From Jack polynomials to minimal model spectra

Abstract

In this note, a deep connection between free field realisations of conformal field theories and symmetric polynomials is presented. We give a brief introduction into the necessary prerequisites of both free field realisations and symmetric polynomials, in particular Jack symmetric polynomials. Then we combine these two fields to classify the irreducible representations of the minimal model vertex operator algebras as an illuminating example of the power of these methods. While these results on the representation theory of the minimal models are all known, this note exploits the full power of Jack polynomials to present significant simplifications of the original proofs in the literature.

14 pages, corrected typos and added comment on connections to the AGT conjecture in introduction, version to appear in J. Phys. A

References in corpus (4)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• On Staggered Indecomposable Virasoro Modules
• On the structure of Verma module over Virasoro and Neveu-Schwarz algebras
• Finalizing the proof of AGT relations with the help of the generalized Jack polynomials

Cited by in corpus (11)

• Relaxed singular vectors, Jack symmetric functions and fractional level $\widehat{\mathfrak{sl}}(2)$ models
• 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
• Superconformal minimal models and admissible Jack polynomials
• 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
• Tensor Categories arising from the Virasoro Algebra
• From Jack to Double Jack Polynomials via the Supersymmetric Bridge