Journal of High Energy Physics (Oct 2022)

Holographic tensor networks from hyperbolic buildings

  • Elliott Gesteau,
  • Matilde Marcolli,
  • Sarthak Parikh

DOI
https://doi.org/10.1007/JHEP10(2022)169
Journal volume & issue
Vol. 2022, no. 10
pp. 1 – 29

Abstract

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Abstract We introduce a unifying framework for the construction of holographic tensor networks, based on the theory of hyperbolic buildings. The underlying dualities relate a bulk space to a boundary which can be homeomorphic to a sphere, but also to more general spaces like a Menger sponge type fractal. In this general setting, we give a precise construction of a large family of bulk regions that satisfy complementary recovery. For these regions, our networks obey a Ryu-Takayanagi formula. The areas of Ryu-Takayanagi surfaces are controlled by the Hausdorff dimension of the boundary, and consistently generalize the behavior of holographic entanglement entropy in integer dimensions to the non-integer case. Our construction recovers HaPPY-like codes in all dimensions, and generalizes the geometry of Bruhat-Tits trees. It also provides examples of infinite-dimensional nets of holographic conditional expectations, and opens a path towards the study of conformal field theory and holography on fractal spaces.

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