Berry curvature, horocycles and scattering states in $AdS_3/CFT_2$

TL;DR

This paper explores the geometric and algebraic structures underlying $AdS_3/CFT_2$ holography, linking geodesic spaces, Berry curvature, and cluster algebras to boundary entanglement and scattering data.

Contribution

It introduces a novel geometric framework involving horocycles and kinematic space, connecting entanglement, scattering, and algebraic structures like cluster algebras in holography.

Findings

Berry curvature on kinematic space relates to scattering energy.

Horocycles provide a geometric interpretation of strong subadditivity.

Cluster variables encode boundary entanglement patterns via lambda lengths.

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.