Scanning space-time with patterns of entanglement

In the ${\mathrm{A}\mathrm{d}\mathrm{S}}_3\mathrm{/}{\mathrm{C}\mathrm{F}\mathrm{T}}_2$ 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 $A_{N-3}$, $N\ge 4$ cluster algebra. In the static case this dynamics of encoding manifests itself in kinematic space, which is a copy of de Sitter space ${\mathrm{d}\mathrm{S}}_2$, in a particularly instructive manner. For a choice of partition of the boundary into $N$ regions the patterns of entanglement, associated with conditional mutual informations 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 ${\mathcal{K}}^{N-3}$ an object well-known from previous studies on scattering amplitudes. On this space of causal patterns cluster dynamics acts by a recursion provided by a Zamolodchikov’s $Y$-system of type $(A_{N-3},A_1)$. 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 ${\mathrm{d}\mathrm{S}}_2$.