CFT$_D$ from TQFT$_{D+1}$ via Holographic Tensor Network, and Precision Discretisation of CFT$_2$

TL;DR

The paper constructs an explicit RG operator from TQFT$_{D+1}$ data (via Turaev–Viro/DW frameworks) that acts as an exact holographic network, enabling the path-integral of conformal field theories in $D$ dimensions to be realized as overlaps $Z(\Omega,\mathcal{C})=\langle\Omega|\Psi\rangle$ with a topological ground state $|\Psi\rangle$. For $D=2$, it provides analytic, exact RCFT partition functions as strange correlators using Frobenius-algebra data in the input fusion category, and demonstrates numerical RG-based searches for CFTs as phase transitions between fixed points. The framework generalizes to $D=3$ and $D=4$, introducing higher Frobenius algebra structures that yield topological fixed points and 2+1D/3+1D CFTs via boundary RG, with concrete Ising-like examples and explicit tensor-network constructions. The work finds preliminary numerical evidence that bulk-boundary propagators in these holographic networks align with AdS/CFT expectations, suggesting a concrete discretized realization of holography that connects lattice TQFT data to continuous CFTs. Collectively, the approach offers a systematic, symmetry-preserving route to derive CFT path-integrals from higher-dimensional TQFTs and to explore AdS/CFT-like relations in a controlled, discretized setting.

Abstract

We show that the path-integral of conformal field theories in $D$ dimensions (CFT$_D$) can be constructed by solving for eigenstates of an RG operator following from the Turaev-Viro formulation of a topological field theory in $D+1$ dimensions (TQFT$_{D+1}$), explicitly realising the holographic sandwich relation between a symmetric theory and a TQFT. Generically, exact eigenstates corresponding to symmetric-TQFT$_D$ follow from Frobenius algebra in the TQFT$_{D+1}$. For $D=2$, we constructed eigenstates that produce 2D rational CFT path-integral exactly, which, curiously connects a continuous field theoretic path-integral with the Turaev-Viro state sum. We also devise and illustrate numerical methods for $D=2,3$ to search for CFT$_D$ as phase transition points between symmetric TQFT$_D$. Finally since the RG operator is in fact an exact analytic holographic tensor network, we compute ``bulk-boundary'' correlator and compare with the AdS/CFT dictionary at $D=2$. Promisingly, they are numerically compatible given our accuracy, although further works will be needed to explore the precise connection to the AdS/CFT correspondence.

Figures (46)

• Figure 1: A triangle with this type of orientation is assigned $\alpha_2(g_1,g_2)$ with $g_3=g_1\times g_2$.
• Figure 2: A simple triangulation of a disk and a tensor network representation of the state $\langle \Omega|$.
• Figure 3: The triangulation of the disk can be converted to a tree-like tensor network by repeated use of the associativity condition (\ref{['eq:alex1']}) which is also illustrated pictorially in the lower half of the figure.
• Figure 4: The associativity relation defines the F symbols $\left[F^{abc}_{d}\right]_{ef}$. And the corresponding 6j symbol is defined as $\left[abecdf\right]=\left[F^{abc}_{d}\right]_{ef}/\sqrt{d_ed_f}$. The details can be found in section 2 of Aasen:2020jwb. This type of change of a diagram is typically referred to as an F-move.
• Figure 5: Tensor network representation of the ground state wave-function. The blue lines are "physical sites" of the topological order, and each colored by $i \in \mathcal{C}$. The red dots represent edges orthogonal to the $S_2$ surface which are weighted by a factor given in (\ref{['eq:dweight']}), and they are summed. The black lines are the dual graph of the blue lines.