  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.
