A microscopic Normal Matrix Model for $(A)dS_2$

TL;DR

This work establishes a nonperturbative bridge between a gravitating $c=1$ Sine-Gordon theory and a Normal Matrix Model, showing how AdS$_2$ and dS$_2$ saddles arise from a periodic potential and how a microscopic large-$N$ limit in the NMM yields all-genus results. By developing both macroscopic and microscopic large-$N$ analyses, it demonstrates that the dual NMM furnishes a finite-$N$ framework in which loop observables and the Wheeler-DeWitt wavefunction can be computed exactly (e.g., at $R=1$ via Laguerre polynomials). The study reveals a rich structure of wormhole geometries (including wineglass wormholes) and a nuanced cosmological sector where the density of states can become negative, hinting that a single-loop boundary dual may be inappropriate in cosmological regimes. Overall, the results illuminate how microscopic matrix-model descriptions can capture quantum-gravitational and stringy features of low-dimensional de-Sitter spacetimes, offering a concrete laboratory for nonperturbative quantum gravity and holography in cosmological settings.

Abstract

We describe the duality between the gravitating $c=1$ (compact) Sine-Gordon model and a normal matrix model. From a two-dimensional quantum gravity perspective and due to the periodic nature of the potential, this model admits both anti-de Sitter and de-Sitter saddles, similarly to simpler models of Sine-Dilaton gravity, as well as more complicated interpolating "wineglass wormhole" geometries. From a string theory perspective the Euclidean de-Sitter (genus zero) saddles are related to the presence of a classical entropic contribution associated to the target space geometry. The gravitating Sine-Gordon model corresponds to a well defined CFT by construction and the eigenvalues of the dual normal matrix model are supported in a compact region of the complex plane. The duality with the normal matrix model is operationally defined even for a finite, but sufficiently large matrix size $N$, depending on the precise observable to be determined. We define and study a "microscopic" version of the large-N limit that allows us to recover non-perturbative results for all physical observables.

Figures (10)

• Figure 1: Euclidean geometries (for $\xi_n = 1$), that oscillate and (generically) end in a singularity ($e^{\phi} \sim e^{- c \tau} \rightarrow 0$). The field $X$ initially oscillates, but never reaches any critical point of the potential (runaway). Forming a periodic de Sitter wormhole such as the ones found in Halliwell:1989puRey:1989thAguilar-Gutierrez:2023rilBlommaert:2025bgd, is an extremely fine-tuned unstable case.
• Figure 2: Top ($\xi_n = 1$): Euclidean "wineglass" $AdS$ wormhole Betzios:2024oliBetzios:2024iay. Bottom ($\xi_n = 1$): Bouncing $EAdS$ wormhole Ghodsi:2024jxe. The scalar field $X$ asymptotes to an attractor $EAdS$ critical point (Here $X_{crit.} = 2 \pi$). Both solutions are $Z_2$ symmetric under $\tau \leftrightarrow - \tau$, and satisfy $\phi'(0) = X'(0) = 0$.
• Figure 3: The Lorentzian continuation of the "wineglass" wormhole (top fig. \ref{['fig:bouncing']}). The Lorentzian Universe is expanding and the field $X(T)$, reaches a de-Sitter critical point (Here $X_{crit.} = \pi$).
• Figure 4: The Gaussian NMM eigenvalue distribution in the complex plane. It forms a disk droplet of area $\pi \mu$ (upon identifying the 't Hooft coupling $\lambda \equiv \mu$).
• Figure 5: An elliptically shaped droplet, for the deformed normal matrix model with $V(z)= N t_2 z^2/2 \, ,$$\overline V(\bar{z})= N \bar{t}_2 \bar{z}^2/2$. We also depict the size of the two axes of the ellipse. One should also perform the replacement $t_2 \rightarrow - i t/\mu$ to connect with the gravitational Sine-Gordon model.