2D Conformal Field Theories and Holography

TL;DR

This paper explores a holographic relationship between 2D conformal field theories and 3D topological quantum field theories, generalizing known chiral results to the full CFT and relating correlation functions to quantum states in Turaev-Viro theory.

Contribution

It establishes a novel CFT/TQFT correspondence for the full 2D CFT, extending the chiral case, and interprets correlation functions as quantum states in Turaev-Viro theory.

Findings

Derived a generalized Verlinde formula from Turaev-Viro states.

Established an operator/state correspondence linking chiral TQFT operators to Turaev-Viro states.

Provided a holographic interpretation of 2D CFT correlation functions.

Abstract

It is known that the chiral part of any 2d conformal field theory defines a 3d topological quantum field theory: quantum states of this TQFT are the CFT conformal blocks. The main aim of this paper is to show that a similar CFT/TQFT relation exists also for the full CFT. The 3d topological theory that arises is a certain ``square'' of the chiral TQFT. Such topological theories were studied by Turaev and Viro; they are related to 3d gravity. We establish an operator/state correspondence in which operators in the chiral TQFT correspond to states in the Turaev-Viro theory. We use this correspondence to interpret CFT correlation functions as particular quantum states of the Turaev-Viro theory. We compute the components of these states in the basis in the Turaev-Viro Hilbert space given by colored 3-valent graphs. The formula we obtain is a generalization of the Verlinde formula. The later is obtained from our expression for a zero colored graph. Our results give an interesting ``holographic'' perspective on conformal field theories in 2 dimensions.