Holographic Dual of Crosscap Conformal Field Theory

TL;DR

This paper develops a holographic duality for 2D conformal field theories on non-orientable surfaces like the real projective plane and Klein bottle, using AdS$_3$ spacetime with dS$_2$ branes, and verifies it through explicit calculations.

Contribution

It introduces a novel holographic dual for crosscap CFTs on non-orientable manifolds, connecting them to AdS$_3$ with dS$_2$ branes, and confirms the duality via partition function and correlation function computations.

Findings

Holographic $ p^2$ and Klein bottle partition functions match CFT results.

Lorentzian spacetime with dS$_2$ brane describes unitary evolution of crosscap states.

Established a holographic $p$-theorem for non-orientable CFTs.

Abstract

We propose a holographic dual for 2D CFT defined on closed non-orientable manifolds, such as the real projective plane $\mathbb{RP}^2$ 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 $\mathbb{RP}^2$ 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.