Superconformal minimal models and admissible Jack polynomials

Abstract

We give new proofs of the rationality of the N=1 superconformal minimal model vertex operator superalgebras and of the classification of their modules in both the Neveu-Schwarz and Ramond sectors. For this, we combine the standard free field realisation with the theory of Jack symmetric functions. A key role is played by Jack symmetric polynomials with a certain negative parameter that are labelled by admissible partitions. These polynomials are shown to describe free fermion correlators, suitably dressed by a symmetrising factor. The classification proofs concentrate on explicitly identifying Zhu's algebra and its twisted analogue. Interestingly, these identifications do not use an explicit expression for the non-trivial vacuum singular vector. While the latter is known to be expressible in terms of an Uglov symmetric polynomial or a linear combination of Jack superpolynomials, it turns out that standard Jack polynomials (and functions) suffice to prove the classification.

arXiv comment32 pages, minor tweak to the explanation of notation on p15, updated references, version to appear in Adv. Math

References in corpus (15)

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Cited by in corpus (11)

• Fermionic CFTs and classifying algebras
• A slow review of the AGT correspondence
• Cosets, characters and fusion for admissible-level $\mathfrak{osp}(1|2)$ minimal models
• An admissible level $\widehat{\mathfrak{osp}}\left(1|2\right)$-model: modular transformations and the Verlinde formula
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Classifying relaxed highest-weight modules for admissible-level Bershadsky-Polyakov algebras
• Realizations of simple affine vertex algebras and their modules: the cases $\widehat{sl(2)}$ and $\widehat{osp(1,2)}$
• Admissible level $\mathfrak{osp}(1|2)$ minimal models and their relaxed highest weight modules
• The super-Virasoro singular vectors and Jack superpolynomials relationship revisited
• Singular vectors for the $W_N$ algebras
• Whittaker modules for the super-Virasoro algebras