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 Ẑ0(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∈N$k \in \mathbb {N}$ there exists a full asymptotic expansion of Ẑ0(X;q)$ \hat{\operatorname{Z}}_0(X;q)$ for q$q$ tending to e2πi/k$e^{2\pi i/k}$, and in particular that the limit Ẑ0(X;e2πi/k)$\hat{\operatorname{Z}}_0(X;e^{2\pi i/k})$ exists and is equal to the Witten–Reshetikhin–Turaev quantum invariant τk(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.