Resurgence analysis of quantum invariants of Seifert fibered homology spheres

For a Seifert fibered homology sphere X$X$, we show that the q$q$-series invariant Z0(X;q)$\hat{\operatorname{Z}}_0(X;q)$, introduced by Gukov-Pei-Putrov-Vafa, is a resummation of the Ohtsuki series Z0(X)$\operatorname{Z}_0(X)$. We show that for every even k is an element of N$k \in \mathbb {N}$ there exists a full asymptotic expansion of Z0(X;q)$ \hat{\operatorname{Z}}_0(X;q)$ for q$q$ tending to e2 pi i/k$e<^>{2\pi i/k}$, and in particular that the limit Z0(X;e2 pi i/k)$\hat{\operatorname{Z}}_0(X;e<^>{2\pi i/k})$ exists and is equal to the Witten-Reshetikhin-Turaev quantum invariant tau k(X)$\tau _k(X)$. We show that the poles of the Borel transform of Z0(X)$\operatorname{Z}_0(X)$ coincide with the classical complex Chern-Simons values, which we further show classifies the corresponding components of the moduli space of flat SL(2,C)$\rm {SL}(2,\mathbb {C})$-connections.