Modular Transformations and Verlinde Formulae for Logarithmic \((p_+,p_−)\)-Models
Published in Nuclear Physics B, 2013
Computes the modular transformations of the characters of the uncountably many simple modules of the logarithmic \((p_+,p_-)\) singlet algebras and derives their Grothendieck fusion rules from a continuous Verlinde formula. Lifting to the triplet algebras via simple current extensions recovers the previously proposed triplet fusion rules.
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.
Recommended citation: D. Ridout, and S. Wood, Nucl. Phys. B 880, 175-202 (2014)
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