Quantum fields on time-periodic AdS3/ℤ

A bstract We consider a free complex massive scalar on the quotient spacetime AdS 3 / 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 SO(2, 2) 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 / 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 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.