Operator-state correspondence in simple current extended conformal field theories: Toward a general understanding of chiral conformal field theories and topological orders

TL;DR

The paper analyzes operator-state correspondence in the Majorana CFT with a $Z_{2}$ simple-current extension, revealing subtleties in chiral blocks arising from the anomalous dimension $h_\sigma=1/16$ and proposing a chiral-bulk (CCFT/DCFT) correspondence via Schottky doubling. It develops a fermionization-based framework to extract asymptotic behaviors of multipoint correlators, clarifies when chiral conformal blocks correspond to bulk correlators, and applies these ideas to Moore–Read states, including a bosonic wavefunction construction and a proposed second-quantized description of non-abelian quasiholes. The work then generalizes to $Z_N$ simple-current extensions, explaining how bulk disorder operators emerge and how invariant vs noninvariant sectors and KW duality shape the operator-state structure, with implications for topological order and tensor-network formulations. Overall, the CCFT/DCFT perspective provides a principled route to define and analyze wavefunctions and edge states for topologically ordered systems, beyond the traditional CFT/VOA framework, and highlights open problems in extending these ideas to broader simple-current extensions and higher dimensions.

Abstract

In this work, we revisit the operator-state correspondence in the Majorana conformal field theory (CFT) with emphasis on its semion representation. Whereas the semion representation (or $Z_{2}$ extension of the chiral Ising CFT) gives a concise ``abelian" (or invertible) representation in the level of fusion rule and quantum states, there exists subtlety when considering the chiral multipoint correlation function. In this sense, the operator-state correspondence in the semion sector of the fermionic theory inevitably contains difficulty coming from its anomalous conformal dimension $1/16$ as a $Z_{2}$ symmetry operator. By analyzing the asymptotic behaviors of the existing correlation functions, we propose a nontrivial correspondence between the chiral conformal blocks and bulk correlation functions containing both order and disorder fields. One can generalize this understanding to $Z_{N}$ models or fractional supersymmetric models (in which there exist long-standing open problems). We expect this may improve our understanding of the simple current extension of CFT which can appear commonly in the studies of topologically ordered systems.

Figures (1)

• Figure 1: Relations between $D$ dimensional bulk CFT, ACFT, and $D+1$ dimensional topological order (TO). For the $2+1$ dimensional topological order we have mainly discussed, the ACFT is called CCFT. It should be stressed that ACFT itself can be "unphysical" to some extent depending on its anomaly, but can be interpreted as a "physical" object in its correspondence to the TO. This gives a modern understanding of bulk-edge or CFT/TQFT correspondence, at the level of the correlation functions and the wavefunctions.