Letters in Mathematical Physics

, Volume 108, Issue 12, pp 2543–2587 | Cite as

Modularity of logarithmic parafermion vertex algebras

  • Jean Auger
  • Thomas CreutzigEmail author
  • David Ridout


The parafermionic cosets \(\mathsf {C}_{k} = {\text {Com}} ( \mathsf {H} , \mathsf {L}_{k}(\mathfrak {sl}_{2}) )\) are studied for negative admissible levels k, as are certain infinite-order simple current extensions \(\mathsf {B}_{k}\) of \(\mathsf {C}_{k}\). Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to \(\mathsf {C}_{k}\), irreducible \(\mathsf {C}_{k}\)- and \(\mathsf {B}_{k}\)-modules are obtained from those of \(\mathsf {L}_{k}(\mathfrak {sl}_{2})\). Assuming the validity of a certain Verlinde-type formula likewise gives the Grothendieck fusion rules of these irreducible modules. Notably, there are only finitely many irreducible \(\mathsf {B}_{k}\)-modules. The irreducible \(\mathsf {C}_{k}\)- and \(\mathsf {B}_{k}\)-characters are computed and the latter are shown, when supplemented by pseudotraces, to carry a finite-dimensional representation of the modular group. The natural conjecture then is that the \(\mathsf {B}_{k}\) are \(C_2\)-cofinite vertex operator algebras.


Vertex algebras Conformal field theory Modular transformations Parafermions Coset constructions 

Mathematics Subject Classification

Primary 17B69 Secondary 13A50 



We thank Shashank Kanade and Andrew Linshaw for discussions relating to the results presented here. J. A. is supported by a Doctoral Research Scholarship from the Fonds de Recherche Nature et Technologies de Québec (184131). T. C. is supported by the Natural Sciences and Engineering Research Council of Canada (RES0020460). 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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Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.School of Mathematics and StatisticsThe University of MelbourneParkvilleAustralia

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