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We introduce a unifying framework for the construction of holographic tensornetworks, based on the theory of hyperbolic buildings. The underlying dualitiesrelate a bulk space to a boundary which can be homeomorphic to a sphere, butalso to more general spaces like a Menger sponge type fractal. In this generalsetting, we give a precise construction of a large family of bulk regions thatsatisfy complementary recovery. For these regions, our networks obey aRyu--Takayanagi formula. The areas of Ryu--Takayanagi surfaces are controlledby the Hausdorff dimension of the boundary, and consistently generalize thebehavior of holographic entanglement entropy in integer dimensions to thenon-integer case. Our construction recovers HaPPY--like codes in alldimensions, and generalizes the geometry of Bruhat--Tits trees. It alsoprovides examples of infinite-dimensional nets of holographic conditionalexpectations, and opens a path towards the study of conformal field theory andholography on fractal spaces.

Holographic tensor networks from hyperbolic buildings