## Abstract

We consider the asymptotics of the Turaev–Viro and the Reshetikhin–Turaev

invariants of a hyperbolic 3-manifold, evaluated at the root of unity exp(2π√−1/r) instead of the standard exp(π√−1/r). We present evidence that, as r tends to ∞, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold. This reveals an asymptotic behavior that is different from that of Witten’s Asymptotic Expansion Conjecture, which predicts polynomial growth of these invariants when evaluated at the standard root of unity. This new phenomenon suggests that the Reshetikhin–Turaev invariants may have a geometric interpretation other than the original one via SU(2) Chern–Simons gauge theory.

invariants of a hyperbolic 3-manifold, evaluated at the root of unity exp(2π√−1/r) instead of the standard exp(π√−1/r). We present evidence that, as r tends to ∞, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold. This reveals an asymptotic behavior that is different from that of Witten’s Asymptotic Expansion Conjecture, which predicts polynomial growth of these invariants when evaluated at the standard root of unity. This new phenomenon suggests that the Reshetikhin–Turaev invariants may have a geometric interpretation other than the original one via SU(2) Chern–Simons gauge theory.

Original language | English |
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Pages (from-to) | 419-460 |

Number of pages | 42 |

Journal | Quantum Topology |

Volume | 9 |

Issue number | 3 |

Publication status | Published - Jul 2018 |