Resurgence analysis of quantum invariants of Seifert fibered homology spheres

For a Seifert fibered homology sphere 𝑋, we show that the 𝑞-series invariant Ẑ 0(𝑋; 𝑞), introduced by Gukov– Pei–Putrov–Vafa, is a resummation of the Ohtsuki series Z0(𝑋). We show that for every even 𝑘 ∈ ℕ there exists a full asymptotic expansion of Ẑ 0(𝑋; 𝑞) for 𝑞 tending to 𝑒2𝜋𝑖∕𝑘, and in particular that the limit Ẑ 0(𝑋; 𝑒2𝜋𝑖∕𝑘) exists and is equal to the Witten–Reshetikhin–Turaev quantum invariant 𝜏𝑘(𝑋). We show that the poles of the Borel transform of Z0(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)-connections.