Letters in Mathematical Physics

, Volume 108, Issue 11, pp 2363–2423 | Cite as

An admissible level \(\widehat{\mathfrak {osp}} \left( 1 \big \vert 2 \right) \)-model: modular transformations and the Verlinde formula

  • John SnaddenEmail author
  • David Ridout
  • Simon Wood


The modular properties of the simple vertex operator superalgebra associated with the affine Kac–Moody superalgebra \(\widehat{{\mathfrak {osp}}} (1|2)\) at level \(-\frac{5}{4}\) are investigated. After classifying the relaxed highest-weight modules over this vertex operator superalgebra, the characters and supercharacters of the simple weight modules are computed and their modular transforms are determined. This leads to a complete list of the Grothendieck fusion rules by way of a continuous superalgebraic analog of the Verlinde formula. All Grothendieck fusion coefficients are observed to be non-negative integers. These results indicate that the extension to general admissible levels will follow using the same methodology once the classification of relaxed highest-weight modules is completed.


Conformal field theory Vertex operator superalgebras Lie superalgebras Verlinde formula Modular transformations Zhu’s algebra 

Mathematics Subject Classification

Primary 17B69 81T40 Secondary 17B10 17B67 



DR thanks Kenji Iohara for illuminating discussions on the structure of Verma modules over \(\widehat{{\mathfrak {osp}}} (1|2)\). We would like to thank the anonymous referee whose careful reading of the original manuscript and many suggestions significantly improved the article. JS’s research is supported by a University Research Scholarship from the Australian National University. DR’s research is supported by the Australian Research Council Discovery Projects DP1093910 and DP160101520 as well as the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers CE140100049. SW’s research is supported by Australian Research Council Discovery Early Career Researcher Award DE140101825 and the Australian Research Council Discovery Project DP160101520.


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Authors and Affiliations

  1. 1.Mathematical Sciences InstituteAustralian National UniversityActonAustralia
  2. 2.School of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia
  3. 3.School of MathematicsCardiff UniversityCardiffUK

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