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 $c_{ p,p'}=3\left(1-\frac{ 2p'}{ p}\right),$ where $(p,p')$ is a pair of coprime positive integers such that $p\geq2$. When $p'=1$, 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 $p'\geq2$.
Categories: math.QA math-ph math.MP math.RT