An admissible level \(\widehat{\mathfrak{osp}}(1\vert2)\)-model: modular transformations and the Verlinde formula

A detailed case study of the affine \(\mathfrak{osp}(1\vert 2)\) vertex operator superalgebra at admissible level \(-\tfrac{5}{4}\), testing the standard module formalism for logarithmic conformal field theories with fermions. Classifies the relaxed highest-weight modules, computes their modular transformations and obtains Grothendieck fusion rules with non-negative integer coefficients from a Verlinde-type formula.

Abstract

The modular properties of the simple vertex operator superalgebra associated to the affine Kac-Moody superalgebra \(\widehat{\mathfrak{osp}} \left( 1 \middle\vert 2 \right)\) 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 analogue 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.

Recommended citation: D. Ridout, J. Snadden and S. Wood, Lett. Math. Phys. 108, 2363-2423 (2018)
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