Holographic Dual of Crosscap Conformal Field Theory

We propose a holographic dual for 2D CFT defined on closed non-orientable manifolds, such as the real projective plane ${\mathbb{R}\mathbb{P}}^2$ and the Klein bottle ${\mathbb{K}}^2$. Such CFT can be constructed by introducing antipodally identified cuttings, i.e. crosscaps, to a sphere and hence called crosscap CFT (XCFT). The gravity dual is AdS${}_3$ spacetime with dS${}_2$ end-of-the-world branes. In particular, the Lorentzian spacetime with a global dS${}_2$ brane is dual to the unitary time evolution of a crosscap state in CFT, post-selected on the CFT ground state. We compute the holographic ${\mathbb{R}\mathbb{P}}^2$ partition function (or the $p$-function), one-point function, and ${\mathbb{K}}^2$ partition function, and see that they successfully reproduce the XCFT results. We also show a holographic $p$-theorem as an application.