On the possibility of $Z_N$ exotic supersymmetry in two dimensional Conformal Field Theory

TL;DR

This paper explores the theoretical possibility of constructing extended parafermionic conformal algebras with exotic supersymmetry in two-dimensional conformal field theories, finding significant restrictions on their existence.

Contribution

It demonstrates that such exotic supersymmetric algebras are only possible in very limited cases, providing new insights into their non-existence for most values of K.

Findings

For K=4, only models at c=1 are possible.

For K=5, the algebra reduces to a known Z_5 parafermionic model.

For K=6,7, and larger K, such algebras do not exist.

Abstract

We investigate the possibility to construct extended parafermionic conformal algebras whose generating current has spin $1+\frac{1}{K}$, generalizing the superconformal (spin 3/2) and the Fateev Zamolodchikov (spin 4/3) algebras. Models invariant under such algebras would possess $Z_K$ exotic supersymmetries satisfying (supercharge)$^K$ = (momentum). However, we show that for $K=4$ this new algebra allows only for models at $c=1$, for $K=5$ it is a trivial rephrasing of the ordinary $Z_5$ parafermionic model, for $K=6,7$ (and, requiring unitarity, for all larger $K$) such algebras do not exist. Implications of this result for existence of exotic supersymmetry in two dimensional field theory are discussed.