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

TL;DR

This paper develops a method to construct conformal field theories in various dimensions using topological quantum field theories and holographic tensor networks, providing exact and numerical tools to connect these theories with holography and AdS/CFT correspondence.

Contribution

It introduces a novel approach to derive CFT$_D$ from TQFT$_{D+1}$ via eigenstates of an RG operator, including exact constructions for D=2 and numerical methods for D=2,3.

Findings

Exact 2D rational CFT path-integral from TQFT$_3$

Numerical phase transition search for CFT$_D$ in D=2,3

Bulk-boundary correlator consistent with AdS/CFT at D=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 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.