Cohomology ring of unitary $N=(2,2)$ full vertex algebra and mirror symmetry
Abstract
The mirror symmetry among Calabi-Yau manifolds is mysterious, however, the mirror operation in 2d N=(2,2) supersymmetric conformal field theory (SCFT) is an elementary operation. In this paper, we mathematically formulate SCFTs using unitary full vertex operator superalgebras (full VOAs) and develop a cohomology theory of unitary SCFTs (aka holomorphic / topological twists). In particular, we introduce cohomology rings, Hodge numbers, and the Witten index of a unitary $N=(2,2)$ full VOA, and prove that the cohomology rings determine 2d topological field theories and give relations between them (Hodge duality and T-duality). Based on this, we propose a possible approach to prove the existence of mirror Calabi-Yau manifolds for the Hodge numbers using SCFTs. For the proof, one need a construction of sigma models connecting Calabi-Yau manifolds and SCFTs which is still not rigorous, but expected properties are tested for the case of Abelian varieties and a special K3 surface based on some unitary $N=(2,2)$ full VOAs.