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

Abstract

By studying the space of geodesics in ADS3/CF T2 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 An 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.

Cluster algebraic description of entanglement patterns for the BTZ black hole

“…For R = 1 this quantity corresponds to the so called lambda length introduced by Penner [15][16][17]. Using the notion of the lambda length in [15] it was shown that it is rewarding to regularize the length of a geodesic by introducing horocycles.…”

“…For R = 1 this quantity corresponds to the so called lambda length introduced by Penner [15][16][17]. Using the notion of the lambda length in [15] it was shown that it is rewarding to regularize the length of a geodesic by introducing horocycles. A horocycle associated to a boundary point is a circle in the bulk touching the boundary merely at this boundary point.…”

“…Then the regularized length of a geodesic is that finite length part of it, which is lying entirely outside (or inside) both of the circles associated to the endpoints of the geodesic. It is known that choosing a horocycle associated to a boundary point can be interpreted as a choice of gauge [14,15]. Now the regularization of ( 16) usually showing up in the literature is a uniform one corresponding to a special uniform choice for the horocycles with their infinitesimally small radii being equal and related to the large value of ρ 0 .…”

“…In arriving at this result another space, the space of horocycles related to the gauge degree of freedom [14] for choosing a cutoff for regularizing the divergent entanglement entropies has also appeared [15]. Horocycles are geometric objects regularizing the diverging length of geodesics.…”

“…Employing them resulted in the observation that the lambda lengths of Penner [16,17], encapsulating this regularization in a geometric manner, are direcly related to entanglement entropies via the Ryu-Takayanagi formula [1]. An extra bonus of this observation was the realization [15] that lambda lengths also provide a link to cluster algebras [18][19][20], algebraic structures that are under intense scrutiny in the mathematics literature.…”