Liouville theory and Matrix models: A Wheeler DeWitt perspective

TL;DR

This work investigates how Liouville theory with $c=1$ matter, treated as bulk 2D quantum gravity, holographically relates to a boundary theory via macroscopic loops and a non-perturbative Matrix Quantum Mechanics (MQM) description. The authors deploy a dual fermionic field theory to compute partition functions, densities of states, and two-boundary correlators, revealing a non-perturbative spectrum with a high-energy Wigner-like envelope and low-energy exponential growth, as well as oscillatory features signaling non-disorder-averaged physics. They show that Euclidean wormholes and connected geometries contribute non-factorising terms to the spectral form factor, suggesting that a single boundary dual cannot capture all bulk information, and argue for a third-quantised Hilbert space where bulk topology is dynamical. Analytic continuations to cosmological regimes yield a rich set of WdW wavefunctions with multiple semiclassical limits, including no-boundary and tunneling proposals, connected through Stokes phenomena. The results motivate considering “geometries inside geometries” and point toward higher-dimensional generalisations and tensor-model frameworks for a fuller holographic understanding of bulk quantum gravity in this setup.

Abstract

We analyse the connections between the Wheeler DeWitt approach for two dimensional quantum gravity and holography, focusing mainly in the case of Liouville theory coupled to $c=1$ matter. Our motivation is to understand whether some form of averaging is essential for the boundary theory, if we wish to describe the bulk quantum gravity path integral of this two dimensional example. The analysis hence, is in a spirit similar to the recent studies of Jackiw-Teitelboim (JT)-gravity. Macroscopic loop operators define the asymptotic region on which the holographic boundary dual resides. Matrix quantum mechanics (MQM) and the associated double scaled fermionic field theory on the contrary, is providing an explicit "unitary in superspace" description of the complete dynamics of such two dimensional universes with matter, including the effects of topology change. If we try to associate a Hilbert space to a single boundary dual, it seems that it cannot contain all the information present in the non-perturbative bulk quantum gravity path integral and MQM.

Figures (13)

• Figure 1: The perturbative expansion of the WdW wavefunction in powers of $g_{st} \sim 1/\mu$. The dashed lines indicate that we keep fixed only the overall size of the loop $\ell$.
• Figure 2: Left: The genus-zero WdW wavefunction as a function of the size $\ell$. Right: The non-perturbative wavefunction (computed numerically) exhibiting a slowly decaying envelope with oscillations.
• Figure 3: The density of states for $\mu=10$ as a function of the energy $E$. It exhibits an exponential growth that is then transitioning to a Dyson semicircle law with superimposed oscillations.
• Figure 4: Left: The genus zero connected eigenvalue correlation function for $\mu=1, E_2=0$. Right: The behaviour of the non-perturbative correlation function is quite similar at short spacings.
• Figure 5: The non-perturbative correlator for $\mu =1$ vs. the sine kernel. While they are qualitatively similar, there do exist differences between them.