Modular invariant representations of the $\mathcal{N}=2$ superconformal algebra

Abstract

We compute the modular transformation formula of the characters for a certain family of (finitely or uncountably many) simple modules over the simple $\mathcal{N}=2$ vertex operator superalgebra of central charge $c_{p,p^{\prime }}=3\left(1-\frac{2p^{\prime }}{p}\right),$ where $(p,p^{\prime })$ is a pair of coprime positive integers such that $p\ge 2$. When $p^{\prime }=1$, the formula coincides with that of the $\mathcal{N}=2$ unitary minimal series found by F. Ravanini and S.-K. Yang. In addition, we study the properties of the corresponding "modular $S$-matrix", which is no longer a matrix if $p^{\prime }\ge 2$.

24 pages, definition of Verlinde coefficient in Section 5 is corrected, minor change in Proposition 3.8, references added

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