Cohomology ring of unitary $N=(2,2)$ full vertex algebra and mirror symmetry

We formulate two-dimensional $N=(2,2)$ supersymmetric conformal field theories in terms of unitary full vertex operator superalgebras and develop their cohomology theory. Cohomology rings, Hodge numbers, and the Witten index of a unitary $N=(2,2)$ full VOA are introduced. Using generalized full vertex operator superalgebras, spectral flow is constructed algebraically. Its periodicities are proved to be equivalent to the existence of top-degree cohomology classes, namely volume forms and holomorphic volume forms, and these characterizations yield Poincaré duality, T-duality, and Frobenius algebra structures on the cohomology rings, and thus two-dimensional topological field theories. A mirror construction for full VOAs and its relation to Hodge-theoretic mirror symmetry are also discussed. Finally, examples arising from abelian varieties, a special K3 surface, and a Landau-Ginzburg model are examined.