W-Algebras, New Rational Models and Completeness of the c = 1 Classification

TL;DR

This paper constructs new W-algebras at specific central charges, demonstrates their rationality through detailed representation analysis, and completes the classification of rational theories with central charge c=1, extending to supersymmetric cases.

Contribution

It introduces new series of W-algebras derived from free field representations, proving their rationality and completing the classification of c=1 rational conformal field theories.

Findings

Constructed W(2,3k) and W(2,8k) algebras for specific c values

Calculated all lowest-weight representations, characters, and fusion rules

Proved these theories are rational and complete the c=1 classification

Abstract

Two series of W-algebras with two generators are constructed from chiral vertex operators of a free field representation. If $c = 1 - 24k$, there exists a W(2,3k) algebra for k in $Z_{+}/2$ and a W(2,8k) algebra for k in $Z_{+}/4$. All possible lowest-weight representations, their characters and fusion rules are calculated proving that these theories are rational. It is shown, that these non-unitary theories complete the classification of all rational theories with effective central charge $c_{eff} = 1$. The results are generalized to the case of extended supersymmetric conformal algebras.