A microscopic realization of dS$_3$

TL;DR

This work presents a concrete dS3 holographic duality to a double-scaled matrix model, realized by quantizing the dS3 gravitational phase space to obtain a Liouville-based wavefunction for the universe and relating integrated cosmological correlators to matrix model resolvents. The observables, computed as norms of the Liouville wavefunction and summed over genus and moduli, reproduce the complex Liouville string amplitudes and reveal a nonperturbative matrix-model underpinning, including a precise microstate counting that matches the Gibbons-Hawking entropy of the dS3 static patch. A key methodological thread is the use of a first-order SL(2,C) CS-like description, careful treatment of invertibility and large diffeomorphisms, and a TQFT folding picture that naturally yields Liouville correlators as the cosmological wavefunction. The results provide a robust microscopic framework for de Sitter holography in three dimensions, with explicit nonperturbative structure and a clear link between bulk gravitational data and a dual matrix integral, while also highlighting subtle issues such as the role of topology, observer inclusion, and the precise spherical partition function normalization.

Abstract

We propose a precise duality between pure de Sitter quantum gravity in 2+1 dimensions and a double-scaled matrix integral. This duality unfolds in two distinct aspects. First, by carefully quantizing the gravitational phase space, we arrive at a novel proposal for the quantum state of the universe at future infinity. We compute cosmological correlators of massive particles in the universe specified by this wavefunction. Integrating these correlators over the metric at future infinity yields gauge-invariant observables, which are identified with the string amplitudes of the complex Liouville string arXiv:2409.17246. This establishes a direct connection between integrated cosmological correlators and the resolvents of the matrix integral dual to the complex Liouville string, thereby demonstrating one aspect of the dS$_3$/matrix integral duality. The second aspect concerns the cosmological horizon of the dS static patch and the Gibbons-Hawking entropy it is conjectured to encode. We show that this entropy can be reproduced exactly by counting the entries of the matrix.

Figures (6)

• Figure 1: Penrose diagram of de Sitter. The square denotes the global patch with future infinity $\mathcal{I}^+$. This is a two-dimensional manifold that we take to be hyperbolic, such that the global metric is that of a Milne type universe. An observer today can only see a piece of dS, called the static patch, and realized in blue in the Penrose diagram. They are surrounded by an event horizon marking the boundary of their visible universe.
• Figure 2: A plot of the leading density of eigenvalues of one of the matrices in the two-matrix integral dual of the complex Liouville string. The density exhibits the familiar square-root behavior near the edge of the spectrum, and oscillates on non-perturbative scales. We propose that the de Sitter microstates are enumerated by integrating the density of eigenvalues up to its first zero $E_0$.
• Figure 3: Two splittings of the three-sphere. On the left the sphere is split into two interlocked solid tori glued along their torus boundaries. On the right the sphere is split into equatorial three-balls glued along their two-sphere boundaries, with the interior of one three-ball identified with the exterior of the other.
• Figure 4: A cartoon of the inflating universe (\ref{['eq:inflating universe metric']}). There is a big bang singularity (shown in red) at $t=0$, and constant $t$ slices are given by hyperbolic surfaces $\Sigma_{g,n}$. The gravitational path integral computes the wavefunction of the universe on a late-time slice with the topology of $\Sigma_{g,n}$ to be given by the Liouville correlation function on $\Sigma_{g,n}$. The massive particles correspond to Wilson lines in the three-dimensional bulk, shown in red.
• Figure 5: A sketch of the folding trick. On the left we have the product theory $\text{VTQFT}\times\overline{\text{VTQFT}}$ on $\Sigma_{g,n}\times I$ with dynamical boundary conditions (gray) on one end, and topological or gapped boundary conditions (red) on the other. We unfold this to a single copy of VTQFT on $\Sigma_{g,n}\times I$, but now with dynamical boundary conditions on both ends of the interval. The VTQFT partition function on the latter is computed by the correlation function of Liouville CFT on $\Sigma_{g,n}$Collier:2023fwi.