Superconformal minimal models and admissible Jack polynomials
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| Main Authors: | , , , |
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| Format: | Preprint |
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2016
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| _version_ | 1866913595847081984 |
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| author | Blondeau-Fournier, Olivier Mathieu, Pierre Ridout, David Wood, Simon |
| author_facet | Blondeau-Fournier, Olivier Mathieu, Pierre Ridout, David Wood, Simon |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1606_04187 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Superconformal minimal models and admissible Jack polynomials Blondeau-Fournier, Olivier Mathieu, Pierre Ridout, David Wood, Simon High Energy Physics - Theory Mathematical Physics Quantum Algebra Representation Theory 81T40, 17B69, 17B68 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. |
| title | Superconformal minimal models and admissible Jack polynomials |
| topic | High Energy Physics - Theory Mathematical Physics Quantum Algebra Representation Theory 81T40, 17B69, 17B68 |
| url | https://arxiv.org/abs/1606.04187 |