Quantum Fields on Time-Periodic AdS$_3/\mathbb{Z}$

We consider a free complex massive scalar on the quotient spacetime AdS$_3/\mathbb{Z}$, which has the isometry group SO(2,2) rather than its universal cover. This problem is of interest as a special example of QFT on a spacetime with closed timelike curves (CTCs), as a new context in which to study generalizations of AdS/CFT and for its role in celestial holography. A basis of time-periodic solutions to the Klein-Gordon wave equation is found in terms of hypergeometric functions. They fall into a PT even and a PT odd principal series representation, rather than the more familiar highest-weight representations of the cover of SO(2,2). For masses below the Breitenlohner-Freedman (BF) bound, the modes fall on the unitary principal series. The presence of CTCs precludes the usual canonical quantization, but geometric quantization, which begins with a symplectic form on the phase space of classical solutions, is applicable. Operators, commutators, an \sot invariant vacuum and a Fock space are constructed and transform like those of a CFT$_2$. The Fock space norm is positive below the BF bound. In celestial holography, AdS$_3/\mathbb{Z}$ arises as leaves of a hyperbolic foliation of Klein space. Our analysis determines new entries in the symmetry-constrained celestial bulk-to-boundary dictionary. In particular the Klein space $\mathcal{S}$-matrix is dual to a maximally entangled state in the tensor product of two copies of the 'wedge CFT$_2$' associated to the timelike and spacelike wedges of Klein space. Translation invariance is not present in the wedge CFT$_2$ itself but emerges as a property of this maximally entangled state.