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A review is presented of the recently obtained expressions for conformal blocks for admissible representations in SL(2) current algebra based on the Wakimoto free field construction. In this realization one needs to introduce a second screening charge, one which depends on fractional powers of free fields. The techniques necessary to deal with these complications are developed, and explicit general integral representations for conformal blocks on the sphere are provided. The fusion rules are discussed and as a check it is verified that the conformal blocks satisfy the Knizhnik-Zamolodchikov equations.

Based on our earlier work on free field realizations of conformal blocks for conformal field theories with SL(2) current algebra and with fractional level and spins, we discuss in some detail the fusion rules which arise. By a careful analysis of the 4-point functions, we find that both the fusion rules previously found in the literature are realized in our formulation. Since this is somewhat contrary to our expectations in our first work based on 3-point functions, we reanalyse the 3-point functions and come to the same conclusion. We compare our results on 4-point conformal blocks in particular with a different realization of these found by O. Andreev, and we argue for the equivalence. We describe in detail how integration contours have to be chosen to obtain convenient bases for conformal blocks, both in his and in our own formulation. We then carry out the rather lengthy calculation to obtain the crossing matrix between s-and t-channel blocks, and we use that to determine the monodromy invariant 4-point greens functions. We use the monodromy coefficients to obtain the operator algebra coefficients for theories based on admissible representations.

In this paper we consider Wakimoto free field realizations of simple affine Lie algebras, a subject already much studied. We present three new sets of results. (i) Based on quantizing differential operator realizations of the corresponding Lie algebras we provide general universal very simple expressions for all currents, more compact than has been established so far. (ii) We supplement the treatment of screening currents of the first kind, known in the literature, by providing a direct proof of the properties for screening currents of the second kind. Finally (iii) we work out explicit free field realizations of primary fields with general non-integer weights. We use a formalism where the (generally infinite) multiplet is replaced by a generating function primary operator. These results taken together allow setting up integral representations for correlators of primary fields corresponding to non-integrable degenerate (in particular admissible) representations.