Resurgent analysis of SU(2) Chern-Simons partition function on Brieskorn spheres Σ(2, 3, 6n + 5)

A bstract $$ \hat{Z} $$ Z ̂ -invariants, which can reconstruct the analytic continuation of the SU(2) Chern-Simons partition functions via Borel resummation, were discovered by GPV and have been conjectured to be a new homological invariant of 3-manifolds which can shed light onto the superconformal and topologically twisted index of 3d $$ \mathcal{N} $$ N = 2 theories proposed by GPPV. In particular, the resurgent analysis of $$ \hat{Z} $$ Z ̂ has been fruitful in discovering analytic properties of the WRT invariants. The resurgent analysis of these $$ \hat{Z} $$ Z ̂ -invariants has been performed for the cases of Σ(2 , 3 , 5) , Σ(2 , 3 , 7) by GMP, Σ(2 , 5 , 7) by Chun, and, more recently, some additional Seifert manifolds by Chung and Kucharski, independently. In this paper, we extend and generalize the resurgent analysis of $$ \hat{Z} $$ Z ̂ on a family of Brieskorn homology spheres Σ(2 , 3 , 6 n + 5) where n ∈ ℤ + and 6 n + 5 is a prime. By deriving $$ \hat{Z} $$ Z ̂ for Σ(2 , 3 , 6 n + 5) according to GPPV and Hikami, we provide a formula where one can quickly compute the non-perturbative contributions to the full analytic continuation of SU(2) Chern-Simons partition function.

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