Low-lying states of the six-dimensional fractional superstring

The $K=4\mathrm{}$ fractional superstring Fock space is constructed in terms of ${\mathrm{Z}}_4$ parafermions and free bosons. The bosonization of the ${\mathrm{Z}}_4$ 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 ${\mathrm{Z}}_4$ 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 fractional superstring. 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 $K=4\mathrm{}$ 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.