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In this work, we revisit the operator-state correspondence in the Majoranaconformal field theory (CFT) with emphasis on its semion representation.Whereas the semion representation gives a concise ``abelian" (or invertible)representation in the level of fusion rule and quantum states, there existssubtlety when considering the chiral multipoint correlation function. In thissense, the operator-state correspondence in the semion sector of the fermionictheory inevitably contains difficulty coming from its anomalous conformaldimension $1/16$ as a $Z_{2}$ symmetry operator. By analyzing the asymptoticbehaviors of the existing correlation functions, we propose a nontrivialcorrespondence between the chiral conformal blocks and bulk correlationfunctions containing both order and disorder fields. One can generalize thisunderstanding to $Z_{N}$ models (in which there exist long-standing openproblems). We expect this may improve our understanding of the simple currentextension of CFT which can appear commonly in the studies of topologicallyordered systems.

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