Low-Lying States of the Six-Dimensional Fractional Superstring

TL;DR

This paper constructs and analyzes the low-lying states of the six-dimensional fractional superstring using $Z_4$ parafermions, exploring the associated algebraic structures and their implications for the physical spectrum.

Contribution

It introduces a Fock space framework for the $K=4$ fractional superstring and derives a candidate fractional superconformal algebra for physical state constraints.

Findings

Fock space is larger than the Lorentz-covariant space.

Derived a form of the fractional superconformal algebra.

Applied the algebra to massless states, but not to massive states.

Abstract

The $K=4$ fractional superstring Fock space is constructed in terms of $\bZ_4$ parafermions and free bosons. The bosonization of the $\bZ_4$ 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 $\bZ_4$ 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.