Modular Transformations and Verlinde Formulae for Logarithmic $(p_{+},p_{-})$-Models

Abstract

The $(p_{+},p_{-})$ singlet algebra is a vertex operator algebra that is strongly generated by a Virasoro field of central charge $1-6(p_{+}-p_{-}{)}^2/p_{+}{p}_{-}$ and a single Virasoro primary field of conformal weight $(2p_{+}-1)(2p_{-}-1)$. Here, the modular properties of the characters of the uncountably many simple modules of each singlet algebra are investigated and the results used as the input to a continuous analogue of the Verlinde formula to obtain the "fusion rules" of the singlet modules. The effect of the failure of fusion to be exact in general is studied at the level of Verlinde products and the rules derived are lifted to the $(p_{+},p_{-})$ triplet algebras by regarding these algebras as simple current extensions of their singlet cousins. The result is a relatively effortless derivation of the triplet "fusion rules" that agrees with those previously proposed in the literature.

22 pages, v2 minor changes; added refs

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• Modularity of logarithmic parafermion vertex algebras
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• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
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• Admissible-level ${\mathfrak{sl}}_3$ minimal models
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• On Regularised Quantum Dimensions of the Singlet Vertex Operator Algebra and False Theta Functions
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• Tensor Categories arising from the Virasoro Algebra
• Staggered modules of $N=2$ superconformal minimal models
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras