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C 2-Cofinite \(\mathcal{W}\)-Algebras and Their Logarithmic Representations

  • Conference paper
Conformal Field Theories and Tensor Categories

Part of the book series: Mathematical Lectures from Peking University ((MLPKU))

Abstract

We discuss our recent results on the representation theory of \(\mathcal{W}\)-algebras relevant to Logarithmic Conformal Field Theory. First we explain some general constructions of \(\mathcal{W}\)-algebras coming from screening operators. Then we review the results on C 2-cofiniteness, the structure of Zhu’s algebras, and the existence of logarithmic modules for triplet vertex algebras. We propose some conjectures and open problems which put the theory of triplet vertex algebras into a broader context. New realizations of logarithmic modules for \(\mathcal{W}\)-algebras defined via screenings are also presented.

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Notes

  1. 1.

    n.b. For brevity, we shall often use “algebra” and “vertex algebra” when we mean “superalgebra” and “vertex superalgebra”, respectively. From the context it should be clear whether the adjective “super” is needed.

  2. 2.

    Without the linear term the central charge would be \(\operatorname{rank}(L)\).

  3. 3.

    Here for simplicity we assume strong rationality, meaning that for a given VOA every (weak) module is completely reducible.

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Acknowledgements

This note is based on lectures given by the authors at the conference on “Tensor Categories and Conformal Field Theory”, June 2011, Beijing. Thus, except for Sect. 5 and some constructions in Sect. 2, all the material is based on earlier works by the authors (we should say that some constructions in Sect. 2 were independently introduced in [25]). We are indebted to the organizers for invitation to this wonderful conference. We also thank A. Semikhatov, A. Gaĭnutdinov, A. Tsuchiya, I. Runkel, Y. Arike, J. Lepowsky, L. Kong and Y.-Z. Huang for the interesting discussion during and after the conference. We are also grateful to Jinwei Yang for helping us around in Beijing.

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Authors and Affiliations

  1. Department of Mathematics, University of Zagreb, Zagreb, 10000, Croatia

    Dražen Adamović

  2. Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY, 12222, USA

    Antun Milas

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  1. Dražen Adamović
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  2. Antun Milas
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Correspondence to Dražen Adamović .

Editor information

Editors and Affiliations

  1. Chern Institute of Mathematics, Nankai University, Tianjin, People's Republic of China

    Chengming Bai

  2. Theoretical Physics, Karlstad University, Karlstad, Sweden

    Jürgen Fuchs

  3. Department of Mathematics, Department of Mathematics, Rutgers University, Piscataway, New Jersey, USA

    Yi-Zhi Huang

  4. Institute for Advanced Study, Tsinghua University, Beijing, People's Republic of China

    Liang Kong

  5. Department of Mathematics, University of Hamburg, Hamburg, Germany

    Ingo Runkel

  6. Department of Mathematics, University of Hamburg, Hamburg, Germany

    Christoph Schweigert

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Adamović, D., Milas, A. (2014). C 2-Cofinite \(\mathcal{W}\)-Algebras and Their Logarithmic Representations. In: Bai, C., Fuchs, J., Huang, YZ., Kong, L., Runkel, I., Schweigert, C. (eds) Conformal Field Theories and Tensor Categories. Mathematical Lectures from Peking University. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39383-9_6

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