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= e2πi/r with r odd, in the limit r→ ∞. We provide numerical evidence for our conjecture.