Holographic networks for ($1+1$)-dimensional de Sitter space-time

Holographic tensor networks associated to tilings of ($1+1$)-dimensional de Sitter spacetime are introduced. Basic features of these networks are discussed, compared, and contrasted with conjectured properties of quantum gravity in de Sitter spacetime. Notably, we highlight a correspondence between the quantum information capacity of the network and the cosmological constant.

Figure 1

${\text{dS}}_2$ embedded in Minkowski spacetime (gray lines are null geodesics, black circles are constant time slices).

Figure 2

${\text{dS}}_2$ in conformal coordinates where $\theta =0$ and $\theta =2\pi$ are identified (gray lines are null geodesics).

Figure 3

A causal diamond is the intersection of the causal future of point $p$ and the causal past of point $q$.

Figure 4

Recursive construction of a tessellation of ${\mathrm{dS}}_2$ with causal diamonds using Farey numbers.

Figure 5

Tessellation of ${\mathrm{dS}}_2$ (color indicates which generation the tile belongs to).

Figure 6

Function $\psi (x)$ maps points from the real axis to the unit interval.

Figure 7

Tessellations of ${\mathrm{dS}}_2$ on the hyperbolic sheet embedded in Minkowski spacetime [Dyadic tessellation viewed from the side and top and Farey tessellation viewed from the top (at the bottom)].

Figure 8

Holographic tensor network $M$ corresponding to the tessellation Fig. 5 of ${\mathrm{dS}}_2$ (with $\ell =1$). This network should be understood as an infinite network extending between the temporal boundaries.

Figure 9

Tessellation and holographic network for ${\mathrm{dS}}_2$ with $\ell =2$. The blue lines (edges of $D^1$) divide the initial causal diamonds.

Figure 10

Tessellation and holographic network for ${\mathrm{dS}}_2$ with $\ell =4$. The blue (edges of $D^1$) and light blue (edges of $D^2$) lines divide the initial causal diamonds.

Figure 11

Symmetry action $q$ on the past boundary ${\mathcal{I}}^{-}$ of ${\mathrm{dS}}_2$.

Figure 12

Symmetry transformation on the dyadic rational numbers induced by a Möbius function $f$.

Figure 13

Different symmetry transformations $(?\circ f\circ {?}^{-1})(x)$ with Möbius functions $f$ which act on the dyadic rational numbers of the real axis [transformations $f(x)=-\frac{1}{x}$ (top) and $f(x)=\frac{x-1}{x}$ (bottom)].

Figure 14

Symmetry transformations $(?\circ f\circ {?}^{-1})(x)$ for different Möbius functions $f$ mapped to the unit interval using $\psi (x)$ [transformations $f(x)=-\frac{1}{x}$ (top) and $f(x)=\frac{x-1}{x}$ (bottom)].

Figure 15

Tessellations after the action of transformations $S(x)$ (top) and $C(x)$ (bottom) from the subgroup $\mathrm{PSL}(2,\mathbb{Z})$.