This paper focuses on the semiclassical behavior of the spinfoam quantum gravity in four dimensions. There has been long-standing confusion, known as the flatness problem, about whether the curved geometry exists in the semiclassical regime of the spinfoam amplitude. The confusion is resolved by the present work. By numerical computations, we explicitly find curved Regge geometries that contribute dominantly to the large-$j$ Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam amplitudes on triangulations. These curved geometries are with small deficit angles and relate to the complex critical points of the amplitude. The dominant contribution from the curved geometry to the spinfoam amplitude is proportional to $e^{i\mathcal{I}}$, where $\mathcal{I}$ is the Regge action of the geometry plus corrections of higher order in curvature. As a result, in the semiclassical regime, the spinfoam amplitude reduces to an integral over Regge geometries weighted by $e^{i\mathcal{I}}$, where $\mathcal{I}$ is the Regge action plus corrections of higher order in curvature. As a by-product, our result also provides a mechanism to relax the cosine problem in the spinfoam model. Our results provide important evidence supporting the semiclassical consistency of the spinfoam quantum gravity.

中文翻译：

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四维洛伦兹自旋泡沫量子引力中的复杂临界点和弯曲几何

本文重点研究了自旋泡沫量子引力在四个维度上的半经典行为。关于弯曲几何是否存在于自旋泡沫振幅的半经典状态中，长期以来一直存在混淆，称为平坦度问题。目前的工作解决了混乱。通过数值计算，我们明确地找到了主要贡献于大的弯曲 Regge 几何形状。$j$Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) 三角剖分上的自旋泡沫振幅。这些弯曲的几何形状具有小的缺陷角，并且与振幅的复杂临界点有关。弯曲几何形状对纺丝泡沫振幅的主要贡献与$e^{\mathrm{一世}\mathcal{我}}$， 在哪里$\mathcal{我}$是几何的 Regge 作用加上曲率更高阶的校正。结果，在半经典状态下，自旋泡沫振幅减少到对 Regge 几何加权的积分$e^{\mathrm{一世}\mathcal{我}}$， 在哪里$\mathcal{我}$是雷格动作加上曲率更高阶的校正。作为副产品，我们的结果还提供了一种机制来放松自旋泡沫模型中的余弦问题。我们的结果提供了支持自旋泡沫量子引力半经典一致性的重要证据。