The technique of bosonisation allows complex affine Lie (super) algebras [or current (super) algebras] to be expressed as a composite of more elemental operators. We review the history of free field realization and bosonisation from Wakimoto onwards. We fully bosonize the su(2)k current algebra, and present an explicit bosonic construction for both the positive and negative modes of the su(2)k/u(1) coset parafermion at an arbitrary level k, using Faa di Bruno polynomials of a vector of 2 bosons.
We present an algebraic proof of these general parafermions by inductively calculating the operator product expansion, and applying a kind of Faa di Bruno ``convolution" to the most singular terms generated. We apply the same techniques to more complicated, graded algebras. Using a particular ordering for the roots of basic Lie superalgebras, we present an explicit differential operator representation for the generators of osp(2r|2n) and sl(r|n). Based on this, we present an explicit free field realization of the corresponding current superalgebra osp(2r|2n)k. The energy momentum tensors of these quite general theories are derived. From this, the full bosonization may also be derived.
For the specific case of osp(2|2), we focus on the non-standard basis, with two fermions in its simple root system. We present the full bosonization of the first and last terms of the positive modes of the parafermionic coset osp(2|2)k/u2(1), and one term of the negative modes. By extending algebraic convolution of the Faa di Bruno polynomials to less singular terms, we show that we can calculate the full structure constant, and verify its status as a primary field with respect to the energy momentum tensor of the theory. The conformal dimension is also calculated.