Orientation Reversal and Resurgent Crossing of the Natural Boundary

In this talk, I will introduce a resurgent method that crosses the $|q|=1$ natural boundary for the $q$-series invariants $\widehat{Z}$ of 3-manifolds (Gukov-Pei-Putrov-Vafa) and, at the level of individual false theta building blocks. In our setup, crossing the natural boundary corresponds to the orientation reversal of the 3-manifold $M_3$. Under this operation, the $q$-series invariants for $M_3$ and $\overline{M_3}$ are very different, and usually one of them is much harder to compute. The resurgence approach proposes a solution to systematically computing these invariants and their individual building blocks for a large class of new examples. The talk is based on joint work with Adams, Costin, Dunne, and Gukov.