Resurgence analysis of quantum invariants of Seifert fibered homology spheres

Abstract

For a Seifert fibered homology sphere X $X$ , we show that the q $q$ -series invariant Z ̂ 0 ( X ; q ) $\hat{\operatorname{Z}}_0(X;q)$ , introduced by Gukov–Pei–Putrov–Vafa, is a resummation of the Ohtsuki series Z 0 ( X ) $\operatorname{Z}_0(X)$ . We show that for every even k ∈ N $k \in \mathbb {N}$ there exists a full asymptotic expansion of Z ̂ 0 ( X ; q ) $ \hat{\operatorname{Z}}_0(X;q)$ for q $q$ tending to e 2 π i / k $e^{2\pi i/k}$ , and in particular that the limit Z ̂ 0 ( X ; e 2 π i / k ) $\hat{\operatorname{Z}}_0(X;e^{2\pi i/k})$ exists and is equal to the Witten–Reshetikhin–Turaev quantum invariant τ k ( X ) $\tau _k(X)$ . We show that the poles of the Borel transform of Z 0 ( 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.