Conformal blocks for admissible representations in SL(2) current algebra

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Despite considerable work in the literature on N-point correlators in 2-d conformal WZNW models based on affine ▪ either by using the free field Wakimoto construction or by directly solving the Knizhnik-Zamolodchikov equations, most published results pertain to integrable representations with t = k + 2 integer and all primary fields having integer or half integer spin. Results for admissible representations corresponding to t = k + 2 = p/ q rational, especially as regards their relation to free field techniques, appear less complete, despite their potential interest in various connections, notably in connections to non-critical string theory via 2-d gravity based on hamiltonian reductions. Indeed, surprisingly, even the fusion rules remain a subject of discussion. The reason for this state of affairs may be traced to the need in the free field Wakimoto construction for introducing a second screening charge as discussed by Bershadsky and Ooguri, one which depends on fractional powers of free fields, and such entities have seemed ambiguous in connection with a consistent interpretation in terms of Wick contractions. In this paper we develop the techniques necessary to deal with these complications, and we provide explicit general integral representations for conformal blocks on the sphere. The main virtue is the straightforward way in which standard free field techniques may now be employed to obtain such results, which have the standard structure well known from other free field studies in conformal field theory. We further discuss fusion rules, and as a check we verify explicitly that our conformal blocks satisfy the Knizhnik-Zamolodchikov equations and are projectively invariant.