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