Starting from fractional supersymmetry as an extended Virasoro algebra, we generalize Felder's BRST cohomology method to construct the characters and branching functions — in terms of string functions — of the SU(2) and SU(3) coset models, for both the unitary and nonunitary cases.

The K = 4 fractional superstring Fock space is constructed in terms of Z 4 parafermions and free bosons. The bosonization of the Z 4 parafermion theory and the generalized commutation relations satisfied by the modes of various parafermion fields are reviewed. In this preliminary analysis, we describe a Fock space which is simply a tensor product of Z 4 parafermion and free boson Fock spaces. It is larger than the Lorentz-covariant Fock space indicated by the fractional superstring partition function. We derive the form of the fractional superconformal algebra that may be used as the constraint algebra for the physical states of the FSS. Issues concerning the associativity, modings and braiding properties of the fractional superconformal algebra are also discussed. The use of the constraint algebra to obtain physical state conditions on the spectrum is illustrated by an application to the massless fermions and bosons of the K = 4 fractional superstring. However, we fail to generalize these considerations to the massive states. This means that the appropriate constraint algebra on the fractional superstring Fock space remains to be found. Some possible ways of doing this are discussed. * 1The case K = 2 (D = 10) corresponds to the superstring. The new theories are those with K > 2; for K = 4, 8 and 16 we have the integer critical dimensions D = 6, 4 and 3, respectively.In this paper we will concentrate exclusively on the simplest case after the (K = 2) superstring, the K = 4 FSS. The reasons for this are twofold. The first is that the complexity of these theories increases considerably with increasing K. Although the world-sheet fractional supersymmetry algebra is non-local, the Z 4 parafermion fields that appear in the K = 4 FSS can be simply represented by free bosons, which enables the calculations to be simplified tremendously. This is not the case in the K = 8 and K = 16 theories [10,11]. Furthermore, a close examination shows that the appropriate world-sheet fractional supersymmetry algebra for the K = 8 theory contains two spin-13/5 currents in addition to the spin-6/5 current [11], which further complicate the analysis. The second reason for our emphasis on the K = 4 FSS is because it is potentially the most interesting one from the phenomenological point of view. As argued in [12,13], the requirements of quantum mechanics, Lorentz invariance and locality suggest that the K = 4 FSS may be automatically compactified from six to four space-time dimensions. Furthermore, as argued in [12], the compactification from the critical dimension 6 to the natural dimension 4 offers the possibility of the construction of heterotic type K = 4 FSS models that have chiral space-time fermions. This is encouraging, since the K = 4 FSS, because of its relative simplicity, affords the best prospect for detailed examination in the near future.Let us highlight some of the similarities and differences between the K = 4 FSS and the superstring. In the superstring, the world-sheet superpartner of the space-time coordinate boson X µ...