  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.
