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The algebraic structure of representation theory naturally arises from 2Dfixed-point tensor network states, which conceptually formulates the pattern oflong-range entanglement realized in such states. In 3D, the same underlyingstructure is also shared by Turaev-Viro state-sum topological quantum fieldtheory (TQFT). We show that a 2D fixed-point tensor network state arisesnaturally on the boundary of the 3D manifold on which the TQFT is defined, andthe fact that exactly the same information is needed to construct either thetensor network or the TQFT is made explicit in a form of holography.Furthermore, the entanglement of the fixed-point states leads to an emergenceof pre-geometry in the 3D TQFT bulk. We further extend these ideas to the casewhere an additional global onsite unitary symmetry is imposed on the tensornetwork states.

The structure of fixed-point tensor network states characterizes patterns of long-range entanglement