A de Sitter Farey tail

We consider quantum Einstein gravity in three dimensional de Sitter space. The Euclidean path integral is formulated as a sum over geometries, including both perturbative loop corrections and nonperturbative instanton corrections coming from geometries with nontrivial topology. These nontrivial geometries have a natural physical interpretation. Conventional wisdom states that the sphere is the unique Euclidean continuation of de Sitter space. However, when considering physics only in the causal patch of a single observer other Euclidean geometries, in this case lens spaces, contribute to physical observables. This induces quantum gravitational effects which lead to deviations from the standard thermal behavior obtained by analytic continuation from the three sphere. The sum over these geometries can be formulated as a sum over cosets of the modular group; this is the de Sitter analog of the celebrated “black hole Farey Tail.” We compute the vacuum partition function including the sum over these geometries. Perturbative quantum corrections are computed to all orders in perturbation theory using the relationship between Einstein gravity and Chern-Simons theory, which is checked explicitly at tree and one-loop level using heat kernel techniques. The vacuum partition function, including all instanton and perturbative corrections, is shown to diverge in a way which cannot be regulated using standard field theory techniques.

Figure 1

The causal diagram of de Sitter space, suppressing the angular $\varphi$ coordinate. Horizontal slices of this Penrose diagram are spheres $S^2$, with North and South Poles given by vertical lines on the left and right, respectively. The static patch associated with an observer at the South Pole is the wedge shaped region on the right. In our coordinates the observer is at $r=0$ and the cosmological horizon is at $r=\pi /2$. On this horizon the timelike Killing vector (denoted by an arrow) becomes null.

Figure 2

Lens spaces are viewed as pairs of solid tori glued together along their $T^2$ boundaries using an element of the torus mapping class group $SL(2,\mathbb{Z})$. Top: the gluing which takes the contractible cycle of one torus into the dual cycle of the other torus gives the sphere $S^3$. Bottom: a more complicated map gives a nontrivial lens space.