Communications in Mathematical Physics

, Volume 368, Issue 2, pp 627–663 | Cite as

Relaxed Highest-Weight Modules I: Rank 1 Cases

  • Kazuya KawasetsuEmail author
  • David Ridout


Relaxed highest-weight modules play a central role in the study of many important vertex operator (super)algebras and their associated (logarithmic) conformal field theories, including the admissible-level affine models. Indeed, their structure and their (super)characters together form the crucial input data for the standard module formalism that describes the modular transformations and Grothendieck fusion rules of such theories. In this article, character formulae are proved for relaxed highest-weight modules over the simple admissible-level affine vertex operator superalgebras associated to \({\mathfrak{s}\mathfrak{l}_2}\) and \({\mathfrak{osp} (1 \vert 2)}\). Moreover, the structures of these modules are specified completely. This proves several conjectural statements in the literature for \({\mathfrak{s}\mathfrak{l}_2}\), at arbitrary admissible levels, and for \({\mathfrak{osp} (1 \vert 2)}\) at level \({-\frac{5}{4}}\). For other admissible levels, the \({\mathfrak{osp}(1 \vert 2)}\) results are believed to be new.


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We thank Dražen Adamović, Tomoyuki Arakawa, Thomas Creutzig, Tianshu Liu, and SimonWood for useful discussions aswell as their encouragement.We also thankWill Stewart for pointing out a small error in a previous version and Ryo Sato for clarifying for us the relation between hiswork and relaxed \({\widehat{\mathfrak{sl}}_2}\) characters. KK’s research is supported by the Australian Research Council Discovery Project DP160101520. DR’s research is supported by the Australian Research Council Discovery Project DP160101520 and the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers CE140100049.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia

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