In this article we consider specific bivector geometries which arise in the large-spin limit of the extension of the Engle–Pereira–Rovelli–Livine spin foam model for quantum gravity by Kaminski, Kisielowski and Lewandowski. We address the implementation of volume simplicity constraints, which are required to ensure that a 4d metric can be reconstructed from the bivector geometry. We find that the necessary conditions are closely related, but not quite equal to the Hopf link volume simplicity constraints introduced in earlier works. We estimate the number of independent geometricity conditions for arbitrary bivector geometries, and find that they always agree with the number of Hopf links on the graph minus one, suggesting that the geometricity conditions can generically be formulated by deformation of the Hopf link volume simplicity constraints.

在本文中，我们考虑了在卡明斯基、基西洛夫斯基和莱万多夫斯基的量子引力的 Engle-Pereira-Rovelli-Livine 自旋泡沫模型的扩展的大自旋极限中出现的特定双向量几何。我们解决了体积简单性约束的实现，这是确保 4 d度量可以从双向量几何重构。我们发现必要条件密切相关，但并不完全等于早期作品中引入的 Hopf 链接体积简单性约束。我们估计任意双向量几何的独立几何条件的数量，并发现它们总是与图上的 Hopf 链接数减一一致，这表明几何条件通常可以通过 Hopf 链接体积简单性约束的变形来制定。