## Details

I propose a bottom-up correspondence between a CFT defined on 2D non-orientable manifolds, such as the real projective plane (RP2) and the Klein bottle (K2), and AdS3 Einstein gravity with dS2 end-of-the-world branes. In this correspondence, a global dS2 end-of-the-world brane (a quotient by Z2) is described by the unitary time evolution of a crosscap state in the CFT, post-selected on the CFT ground state.

From the boundary perspective, certain 2D non-orientable manifolds, such as RP2, do not admit a bulk, making it nontrivial to understand how the bulk is filled by an AdS3 dual. From the gravity perspective, while AdS3 with a dS2 end-of-the-world brane is expected to shed light on dS2 quantum gravity, its CFT dual has not yet been identified. In contrast, AdS3 with an AdS2 end-of-the-world brane has long been known to be dual to a 2D boundary CFT, a duality that has played an important role in understanding quantum aspects of AdS2 gravity, including the information paradox.

To establish the correspondence, we compute the holographic RP2 partition function (or p-function), the one-point function, and the Klein bottle partition function, demonstrating that they successfully reproduce the CFT results. If time permits, I will also prove that the holographic p-function is monotonic under a holographic RG flow, whose CFT dual remains to be determined.