  The goal of this paper is to make the vertex operator algebra approach to
mirror symmetry accessible to algebraic geometers. Compared to better-known
approaches using moduli spaces of stable maps and special Lagrangian
fibrations, this approach follows more closely the original line of thinking
that lead to the discovery of mirror symmetry by physicists. The ultimate goal
of the vertex algebra approach is to give precise mathematical definitions of
N=(2,2) superconformal field theories called A and B models associated to any
Calabi-Yau variety and then show that thus constructed theories are related by
the mirror involution for all known examples of mirror symmetric varieties.
