$\mathcal S$-modular transformation of the $\mathcal N=2$ superconformal algebra characters

The $\mathcal N=2$ superconformal algebra characters have to exhibit modular invariance in order to be appropriately applied in quantum superstring theories. The nonunitary characters are given by higher-level Appell functions and different kinds of Jacobi theta functions are involved within their algebraic structures. Evaluating their $\mathcal T$-modular invariance appears to be quite simple, but verifying their $\mathcal S$-modular invariance entails a serious mathematical physics exploration. In this regard, we use a new vocabulary for Jacobi theta functions, namely “spectral theta functions,” which allows us to come up with the $\mathcal S$-modular transformation of nonunitary (nontrivial) $\mathcal N=2$ characters for the central charge $c=3(1-2p/u)$, where $(u,p)$ is a pair of coprime positive integers.