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

In this paper, 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 at the level of fusion rules and quantum states, there exists a 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 a 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 (for which there exist long-standing open problems). We expect that this may improve our understanding of the simple current extension of CFT, which commonly appears in the studies of topologically ordered systems.