Superconformal minimal models and admissible Jack polynomials

TL;DR

This paper provides new proofs for the rationality and module classification of N=1 superconformal minimal models using free field realization and Jack symmetric functions, notably simplifying the process by avoiding explicit vacuum vector expressions.

Contribution

It introduces a novel approach combining free field realization with Jack symmetric functions to classify modules without explicit vacuum vector formulas.

Findings

Jack polynomials with negative parameters describe free fermion correlators.

Classification of modules achieved without explicit vacuum singular vector expressions.

Identification of Zhu's algebra and its twisted version using symmetric functions.

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.