Superconformal minimal models and admissible Jack polynomials

Gives new proofs of the rationality of the \(\mathcal{N}=1\) superconformal minimal model vertex operator superalgebras and classifies their simple modules in both the Neveu–Schwarz and Ramond sectors, without coset constructions. Zhu’s algebras are identified by combining free field realisations with Jack symmetric polynomials, without needing an explicit formula for the vacuum singular vector.

Abstract

We give new proofs of the rationality of the \(\mathcal{N}=1\) 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.

Recommended citation: O. Blondeau-Fournier, P. Mathieu, D. Ridout and S. Wood, Adv. Math. 314, 71-123 (2017)
Download Paper