Low-dimensional de Sitter quantum gravity

TL;DR

This work develops a detailed boundary-centric formulation of quantum gravity with positive cosmological constant in 1+1 and 2+1 dimensions. By recasting nearly dS2 gravity as a PSL(2,R) BF theory and dS3 gravity as a PSL(2,C) Chern-Simons theory, the authors derive Schwarzian- and Virasoro-coadjoint boundary actions that capture all boundary gravitons and topological degrees of freedom, and they connect these to Euclidean AdS counterparts via a cosmological-constant flip. They define a genus expansion for de Sitter JT gravity, relate it to volumes of moduli spaces and to a double-scaled matrix model, and provide a non-perturbative completion framework, while also extending the analysis to dS3 with boundary measures and monodromies. The results illuminate how de Sitter quantum gravity can be approached non-perturbatively through boundary dynamics and analytic continuation from AdS, offering insights into wavefunctions, scattering, and possible holographic duals in low dimensions.

Abstract

We study aspects of Jackiw-Teitelboim (JT) quantum gravity in two-dimensional nearly de Sitter (dS) spacetime, as well as pure de Sitter quantum gravity in three dimensions. These are each theories of boundary modes, which include a reparameterization field on each connected component of the boundary as well as topological degrees of freedom. In two dimensions, the boundary theory is closely related to the Schwarzian path integral, and in three dimensions to the quantization of coadjoint orbits of the Virasoro group. Using these boundary theories we compute loop corrections to the wavefunction of the universe, and investigate gravitational contributions to scattering. Along the way, we show that JT gravity in dS$_2$ is an analytic continuation of JT gravity in Euclidean AdS$_2$, and that pure gravity in dS$_3$ is a continuation of pure gravity in Euclidean AdS$_3$. We define a genus expansion for de Sitter JT gravity by summing over higher genus generalizations of surfaces used in the Hartle-Hawking construction. Assuming a conjecture regarding the volumes of moduli spaces of such surfaces, we find that the de Sitter genus expansion is the continuation of the recently discovered AdS genus expansion. Then both may be understood as coming from the genus expansion of the same double-scaled matrix model, which would provide a non-perturbative completion of de Sitter JT gravity.

Figures (13)

• Figure 1: (a) The Penrose diagram for global dS$_2$ with $\alpha = 1$. In our analysis we consider fluctuating Cauchy surfaces (depicted as blue curves) near past and future infinity. The left and right boundaries are identified. The geometries with $\alpha <1$ are "squashed" horizontally, while those with $\alpha>1$ are "stretched." (b) A cartoon of the geometry used in the Hartle-Hawking construction. The Lorentzian and Euclidean segments are glued together across the circle. The fluctuating boundary is indicated in blue.
• Figure 2: Under the analytic continuation $b\to i \alpha$, $J'_R \to- i J$, $J'_L \to iJ$, the double trumpet of Saad:2019lba is mapped to nearly global dS$_2$.
• Figure 3: A topological annulus, with a Wilson loop depicted by a dotted curve. The monodromy is parameterized by $\lambda$.
• Figure 4: (a) A Witten diagram, with a spatial slice of dS$_2$ drawn as a disc. The solid lines are scalars, and there is a single scalar exchange. This process is encoded in the boundary Schwarzian description as an operator, the reparameterized four-point function, endowing it with a coupling to the reparameterization field. (b) The expectation value of the matter four-point function in the Schwarzian theory dresses the process with an infinite number of corrections involving exchanges of the reparameterization modes, indicated with dotted lines.
• Figure 5: The two one-loop diagrams that correct the two-point function at $O(1/C) = O(G_2)$.