A growing body of evidence suggests that quantum gravity is holographic, meaning that spacetime emerges from a lower-dimensional (non-gravitational) theory. The best-understood example is AdS/CFT, where the lower-dimensional degrees of freedom form a CFT which can be interpreted as residing at the spatial conformal boundary of the Anti-de Sitter space. What about de Sitter, however? Is there a holographic dual, and if so, do the lower-dimensional degrees of freedom live at future or past infinity (as in dS/CFT), on the cosmological horizon (as in static-patch holography), or in no particular place at all (as in ensemble average or matrix model proposals)? As a first step to address this question, we approach the quantization of Jackiw-Teitelboim (JT) gravity from a canonical perspective. In this talk we will construct the classical phase space of JT gravity with positive cosmological constant on spatial slices with circle topology. We identify solutions not previously discussed in the literature, and argue that the correct result for the symplectic form requires being careful about spatial boundary terms in the action (despite the lack of spatial boundary). We find the phase space has many singular points and is not even Hausdorff. Nonetheless, it admits a group-theoretic description which is quite amenable to quantization. We comment on next steps toward quantization.
[Based on 2409.12943 and upcoming work.]