We derive the Kac and new determinant formulas for an arbitrary (integer) level $K$ fractional superconformal algebra using the BRST cohomology techniques developed in conformal field theory. In particular, we reproduce the Kac determinants for the Virasoro ($K=1\mathrm{}$) and superconformal ($K=2\mathrm{}$) algebras. For $K\ge 3$ there always exist modules where the Kac determinant factorizes into a product of more fundamental new determinants. Using our results for general $K$, we sketch the nonunitarity proof for the SU(2) minimal series; as expected, the only unitary models are those already known from the coset construction. We apply the Kac determinant formulas for the spin-4/3 parafermion current algebra (i.e., the $K=4\mathrm{}$ fractional superconformal algebra) to the recently constructed three-dimensional flat Minkowski space-time representation of the spin-4/3 fractional superstring.

• Received 25 October 1993

DOI:https://doi.org/10.1103/PhysRevD.49.4122