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Modularity of Relatively Rational Vertex Algebras and Fusion Rules of Principal Affine W-Algebras

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Abstract

We study modularity of the characters of a vertex (super)algebra equipped with a family of conformal structures. Along the way we introduce the notions of rationality and cofiniteness relative to such a family. We apply the results to determine modular transformations of trace functions on admissible modules over affine Kac–Moody algebras and, via BRST reduction, trace functions on minimal series representations of principal affine W-algebras.

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Acknowledgements

The first author is partially supported by JSPS KAKENHI Grant Nos. 17H01086 and 17K18724. The second author was supported by an Alexander von Humboldt Foundation grant and later by CAPES-Brazil. The second author would like to thank Victor G. Kac for several ideas which go back to discussions had with him in 2011. Both authors would like to thank the referees for their helpful comments. The work has been presented at conferences “Lie and Jordan Algebras VI”, Bento Gonçalves, Brazil, December 2015, “Quántum 2016”, Córdoba, Argentina, February 2016, and “Vertex Algebras and Quantum Groups” Banff, Canada, March 2016. The authors would like to thank the organisers of these conferences.

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Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto, Japan

    Tomoyuki Arakawa

  2. MIT, Cambridge, USA

    Tomoyuki Arakawa

  3. Instituto de Matemática e Estatística (GMA), UFF, Niterói, RJ, Brazil

    Jethro van Ekeren

Authors
  1. Tomoyuki Arakawa
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  2. Jethro van Ekeren
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Correspondence to Jethro van Ekeren.

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Communicated by Y. Kawahigashi

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Arakawa, T., van Ekeren, J. Modularity of Relatively Rational Vertex Algebras and Fusion Rules of Principal Affine W-Algebras. Commun. Math. Phys. 370, 205–247 (2019). https://doi.org/10.1007/s00220-019-03504-6

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  • DOI: https://doi.org/10.1007/s00220-019-03504-6

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