$AdS_3/CFT_2$中的Berry曲率、horocycles和散射态
Berry curvature, horocycles and scattering states in $AdS_3/CFT_2$
摘要：
通过对$ADS_3/CFT_2$中测地线空间的研究和对测地线运动的量子化，我们将散射数据与CFT真空的边界纠缠联系起来。基本思想是使用由测地线空间即运动学空间的坐标参数化的平面波族。这个想法使一个简单的计算Berry曲率生活在运动空间。因此，我们恢复Crofton形式的系数取决于散射能量。在达到这些结果的horocycles空间使用。我们表明，这一新的空间与运动学空间的配合使用，自然采用了规范的自由度负责一个类似的贝里的阶段。Horocycles也给了一个新的几何外观的强次可加性关系的lambda长度产生剪切坐标的测地四边形。测地线多边形的推广，然后揭示了一个有趣的连接与$A_{n}$集群代数。这里的簇变量是通过Ryu-Takayanagi关系与边界的正则化熵相关的lambda长度。这个想法的阐述表明，集团代数可能提供一个自然的代数手段编码的规范不变的纠缠模式的某些边界纠缠态的几何体测地线。最后，使用积分几何的语言，我们展示了如何某些传播连接到体积，边界和运动空间的基本散射问题的数据。我们还提出了一些提示，这些想法可能会推广到更一般的全息场景。
英文摘要：
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.
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