Abstract: We introduce the Gukov-Pei-Putrov-Vafa (GPPV) invariant (or Z-hat invariant) in Chern-Simons theory associated with a Seifert fibered integral homology sphere (SFIHS). The GPPV invariant is a partial theta series and a higher-depth quantum modular form. By philosophy of resurgence theory, Ecalle's alien derivations provide all components of $SL(2,\mathbb{C})$-irreps of the fundamental group of the SFIHS. As the complex variable goes to a rational number, the GPPV invariant becomes the Witten-Reshetikhin-Turaev (WRT) invariant. Simultaneously, only $SU(2)$ components remain in the transseries expansion. This yields a proof of the so-called asymptotic expansion conjecture of the WRT invariant of SFIHS.
The talk is based on a joint work with J. Andersen, L. Han,W. Mistegard, D. Sauzin, S. Sun.