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Modularity of logarithmic parafermion vertex algebras

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Abstract

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.

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Acknowledgements

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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Auger, J., Creutzig, T. & Ridout, D. Modularity of logarithmic parafermion vertex algebras. Lett Math Phys 108, 2543–2587 (2018). https://doi.org/10.1007/s11005-018-1098-4

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