Resurgence Analysis of Quantum Invariants of Seifert Fibered Homology Spheres

Abstract

For a Seifert fibered homology sphere we show that the q-series Z-hat invariant introduced by Gukov, Pei, Putrov and Vafa is a resummation of the Ohtsuki serie. We show that for every even level k there exists a full asymptotic expansion of Z-hat for q tending to a certain k'th root of unity and in particular that the limit exists and is equal to the WRT quantum invariant. We show that the poles of the Borel transform of the Ohtsuki series 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.

arXiv commentWe have added citations to recent complementary work of Fuji, Iwaki, Murakami and Terashima. The comments from the previous update still apply; this versions differs from the original version in that several new results on the Zed-hat invariant is included, and results for WRT invariants of surgeries on the figure 8 knot from the original version will appear fully proven in a new separate paper