Volume conjectures for the reshetikhin-turaev and the turaev-viro invariants

Qingtao Chen ETH Zurich Tian Yang Texas A&M University

Geometric Analysis and Geometric Topology mathscidoc:1910.15001

Quantum Topology, 9, 419–460, 2018
We consider the asymptotics of the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic $3$-manifold, evaluated at the root of unity $exp(2\pi\sqrt{-1}/r)$ instead of the standard $exp(\pi\sqrt{-1}/r)$. We present evidence that, as $r$ tends to $\infty$, 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.
Volume Conjecture
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@inproceedings{qingtao2018volume,
  title={VOLUME CONJECTURES FOR THE RESHETIKHIN-TURAEV AND THE TURAEV-VIRO INVARIANTS},
  author={Qingtao Chen, and Tian Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191007113313710405529},
  booktitle={Quantum Topology},
  volume={9},
  pages={419–460},
  year={2018},
}
Qingtao Chen, and Tian Yang. VOLUME CONJECTURES FOR THE RESHETIKHIN-TURAEV AND THE TURAEV-VIRO INVARIANTS. 2018. Vol. 9. In Quantum Topology. pp.419–460. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191007113313710405529.
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