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

## TL;DR

This paper extends the volume conjecture for closed hyperbolic 3-manifolds to all orders in perturbation theory, linking complex Chern-Simons invariants with quantum invariants at roots of unity.

## Contribution

It introduces a comprehensive perturbative expansion framework for complex Chern-Simons theory on closed 3-manifolds and conjectures its equivalence with Witten-Reshetikhin-Turaev invariants at large roots of unity.

## Key findings

- Formulas for perturbative invariants of closed 3-manifolds
- Conjecture relating perturbative expansion to quantum invariants
- Numerical evidence supporting the conjecture

## 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.