CFT𝐷fromTQFT𝐷+1via Holographic Tensor Network, and Precision Discretization ofCFT2

Abstract

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.