Fractional Supersymmetry in Conformal Field Theory and String Theory

In the first part of this thesis, we study the structure of the fractional superconformal algebras. These are generalizations of the ordinary superconformal algebra, which generate a symmetry between bosonic fields and parafermionic fields of fractional spin. We show that the SU(2) minimal coset conformal field theories have fractional supersymmetry by explicitly constructing the irreducible representations of the algebra, employing a generalization of the BRST cohomology method introduced by Felder. We then calculate the characters and branching functions of the irreducible representations and show that they agree with the known results for SU(2) coset models in the unitary case. Finally, we show how this method can be extended to study the irreducible representations of SU(N) minimal coset models (N>2), without detailed knowledge of the structure of the symmetry algebra. In the second part of this thesis, we study string theories with world-sheet fractional supersymmetry, restricting ourselves, for simplicity, to the SU(2), K = 4 case, which recent work suggests is consistent in six space-time dimensions. We construct the K = 4 fractional superstring Fock space in terms of {bf Z}_4 parafermions and free bosons. Employing a bosonization of the parafermion theory, we compute the operator product expansions of the primary fields, and then obtain the generalized commutation relations satisfied by the modes of these fields. We consider the simplest representation of the string theory, in which the Fock space is a tensor product of {bf Z}_4 parafermion and free boson Fock spaces. This Fock space is larger than the Lorentz-covariant one indicated by the fractional superstring partition function. We derive the form of the fractional superconformal algebra appropriate for the tensor product theory that may generate 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 the physical state conditions that the spectrum must satisfy is illustrated by an application to the massless states of the K = 4 fractional superstring. However, the appropriate constraint algebra on the tensor product Fock space for the whole spectrum remains to be found.