The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new boson-fermion correspondence, of type D-A. Further, we define a new concept of twisted vertex algebra of order $N$, which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing classes of examples of twisted vertex algebras, as well as for deriving formulas for the operator product expansions (OPEs), analytic continuations and normal ordered products. By using the underlying Hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for three important groups of examples. We show that the correspondences of type B, C and D-A are isomorphisms of twisted vertex algebras.Comment: Intermediate version---shorter than version 2, but longer than the version which is to appear in the Journal of Mathematical Physics. The name of the new correspondence related to the bosonization on the single neutral fermion Fock space is changed to "boson-fermion correspondence of type D-A". Part of the definition of a twisted vertex algebra revised. Minor changes to language, typos correcte
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