wormhole dynamics and providing a precise framework for integrating in/out wormholes via matrix-model techniques.

Abstract

We continue our study of factorizing theories of dilaton gravity, characterized by a universal bilocal interaction. All such factorizing theories can be shown to have discrete spectra, distinguished only by their local dilaton potentials. We show how such theories can be used to construct all alpha-states in the Hilbert space of baby universes of ordinary JT gravity. Large classes of these theories with different local potentials are found to be non-perturbatively equivalent and have identical discrete spectra. This is a concrete example of how different bulk descriptions can give rise to the same boundary theory. Such equivalences manifest themselves as null states, which have to be quotiented out in order to construct a proper baby universe Hilbert space. Our results also allow us to revisit the mechanism discussed by Coleman, Giddings, and Strominger, and concretely link ensemble averaging to the appearance or disappearance of spacetime wormholes. We then investigate JT gravity deformed only by the universal bilocal interaction. In this theory, the only terms that do not cancel in a topological expansion are disks, which capture perturbative fluctuations around a two-dimensional black hole saddle. We find that this theory of black holes has an evenly spaced spectrum, instead of a quantum chaotic one. We present a dual quantum mechanical system with exactly the same discrete spectrum, and propose that this is an example of a new holographic duality between a two-dimensional theory of quantum gravity and a conventional quantum mechanics.

Figures (6)

• Figure 1: The spacing distribution $P(s)$ of adjecent energy levels obtained from a Gaussian Hermitian random matrix at $L = 10^4$. The histogram is a function of the difference between adjecent eigenvalues, denoted by $s$ and we focussed on $1272$ eigenvalues $E$ within the bin $[-1/10,1/10]$. The solid blue curve is the infinite $L$ result given by $\frac{32s^2}{\pi^2}e^{-4s^2/\pi}$.
• Figure 2: Left: Microcanonical spectral form factor for GUE as a function of time for a bin of size $1/10$ centered around zero energy. Right: Same microcanonical spectral form factor but with a spectrum obtained from the electrostatics problem \ref{['51']} with a Gaussian potential $V = 1/2 x^2$. We clearly see a huge difference between the plot on the right, not only are there the Poincare recurrences, but there is no ramp region as well. In both plots we used $L = 10^4$ and the bin contained $1272$ eigenvalues. We verified that the recurrence time is indeed $2\pi \rho(0) = 4\times 10^4$ even if appearing non-uniform in the figure above due to the log-log scale.
• Figure 3: Spacing distribution typical for clock behaviour. The right is a zoom-in of the left plot that emphasizes the small spread of the distribution. Here we used the electrostatics problem for a quartic potential $V = -1/2x^2 + x^4$, again we used the binsize $1/10$ centered around zero and took $L=10^4$.
• Figure 4: Left: First seven normalized eigenstates $\psi_i(x)$ and energies of $\widehat{Q}$ for $\textsf{S}_0 = 5$. We also plotted the potential $u(x)$ in black. The more red the coloring the higher the energy. The left vertical axis is for $E_i$ and $u(x)$, whereas the right one is for $\psi_i$. Right: Integrated spectral density $n(E) = \int^E_0 \rho(E')\mathrm{d} E'$ for the first 1500 eigenvalues of ${\textsf{H}_\text{JT}}$ (orange dots) and the JT gravity result (blue).
• Figure 5: Metric function $G(\Phi)$ as a function of the radial coordinate (equal to the dilaton $\Phi$) for the potential $U(\Phi,s_0)$ with $s_0 = 1$ (orange) alongside with the undeformed metric potential $\Phi^2 - \Phi_h^2$. We used $\Phi_h = 0.7$ and took the first $1500$ eigenvalues of ${\textsf{H}_\text{JT}}$ at $S_0 = 5$. The inset shows a clear difference between the two cases, but on 'average' (average over a small $\Phi$ window) the blue and orange curve lie on top of each other.