Abstract
We propose an extension of the recently-proposed volume conjecture for closed hyperbolic 3-manifolds, to all orders in perturbative expansion. We first derive formulas for the perturbative expansion of the partition function of complex Chern–Simons theory around a hyperbolic flat connection, which produces infinitely-many perturbative invariants of the closed oriented 3-manifold. The conjecture is that this expansion coincides with the perturbative expansion of the Witten–Reshetikhin–Turaev invariants at roots of unity \({q=e^{2\pi i/r}}\) with r odd, in the limit \({r\to \infty}\). We provide numerical evidence for our conjecture.
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Gang, D., Romo, M. & Yamazaki, M. All-Order Volume Conjecture for Closed 3-Manifolds from Complex Chern–Simons Theory. Commun. Math. Phys. 359, 915–936 (2018). https://doi.org/10.1007/s00220-018-3115-y
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DOI: https://doi.org/10.1007/s00220-018-3115-y