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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.

Heisenberg VOAs over fields of prime characteristic and their representations