We show that the path integral of conformal field theories in 𝐷 dimensions (CFT𝐷) can be constructed by solving for eigenstates of a renormalization group (RG) operator following from the Turaev-Viro formulation of a topological field theory (topological quantum field theory) (TQFT) in 𝐷+1 dimensions (TQFT𝐷+1), explicitly realizing the holographic sandwich relation between a symmetric theory and a TQFT. Generically, exact eigenstates corresponding to symmetric TQFT𝐷 follow from Frobenius algebra in TQFT𝐷+1. For 𝐷=2, we construct eigenstates that produce 2D rational CFT path integrals exactly, which curiously connect 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 CFT𝐷 as phase transition points between symmetric TQFT𝐷. Finally, since the RG operator is in fact an exact analytic holographic tensor network, we compute “bulk-boundary” correlators and compare them 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.

中文翻译：

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CFTDfromTQFTD+1，以及 CFT2 的精确离散化


我们表明，D维共形场论（CFTD）的路径积分可以通过求解重整化群（RG）算子的特征态来构建，遵循拓扑场论（拓扑量子场论）（TQFTD+1）的Turaev-Viro公式（TQFTD+1），明确实现了对称理论和 TQFT 之间的全息三明治关系。通常，对应于对称 TQFTD 的精确特征态遵循 TQFTD+1 中的 Frobenius 代数。当 D=2 时，我们构建了精确产生 2D 有理 CFT 路径积分的特征态，它奇怪地将连续场理论路径积分与 Turaev-Viro 状态和联系起来。我们还设计并说明了 D=2、3 的数值方法，以搜索 CFTD 作为对称 TQFTD 之间的相变点。最后，由于 RG 算子实际上是一个精确的解析全息张量网络，因此我们计算了 “bulk-boundary” 相关器，并将其与 D=2 处的 AdS/CFT 字典进行比较。有希望的是，鉴于我们的准确性，它们在数值上是兼容的，尽管需要进一步的工作来探索与 AdS/CFT 对应关系的确切联系。