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In this paper, inspired by [1-3], we proposed a class of latticerenormalization group (RG) operators, each operator determined by a topologicalorder $T$ in $D+1$ space-time dimensions. Taking the overlap between aneigenstate $\langle\Omega|$ of the RG operator with the ground statewave-function $|\Psi\rangle$ of $T$ (i.e. $\langle\Omega|\Psi\rangle$) givesrise to partition functions of conformal (including topological) theories in$D$ dimensions with categorical symmetry related to $T$, realizing aholographic relation discussed in the literature explicitly. We illustrate thisin explicit examples at $D=1,2,3$. Exact eigenstates of the RG operator can besolved explicitly from (higher) separable Frobenius algebra of the (higher)input fusion category $\mathcal{C}$ defining the lattice model of $T$, and theygive the $D$ dimensional symmetric TQFTs. Eigenstates corresponding to actualconformal theories describe phase transitions between these topological fixedpoints. The critical points can be searched numerically and we demonstrate thatknown critical couplings of $SU(2)_k$ integrable lattice models are numericallyrecovered from our procedure, alongside a curious tricritical point that wefound at $k=4$. We demonstrate that the 2+1 D Ising model can also be obtainedas a strange correlator with the associated 4D topological order being the 4Dtoric code. The numerical procedure that we devise to search for the 3Dcritical temperature is a novel tensor renormalization group algorithm, thatfully harnesses the algebraic and geometric properties of the RG operator.Finally since the RG operator is in fact an exact analytic holographic tensornetwork, we compute ``bulk-boundary'' correlator and compare with AdS/CFT.Promisingly, they are numerically compatible given our accuracy, althoughfurther works will be needed to explore the precise connection to the AdS/CFTcorrespondence.

Exact Holographic Tensor Networks -- Constructing CFT$_D$ from TQFT$_{D+1}$