Two dimensional de-Sitter and deformed CFTs

We present an alternative dimensional reduction that yields an effective theory of dilatons in a two-dimensional de Sitter background. Specifically, by performing an S-wave reduction of higher-dimensional Einstein gravity, we obtain free massless dilatons in the Nariai static patch, and a dynamically evolving dilatons in the past Milne wedge. We then propose a (Nariai) static patch worldsheet formulation in terms of CFTs with SL(2,$\mathbb{R}$) deformed Hamiltonians on the cylinder. A key feature of this construction is that a stretched horizon in the (Nariai) static patch, equipped with an emergent UV boundary condition, acts as a gravitating observer. Using the similar reduction, we have also obtained a Schwarzian action coupled to free massless dilatons in the near horizon near extremal limit of four dimensional charged AdS black holes. The worldsheet description for the same has been proposed and discussed in \cite{Das:2025cuq}. We also comment on how different notions of worldsheet time may themselves be \textit{emergent}.