Appell–Lerch sums and \(\mathcal {N}=2\) moduli

We study moduli of suitably framed \(\mathcal {N}=2\) $\mathcal{N}=2$ elliptic curves. We introduce the notion of tameness for a family of super-spaces and show that the non-tameness of the resulting universal family is essentially controlled by the Appell–Lerch sum \(\kappa \) $\kappa$ , familiar from the theory of mock modular forms. In this optic, \(\kappa \) $\kappa$ arises when considering purely Fermionic deformations of \(\mathcal {N}=2\) $\mathcal{N}=2$ elliptic curves.