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. Published by the American Physical Society 2024

中文翻译：

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通过全息张量网络从 TQFTD+1 获得 CFTD，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D+1维拓扑场论（拓扑量子场论）（TQFTD+1）的Turaev-Viro公式，明确实现对称理论和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。美国物理学会 2024 年出版