The mirror symmetry of Calabi-Yau manifolds is a mysterious operation in which the complex and Kahler structures are interchanged, but at the level of supersymmetric 2d conformal field theory, which is the origin of its discovery, the mirror is an obvious operation corresponding to an outer automorphism of a Lie algebra.
We formulate supersymmetric 2d conformal field theory using full vertex operator algebras and mathematically construct “topological twist”, a functor from supersymmetric 2d conformal field theories to 2d topological field theories. In this talk, based on arXiv:2504.09919, we propose a possible approach to prove the existence of mirror Calabi-Yau manifolds for the Hodge numbers using vertex operator algebraic (or string-theoretic) method.