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.

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Reshetikhin-Turaev 和 Turaev-Viro 不变量的体积猜想

我们考虑双曲 $3$-流形的 Turaev-Viro 和 Reshetikhin-Turaev 不变量的渐近性，在统一 $\exp({2\pi\sqrt{-1}}/{r})$ 的根处评估而不是标准的 $\exp({\pi\sqrt{-1}}/{r})$。我们提出的证据表明，随着 $r$ 趋向于 $\infty$，这些不变量随着流形的双曲线和复体积分别给出的增长率呈指数增长。这揭示了一种与 Witten 的渐近扩展猜想不同的渐近行为，后者预测了这些不变量在以标准单位根进行评估时的多项式增长。这一新现象表明，通过 $SU(2)$ Chern-Simons 规范理论，Reshetikhin-Turaev 不变量可能具有不同于原始几何解释的几何解释。