Abstract
We establish a dictionary between group field theory (thus, spin networks and random tensors) states and generalized random tensor networks. Then, we use this dictionary to compute the Rényi entropy of such states and recover the Ryu–Takayanagi formula, in two different cases corresponding to two different truncations/approximations, suggested by the established correspondence.
Highlights
Group Field Theory (GFT) [6,7,8,9], a closely related formalism sharing the same type of fundamental degrees of freedom, identify this microstructure with spin networks, which are graphs labeled by group-theoretic data
In GFT models of quantum gravity spin network states arise as many-body states in a 2nd quantised context, whose kinematics and dynamics are governed by a quantum field theory over a group manifold with quanta corresponding to tensor maps associated to nodes of the spin network graphs
A number of results in AdS/CFT suggest that a static AdS space-time, which we expect to be one such state, at the quantum level, satisfies the Ryu-Takayanaki (RT) formula [20] for the entanglement entropy, which is very efficiently computed via random tensor network techniques[23]
Summary
A d-dimensional GFT is a combinatorially non-local field theory living on (d copies of) a group manifold [6,7,8,9]. Where appropriate group theoretic data are used and specific properties are imposed on the states and quantum amplitudes, the same lattice structures can be understood in terms of simplicial geometries. The associated many-body description of such lattice states can be given in terms of a tensor network decomposition. The corresponding (generalized) tensor networks are provided with a field theoretic formulation and a quantum dynamics (and, in specific models, with additional symmetries). After a brief introduction to the GFT formalism, we detail this correspondence between GFT states and (generalized) tensor networks
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