Les Houches Lectures on De Sitter Space

TL;DR

This work surveys the foundations of de Sitter quantum gravity by first revisiting the classical geometry of de Sitter space, then examining quantum field theory on a fixed dS background—including invariant vacua and the Gibbons-Hawking temperature—and finally outlining a dS$_3$/CFT perspective. It derives the de Sitter horizon entropy from Schwarzschild–de Sitter thermodynamics and clarifies the boundary data via the Brown-York stress tensor, reinforcing the area-entropy relation. A central result is that the asymptotic symmetry group for gravity in dS$_3$ is the Euclidean 2D conformal group, leading to a holographic-like description in terms of a boundary CFT on ${\cal I}^{\pm}$. Together, these pieces establish a concrete framework for understanding quantum gravity in de Sitter space and motivate the dS$_3$/CFT correspondence.

Abstract

These lectures present an elementary discussion of some background material relevant to the problem of de Sitter quantum gravity. The first two lectures discuss the classical geometry of de Sitter space and properties of quantum field theory on de Sitter space, especially the temperature and entropy of de Sitter space. The final lecture contains a pedagogical discussion of the appearance of the conformal group as an asymptotic symmetry group, which is central to the dS/CFT correspondence. A (previously lacking) derivation of asymptotically de Sitter boundary conditions is also given.

Figures (8)

• Figure 1: Hyperboloid illustrating de Sitter space. The dotted line represents an extremal volume $S^{d-1}$.
• Figure 2: Penrose diagram for dS${}_d$. The north and south poles are timelike lines, while every point in the interior represents an $S^{d-2}$. A horizontal slice is an $S^{d-1}$. The dashed lines are the past and future horizons of an observer at the south pole. The conformal time coordinate $T$ runs from $-\pi/2$ at ${\cal{I}}^-$ to $+\pi/2$ at ${\cal{I}}^+$.
• Figure 3: These diagrams show the regions $\cal{O}^-$ and $\cal{O}^+$ corresponding respectively to the causal past and future of an observer at the south pole.
• Figure 4: The dashed lines are slices of constant $t$ in planar coordinates. Note that each slice is an infinite flat $d{-}1$-dimensional plane which extends all the way down to ${\cal{I}}^-$.
• Figure 5: This Penrose diagram shows the direction of the flow generated by the Killing vector $\partial/\partial t$ in static coordinates. The horizons (dotted lines) are at $r^2 = 1$, and the southern causal diamond is the region with $0 \le r \le 1$ on the right hand side. Past and future null infinity ${\cal{I}}^\pm$ are at $r = \infty$.