  In this paper, we study Heisenberg vertex algebras over fields of prime
characteristic. The new feature is that the Heisenberg vertex algebras are no
longer simple unlike in the case of characteristic zero. We then study a family
of simple quotient vertex algebras and we show that for each such simple
quotient vertex algebra, irreducible modules are unique up to isomorphism and
every module is completely reducible. To achieve our goal, we also establish a
complete reducibility theorem for a certain category of modules over Heisenberg
algebras.
