Relaxed singular vectors, Jack symmetric functions and fractional level $\widehat{\mathfrak{sl}}(2)$ models

Abstract

The fractional level models are (logarithmic) conformal field theories associated with affine Kac-Moody (super)algebras at certain levels $k\in \mathbb{Q}$. 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 $\widehat{\mathfrak{sl}}(2)$ 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.

33 pages; v2: corrected typos and added references

References in corpus (8)

• Logarithmic extensions of minimal models: characters and modular transformations
• sl^(2)_{-1/2}: A Case Study
• Fusion in Fractional Level sl^(2)-Theories with k=-1/2
• Bosonic Ghosts at $c=2$ as a Logarithmic CFT
• The Verlinde formula in logarithmic CFT
• From Jack polynomials to minimal model spectra
• Resonant local systems on complements of discriminantal arrangements and sl_2 representations
• A construction of admissible $A_1^{(1)}$-modules of level $-4/3$

Cited by in corpus (5)

• Rationalizing CFTs and Anyonic Imprints on Higgs Branches
• Admissible-level ${\mathfrak{sl}}_3$ minimal models
• From Jack to Double Jack Polynomials via the Supersymmetric Bridge
• Staggered modules of $N=2$ superconformal minimal models
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras