We study the discretization of a Poincaré/Euclidean BF theory. Upon the addition of a boundary term this theory is equivalent to the BFCG theory defined in terms of the Poincaré/Euclidean 2-group. At an intermediate step in the discretization, we note that there are multiple options for how to proceed. One option brings us back to recovering the discrete variables and phase space of the BF theory. Another option allows us to rediscover the phase space related to the $G$ networks given by Asante et al.. Indeed, our main result is that we are now able to relate the continuum fields with the discrete variables shown by Asante et al. This relation is important in determining how to implement the simplicity constraints to recover gravity using the BFCG action. In fact, we show that such a relation is not as simple as in the BF discretization: the discretized variable on the triangles actually depend on several of the continuum fields instead of solely the continuum $B$ field. We also compare and contrast the discretized BF and BFCG models as pairs of of dual 2-groups. This work highlights (again) how the choice of boundary term influences the resulting symmetry structure of the discretized theory—and hence ultimately the choice of quantum states.

中文翻译：

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4D Poincaré BF 理论的离散化：从群到 2 群

我们研究了 Poincaré/Euclidean BF 理论的离散化。添加边界项后，该理论等效于根据 Poincaré/Euclidean 2 群定义的 BFCG 理论。在离散化的中间步骤，我们注意到如何进行有多种选择。一种选择让我们回到恢复 BF 理论的离散变量和相空间。另一种选择允许我们重新发现与$G$Asante等人给出的网络。. 事实上，我们的主要结果是我们现在能够将连续域与 Asante等人展示的离散变量联系起来。这种关系对于确定如何使用 BFCG 动作来实现简单约束以恢复重力很重要。事实上，我们证明这种关系并不像 BF 离散化那样简单：三角形上的离散变量实际上取决于几个连续统域，而不仅仅是连续统$乙$场地。我们还将离散化的 BF 和 BFCG 模型作为对偶 2 组进行比较和对比。这项工作（再次）强调了边界项的选择如何影响离散化理论的对称结构，从而最终影响量子态的选择。