Kac and New Determinants for Fractional Superconformal Algebras

TL;DR

This paper derives Kac and new determinant formulas for fractional superconformal algebras at arbitrary levels, revealing new modules and applying results to fractional superstrings, including proving a no-ghost theorem.

Contribution

It introduces new determinant formulas for fractional superconformal algebras at all levels and applies these to analyze fractional superstrings and their physical spectra.

Findings

Derived Kac and new determinants for all levels K

Identified modules where determinants factorize for K≥3

Proved no-ghost theorem for spin-4/3 fractional superstring

Abstract

We derive the Kac and new determinant formulae 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 ($K=1$) and superconformal ($K=2$) algebras. For $K\geq3$ 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 non-unitarity 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 formulae for the spin-4/3 parafermion current algebra ({\em i.e.}, the $K=4$ fractional superconformal algebra) to the recently constructed three-dimensional flat Minkowski space-time representation of the spin-4/3 fractional superstring. We prove the no-ghost theorem for the space-time bosonic sector of this theory; that is, its physical spectrum is free of negative-norm states.