Higher gauge theories play a prominent role in the construction of 4D topological invariants and have been long ago proposed as a tool for 4D quantum gravity. The Yetter lattice model and its continuum counterpart, the BFCG theory, generalize BF theory to 2-gauge groups and—when specialized to 4D and the Poincar 2-group—they provide an exactly solvable topologically-flat version of 4D general relativity. The 2-Poincar Yetter model was conjectured to be equivalent to a state sum model of quantum flat spacetime developed by Baratin and Freidel after work by Korepanov (KBF model). This conjecture was motivated by the origin of the KBF model in the theory of two-representations of the Poincar 2-group. Its proof, however, has remained elusive due to the lack of a generalized Peter–Weyl theorem for 2-groups. In this work we prove this conjecture. Our proof avoids the Peter–Weyl theorem and rather leverages the geometrical content of the Yetter model. Key for the proof is the introduction of a kinematical boundary Hilbert space on which 1- and two-Lorentz invariance is imposed. Geometrically this allows the identification of (quantum) tetrad variables and of the associated (quantum) Levi-Civita connection. States in this Hilbert space are labelled by quantum numbers that match the two-group representation labels. Our results open exciting opportunities for the construction of new representations of quantum geometries. Compared to loop quantum gravity, the higher gauge theory framework provides a quantum representation of the ADM—Regge initial data, including an identification of the intrinsic and extrinsic curvature. Furthermore, it leads to a version of the diffeomorphism and Hamiltonian constraints that acts on the vertices of the discretization, thus providing a prospect for a quantum realization of the hypersurface deformation algebra in 4D.

更高规范的理论在 4D 拓扑不变量的构建中发挥着重要作用，并且很久以前就被提出作为 4D 量子引力的工具。Yetter 晶格模型及其连续统对应物 BFCG 理论将 BF 理论推广到 2 规范群，并且当专门用于 4D 和 Poincar 2 群时，它们提供了 4D 广义相对论的完全可解的拓扑平坦版本。2-Poincar Yetter 模型被推测为等价于 Baratin 和 Freidel 在 Korepanov 工作后开发的量子平面时空状态和模型（KBF 模型）。这个猜想是由 KBF 模型在 Poincar 2 群的二表示理论中的起源所激发的。然而，由于缺乏 2 群的广义 Peter-Weyl 定理，它的证明仍然难以捉摸。在这项工作中，我们证明了这个猜想。我们的证明避免了 Peter-Weyl 定理，而是利用了 Yetter 模型的几何内容。证明的关键是引入运动学边界希尔伯特空间，在该空间上施加了 1 和 2 洛伦兹不变性。在几何上，这允许识别（量子）四分体变量和相关的（量子）Levi-Civita 连接。此希尔伯特空间中的状态由与两组表示标签匹配的量子数标记。我们的结果为构建量子几何的新表示提供了令人兴奋的机会。与圈量子引力相比，更高规范理论框架提供了 ADM-Regge 初始数据的量子表示，包括对内在曲率和外在曲率的识别。此外，