Holography and Kinematic Space for Gravitational Sub-regions in AdS

It is well-known in integral geometry that a maximally symmetric Riemannian manifold, such as a static slice of vacuum AdS spacetime, can be perfectly covered by the geodesics in the Kinematic space, which we call the partial-entanglement-entropy (PEE) threads. In this context, the area of a codimension-one surface in the manifold can be computed by counting its intersections with the PEE threads, which is the celebrated Crofton formula. In this paper, we analyze the Kinematic space for a generic subregion in vacuum AdS space, and propose that the PEE threads emanate from a co-dimension one surface can perfectly cover a subregion in the manifold. Furthermore, we build holographic tensor network models on the network of the PEE threads confined in a subregion, thereby providing a concrete framework that realizes the surface-state correspondence and the generalized entanglement wedges for gravitational subregions.