On the extended W-algebra of type sl_2 at positive rational level

Abstract

The extended W-algebra of type sl_2 at positive rational level, denoted by M_{p_+,p_-}, is a vertex operator algebra that was originally proposed in [1]. This vertex operator algebra is an extension of the minimal model vertex operator algebra and plays the role of symmetry algebra for certain logarithmic conformal field theories. We give a construction of M_{p_+,p_-} in terms of screening operators and use this construction to prove that M_{p_+,p_-} satisfies Zhu's c_2-cofiniteness condition, calculate the structure of the zero mode algebra (also known as Zhu's algebra) and classify all simple M_{p_+,p_-}-modules.

64 pages, 2 figures; version to appear in journal, Int Math Res Notices (2014)

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• Logarithmic Conformal Field Theory: Beyond an Introduction
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• Schur-Weyl Duality for Heisenberg Cosets
• Logarithmic conformal field theory, log-modular tensor categories and modular forms
• Modular Transformations and Verlinde Formulae for Logarithmic $(p_{+},p_{-})$-Models
• Bosonic ghostbusting -- The bosonic ghost vertex algebra admits a logarithmic module category with rigid fusion
• Superconformal minimal models and admissible Jack polynomials
• Admissible level $\mathfrak{osp}(1|2)$ minimal models and their relaxed highest weight modules
• On Regularised Quantum Dimensions of the Singlet Vertex Operator Algebra and False Theta Functions
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• Singular vectors for the $W_N$ algebras
• Simplicity of higher rank triplet W-algebras
• The quantum group dual of the first-row subcategory for the generic Virasoro VOA
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras