Abstract
By studying the space of geodesics in $ADS_3/CFT_2$ and quantizing the geodesic motion, we relate scattering data to boundary entanglement of the CFT vacuum. The basic idea is to use a family of plane waves parametrized by coordinates of the space of geodesics i.e. kinematic space. This idea enables a simple calculation of the Berry curvature living on kinematic space. As a result we recover the Crofton form with a coefficient depending on the scattering energy. In arriving at these results the space of horocycles is used. We show that this new space used in concert with kinematic space incorporates naturally the gauge degrees of freedom responsible for an analogue of Berry's Phase. Horocycles also give a new geometric look to the strong subadditivity relation in terms of lambda lengths giving rise to shear coordinates of geodesic quadrangles. A generalization for geodesic polygons then reveals an interesting connection with $A_{n}$ cluster algebras. Here the cluster variables are the lambda lengths related to the regularized entropies of the boundary via the Ryu-Takayanagi relation. An elaboration of this idea indicates that cluster algebras might provide a natural algebraic means for encoding the gauge invariant entanglement patterns of certain boundary entangled states in the geometry of bulk geodesics. Finally using the language of integral geometry we show how certain propagators connected to the bulk, boundary and kinematic spaces are related to data of elementary scattering problems. We also present some hints how these ideas might be generalized for more general holographic scenarios.
Highlights
Due to the progress in the field of AdS=CFT it has become clear that entanglement patterns of certain quantum states associated to one type of space are encoded into geometric structures of another one [1,2,3,4,5,6]
Most recently a new type of space, the space of geodesics called kinematic space, has been invoked in the hope to act as an interpreter between the original spaces via the universal language of integral geometry [7,8]
VIII using the language of integral geometry we show how certain propagators of the literature connected to our spaces D, K, ∂D are related to elementary scattering problems
Summary
Due to the progress in the field of AdS=CFT it has become clear that entanglement patterns of certain quantum states associated to one type of space (boundary) are encoded into geometric structures of another one (bulk) [1,2,3,4,5,6]. Freedom are away from satisfying the strong subadditivity relation to the shear coordinates of geodesic quadrangles in the bulk This boundary measure, the conditional mutual information is related to the geodesic distance between the geodesics forming the opposite sides of the quadrangle, ones having timelike separation as points in kinematic space. Since boundary intervals with their associated geodesics of the bulk are organized according to the causal structure of their corresponding points in kinematic space [7], in this manner the underlying quivers of cluster algebras are connected to structures of causality. According to Gutzwiller [16] the horocycles can be regarded as some sort of monitoring stations where the test particles following the geodesics are registered after (and before) they are being scattered, i.e., probing the deeper regions of the bulk
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