CFT_{D} from TQFT_{D+1} via Holographic Tensor Network, and Precision Discretization of CFT_{2}

Abstract

We show that the path integral of conformal field theories in D dimensions (CFT_{D}) 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 (TQFT_{D+1}), explicitly realizing the holographic sandwich relation between a symmetric theory and a TQFT. Generically, exact eigenstates corresponding to symmetric TQFT_{D} follow from Frobenius algebra in TQFT_{D+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 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” 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.