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

Abstract

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.