Abstract
The fractional level models are (logarithmic) conformal field theories associated with affine Kac–Moody (super)algebras at certain levels k∈Q. They are particularly noteworthy because of several longstanding difficulties that have only recently been resolved. Here, Wakimoto's free field realisation is combined with the theory of Jack symmetric functions to analyse the fractional level slˆ(2) models. The first main results are explicit formulae for the singular vectors of minimal grade in relaxed Wakimoto modules. These are closely related to the minimal grade singular vectors in relaxed (parabolic) Verma modules. Further results include an explicit presentation of Zhu's algebra and an elegant new proof of the classification of simple relaxed highest weight modules over the corresponding vertex operator algebra. These results suggest that generalisations to higher rank fractional level models are now within reach.
Highlights
We begin by briefly summarising some historical details regarding fractional level sl (2) models and the application of Jack symmetric functions to conformal field theory
When g is the Virasoro algebra, the vacuum module of the universal vertex operator algebra is the quotient of the Verma module whose highest weight vector has conformal weight 0 by the Verma submodule generated by the singular vector of conformal weight 1
For g = sl (2), one quotients the Verma module generated by the highest weight vector of sl 2 -weight 0 by the Verma submodule generated by the singular vector of sl 2 -weight −2
Summary
We begin by briefly summarising some historical details regarding fractional level sl (2) models and the application of Jack symmetric functions to conformal field theory. The work presented here forms a part of an ambitious project aimed at elucidating the properties of general fractional. Wood / Nuclear Physics B 894 (2015) 621–664 level models as fundamental examples of logarithmic conformal field theories. A short digression on the notion of admissibility follows as we use the term in a non-standard manner as compared to much of the literature. This provides us with an opportunity to fix some notation. We outline the main results of the research reported here
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