All-Order Volume Conjecture for Closed 3-Manifolds from Complex Chern-Simons Theory

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.

22 pages, 2 figures; v2: published version

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