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

Abstract

We study moduli of suitably framed \(\mathcal {N}=2\) 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 \(\kappa \), familiar from the theory of mock modular forms. In this optic, \(\kappa \) arises when considering purely Fermionic deformations of \(\mathcal {N}=2\) elliptic curves.