Scanning spacetime with patterns of entanglement

Abstract

In the AdS3/CFT2 setup, we elucidate how gauge-invariant boundary patterns of entanglement of the CFT vacuum are encoded into the bulk via the coefficient dynamics of an AN-3, N≥4, cluster algebra. In the static case, this dynamics of encoding manifests itself in kinematic space, which is a copy of de Sitter space dS2, in a particularly instructive manner. For a choice of partition of the boundary into N regions, the patterns of entanglement, associated with conditional mutual information of overlapping regions, are related to triangulations of geodesic N-gons. Such triangulations are then mapped to causal patterns in kinematic space. For a fixed N, the space of all causal patterns is related to the associahedron KN-3, an object well known from previous studies on scattering amplitudes. On this space of causal patterns, cluster dynamics acts by a recursion provided by Zamolodchikov's Y system of type (AN-3,A1). We observe that the space of causal patterns is equipped with a partial order and is isomorphic to the Tamari lattice. The mutation of causal patterns can be encapsulated by a walk of N-3 particles interacting in a peculiar manner in the past light cone of a point of dS2.