A class of three-manifold invariants, called Z-hat invariants, are q-series with peculiar modular properties whose original definition comes from three-dimensional N=2 supersymmetric quantum field theories associated to the manifold. A formal operation on the q-series, commonly referred to as “going to the other side,” is expected to relate invariants of manifolds with opposite orientation, heuristically corresponding to the transformation q to q inverse.
In this talk, I will explain how this operation can be made precise when the invariants are expressed as indefinite theta functions defining pure mock modular forms. I will review the relevant background and clarify how orientation reversal is reflected in the relation between mock and false theta functions. Based on ongoing work with M. Cheng and I. Coman.