$\hat{Z}$ invariants at rational $\tau$

Abstract

$\hat{Z}$ invariants of 3-manifolds were introduced as series in $q=e^{2\pi i\tau }$ in order to categorify Witten-Reshetikhin-Turaev invariants corresponding to $\tau =1/k$. However modularity properties suggest that all roots of unity are on the same footing. The main result of this paper is the expression connecting Reshetikhin-Turaev invariants with $\hat{Z}$ invariants for $\tau \in \mathbb{Q}$. We present the reasoning leading to this conjecture and test it on various 3-manifolds.

arXiv comment15 pages

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• 3d-3d correspondence for mapping tori
• Quivers for 3-manifolds: the correspondence, BPS states, and 3d $\mathcal{N}$=2 theories
• Cobordism invariants from BPS q-series
• Resurgent Analysis for Some 3-manifold Invariants
• Resurgent Analysis of SU(2) Chern-Simons Partition Function on Brieskorn Spheres $\Sigma (2,3,6n+5)$