Phase space of Jackiw-Teitelboim gravity with positive cosmological constant

Abstract

In this paper we construct the classical phase space of Jackiw-Teitelboim gravity with positive cosmological constant on spatial slices with circle topology. This turns out to be somewhat more intricate than in the case of negative cosmological constant; this phase space has many singular points and is not even Hausdorff. Nonetheless, it admits a group-theoretic description which is quite amenable to quantization.

Figures (10)

• Figure 1: Global information in a $dS_2$ geometry: we glue a timelike geodesic to its image under a $dS_2$-isometry in the extended $dS_2$ solution, with gluings in different conjugacy classes being physically distinct solutions (we also need to identify each gluing isometry with its inverse). This gives one of the two phase space parameters of the theory, with the other coming from the magnitude of the dilaton.
• Figure 2: The infinite strip, with $\tau\in(-\pi/2,\pi/2)$ and $\sigma\in\mathbb{R}$, is the universal cover of the de Sitter hyperboloid, (\ref{['hyp']}).
• Figure 3: The region $\mathcal{R}$ in three-dimensional Minkowski space which is in one-to-one correspondence with elements of $SO^+(2,1)$. The blue points are rotations, the yellow points are boosts, and the green points are null rotations. $\mathcal{R}$ lies between the sheets of the hyperboloid $Q^iQ_i=-\pi^2$ (dark blue), with the upper sheet included but not the lower one, since topologically points on the upper sheet are identified with their opposites on the lower sheet. This identification illustrates the $\mathbb{S}^1\times \mathbb{R}^2$ topology of $SO^+(2,1)$.
• Figure 4: Integral curves of Killing vector fields (KVFs) corresponding to the three nontrivial types of conjugacy classes of $SO^+(2,1)$. (a) rotation, with global-coordinate components $(0,1)^\mu$; (b) null rotation, with components $(\cos\tau\sin\sigma,1+\sin\tau\cos\sigma)^\mu$; (c) boost, with components $(\cos\tau\sin\sigma,\sin\tau\cos\sigma)^\mu$. Motion along these vector fields corresponds to elements $(Q^i,n)$ of $\widetilde{G}$ where $Q^i$ is (a) a timelike vector proportional to $(1,0,0)$, (b) a null vector proportional to $(1,-1,0)$, and (c) a spacelike vector proportional to $(0,-1,0)$. Note that all KVFs are invariant under the central $2\pi$ translations in the $\sigma$ direction.
• Figure 5: Representative actions on the infinite strip by elements of the various conjugacy classes of $\widetilde{G}$. The colorful dashed line is the image of the black dashed line $(\sigma=0)$ under the isometry. The vectors of Lie-algebra charges that generate each transformation (aside from $2\pi n$ translations) are (a) $Q^i=(a,0,0)$, timelike; (b) $Q^i=(b,-b,0)$, null; and (c) $Q^i=(0,-c,0)$, spacelike.